Evolution of solitary waves and undular bores in
shallow-water flows over a gradual slope with bottom
friction
G.A. El1, R.H.J. Grimshaw2
Department of Mathematical Sciences, Loughborough University,
Loughborough LE11 3TU, UK
1 e-mail: G.El@lboro.ac.uk 2 e-mail: R.H.J.Grimshaw@lboro.ac.uk
A.M. Kamchatnov
Institute of Spectroscopy, Russian Academy of Sciences,
Troitsk, Moscow Region, 142190 Russia
e-mail: kamch@isan.troitsk.ru
Abstract
This paper considers the propagation of shallow-water solitary and nonlinear peri-
odic waves over a gradual slope with bottom friction in the framework of a variable-
coefficient Korteweg-de Vries equation. We use the Whitham averaging method, using
a recent development of this theory for perturbed integrable equations. This general
approach enables us not only to improve known results on the adiabatic evolution of
isolated solitary waves and periodic wave trains in the presence of variable topography
and bottom friction, modeled by the Chezy law, but also importantly, to study the
effects of these factors on the propagation of undular bores, which are essentially un-
steady in the system under consideration. In particular, it is shown that the combined
action of variable topography and bottom friction generally imposes certain global
restrictions on the undular bore propagation so that the evolution of the leading soli-
tary wave can be substantially different from that of an isolated solitary wave with
the same initial amplitude. This non-local effect is due to nonlinear wave interactions
within the undular bore and can lead to an additional solitary wave amplitude growth,
which cannot be predicted in the framework of the traditional adiabatic approach to
the propagation of solitary waves in slowly varying media.
1 Introduction
There have been many studies of the propagation of water waves over a slope, sometimes
also subject to the effects of bottom friction. Many of these works have considered linear
waves, or have been numerical simulations in the framework of various nonlinear long-wave
model equations. Our interest here is in the propagation of weakly nonlinear long water
http://arxiv.org/abs/0704.0045v1
waves over a slope, simultaneously subject to bottom friction, a combination apparently
first considered by Miles (1983a,b) albeit for the special case of a single solitary wave, or a
periodic wavetrain. An appropriate model equation for this scenario is the variable-coefficient
perturbed Korteweg-de Vries (KdV) equation (see Grimshaw 1981, Johnson 1973a,b),
At + cAx +
AAx +
Axxx = −CD
|A|A. (1)
Here A(x, t) is the free surface elevation above the undisturbed depth h(x) and c(x) =
gh(x) is the linear long wave phase speed. The bottom friction term on the right-hand
side is represented by the Chezy law, modelling a turbulent boundary layer. Here CD is a
non-dimensional drag coefficient, often assumed to have a value around 0.01 (Miles 1983a,b).
Other forms of friction could be used (see, for instance Grimshaw et al 2003) but the Chezy
law seems to be the most appropriate for water waves in a shallow depth. In (1) the first
two terms on the left-hand side are the dominant terms, and by themselves describe the
propagation of a linear long wave with speed c. The remaining terms on the left-hand
side represent, respectively, the effect of varying depth, weakly nonlinear effects and weak
linear dispersion. The equation is derived using the usual KdV balance in which the linear
dispersion, represented by ∂2/∂x2, is balanced by nonlinearity, represented by A. Here we
have added to this balance weak inhomogeneity so that cx/c scales as h
2∂3/∂x3, and weak
friction so that CD scales with h∂/∂x. Within this basic balance of terms, we can cast (1)
into the asymptotically equivalent form
AAX +
AXXX = −CD
|A|A, (2)
where τ =
c(x′)
, X = τ − t. (3)
Here we have h = h(x(τ)), explicitly dependent on the variable τ which describes evolution
along the path of the wave.
The governing equation (2) can be cast into several equivalent forms. That most com-
monly used is the variable-coefficient KdV equation, obtained here by putting
B = (gh)1/4A (4)
so that Bτ +
2g1/4h5/4
BBX +
BXXX = −CD
|B|B . (5)
This form shows that, in the absence of friction term, i.e. when CD ≡ 0, equation (2)
has two integrals of motion with the densities proportional to h1/4A and h1/2A2. These
are often referred to as laws for the conservation of “mass” and “momentum”. However,
these densities do not necessarily correspond to the corresponding physical entities. Indeed,
to leading order, the “momentum” density is proportional to the wave action flux, while
the “mass” density differs slightly from the actual mass density. This latter issue has been
explored by Miles (1979), where it was shown that the difference is smaller than the error
incurred in the derivation of equation (4), and is due to reflected waves.
Our main concern in this paper is with the behaviour of an undular bore over a slope
in the presence of bottom friction, using the perturbed KdV equation (2), where we were
originally motivated by the possibility that the behaviour of a tsunami approaching the
shore might be modeled in this way. The undular bore solution to the unperturbed KdV
equation can be constructed using the well-known Gurevich-Pitaevskii (GP) (1974) approach
(see also Fornberg and Whitham 1978). In this approach, the undular bore is represented
as a modulated nonlinear periodic wave train. The main feature of this unsteady undular
bore is the presence of a solitary wave (which is the limiting wave form of the periodic
cnoidal wave) at its leading edge. The original initial-value problem for the KdV equation is
then replaced by a certain boundary-value problem for the associated modulation Whitham
equations. We note, however, that so far, the simplest, “(x/t)”-similarity solutions of the
modulation equations have been used for the modelling of undular bores in various contexts
(see Grimshaw and Smyth 1986, Smyth 1987 or Apel 2003 for instance). These solutions,
while effectively describing many features of undular bores, are degenerate and fail to cap-
ture, even qualitatively, some important effects associated with non-self-similar modulation
dynamics. In particular, in the classical GP solution for the resolution of an initial jump
in the unperturbed KdV equation, the amplitude of the lead solitary wave in the undular
bore is constant (twice the value of the initial jump). On the other hand, the modulation
solution for the undular bore evolving from a general monotonically decreasing initial profile
shows that the lead solitary wave amplitude in fact grows with time (Gurevich, Krylov and
Mazur 1989; Gurevich, Krylov and El 1992; Kamchatnov 2000). As we shall see, the very
possibility of such variations in the modulated solutions of the unperturbed KdV equation
has a very important fluid dynamics implication: in a general setting, the undular bore lead
solitary wave cannot be treated as an individual KdV solitary wave but rather represents a
part of the global nonlinear wave structure. In other words, while at every particular moment
of time the lead solitary wave has the spatial profile of the familiar KdV soliton, generally,
the temporal dependence of its amplitude cannot be obtained in the framework of single
solitary wave perturbation theory.
In the unperturbed KdV equation, the growth of the lead solitary wave amplitude is
caused by the spatial inhomogeneity of the initial data. Here, however, the presence of a
perturbation due to topography and/or friction serves as an alternative and/or additional
cause for variation of the lead solitary wave amplitude. Thus, in the present case, the
variation in the amplitude will have two components (which generally, of course, cannot be
separated because of the nonlinear nature of the problem); one is local, described by the
adiabatic perturbation theory for a single solitary wave, and the other one is nonlocal, which
in principle requires the study of the full modulation solution. Depending on the relative
values of the small parameters associated with the slope, friction and spatial non-uniformity
of the initial modulations, we can take into account only one of these components, or a
combination of them.
The structure of the paper is as follows. First, in Section 2, we reformulate the basic model
(1) as a constant-coefficient KdV equation perturbed by terms representing topography
and friction. Then we derive in Section 3 the associated perturbed Whitham modulation
equations using methods recently developed by Kamchatnov (2004). Next, in Section 4, this
Whitham system is integrated in the solitary-wave limit. Our purpose here is primarily to
obtain the equation of a multiple characteristic, which defines the leading edge of a shoaling
undular bore in the case when the modulations due to the combined action of the slope and
bottom friction are small compared to the existing spatial modulations due to non-uniformity
of the initial data. As a by-product of this integration, we reproduce and extend the known
results on the adiabatic variation of a single solitary wave (Miles 1983a,b). Then, in Section
5, we carry out an analogous study of a cnoidal wave, propagating over a gradual slope and
subject to friction, a case studied previously by Miles (1983 b) but under the restriction of
zero mean flow, which is removed here. Finally, in Section 6 we study effects of a gradual
slope and bottom friction on the front of an undular bore which represents a modulated
cnoidal wave transforming into a system of weakly interacting solitons near its leading edge.
2 Problem formulation
For the purpose of the present paper it is convenient to recast (2) into the standard KdV
equation form with constant coefficients, modified by certain perturbation terms. Thus we
introduce the new variables
A, T =
hdτ =
6g3/2
h(x)dx. (6)
so that UT + 6UUX + UXXX = R = F (T )U −G(T )|U |U, (7)
where F (T ) = −9hT
, G(T ) = 4CD
. (8)
In this form, the governing equation (7) has the structure of the integrable KdV equation
on the left-hand side, while the separate effects of the varying depth and the bottom friction
are represented by the two terms on the right-hand side. This structure enables us to use
the general theory developed in Kamchatnov (2004) for perturbed integrable systems.
For much of the subsequent discussion, it is useful to assume that h(x) = constant,
CD = 0 for x < 0 in the original equation (1), which corresponds to F (T ) = G(T ) = 0 for
T < 0 in (7). We shall also assume that A = 0 for x > 0 at t = 0, which corresponds to
U = 0 for X > 0 on X = τ(T ) (see (6)). Then we shall propose two types of initial-value
problem for (1), and correspondingly for (7).
(a) Let a solitary wave of a given amplitude a0 initially propagating over a flat bottom
without friction (i.e a soliton described by an unperturbed KdV equation), enter the variable
topography and bottom friction region at t = 0, x = 0 (Fig. 1 a).
(b) Let an undular bore of a given intensity propagate over a flat bottom without friction
(the corresponding solution of the unperturbed KdV equation will be discussed in Section
5). Let the lead solitary wave of this undular bore have the same amplitude a0 and enter
the variable topography and bottom friction region at t = 0, x = 0 (Fig. 1b).
In particular, we shall be interested in the comparison of the slow evolution of these
two, initially identical, solitary waves in the two different problems described above. The
expected essential difference in the evolution is due to the fact that the lead solitary wave
in the undular bore is generally not independent of the remaining part of the bore and can
exhibit features that cannot be captured by a local perturbation analysis. The well-known
example of such a behaviour, when a solitary wave is constrained by the condition of being
a part of a global nonlinear wave structure, is provided by the undular bore solution of the
KdV-Burgers (KdV-B) equation
ut + 6uux + uxxx = µuxx, µ ≪ 1 . (9)
( )h x
  a) 
( )h x
Figure 1: Isolated solitary wave (a) and undular bore (b) entering the variable topogra-
phy/bottom friction region.
Indeed, the undular bore solution of the KdV-B equation (9) is known to have a solitary
wave at its leading edge (see Johnson 1970; Gurevich & Pitaevskii 1987; Avilov, Krichever
& Novikov 1987) and this solitary wave: (a) is asymptotically close to a soliton solution
of the unperturbed KdV equation; and (b) has the amplitude, say a0, that is constant in
time. At the same time, it is clear that if one takes an isolated KdV soliton of the same
amplitude a0 as initial data for the KdV-Burgers equation it would damp with time due
to dissipation. The physical explanation of such a drastic difference in the behaviour of an
isolated soliton and a lead solitary wave in the undular bore for the same weakly dissipative
KdV-B equation is that the action of weak dissipation on an expanding undular bore is
twofold: on the one hand, the dissipation tends to decrease the amplitude of the wave
locally but, on the other hand, it “squeezes” the undular bore so that the interaction (i.e.
momentum exchange) between separate solitons within the bore becomes stronger than in
the absence of dissipation and this acts as the amplitude increasing factor. The additional
momentum is extracted from the upstream flow with a greater depth (see Benjamin and
Lighthill 1954). As a result, in the case of the KdV-B equation, an equilibrium non-zero
value for the lead solitary wave amplitude in the undular bore is established. Of course,
for other types of dissipation, a stationary value of the lead soliton amplitude would not
necessarily exist, but in general, due to the expected increase of the soliton interactions near
the leading edge, the amplitude of the lead soliton of the undular bore would decay slower
than that of an isolated soliton. Indeed, the presence here of variable topography as well
can result in an additional “nonlocal” amplitude growth.
While the problem (a) can be solved using traditional perturbation analysis for a single
solitary wave, which leads to an ordinary differential equation along the solitary wave path
(see Miles 1983a,b), the undular bore evolution problem (b) requires a more general approach
which can be developed on the basis of Whitham’s modulation theory leading to a system of
three nonlinear hyperbolic partial differential equations of the first order. Since the Whitham
method, being equivalent to a nonlinear multiple scale perturbation procedure, contains the
adiabatic theory of slow evolution of a single solitary wave as a particular (albeit singular)
limit, it is instructive for the purposes of this paper to treat both problems (a) and (b) using
the general Whitham theory.
3 Modulation equations
The original Whithammethod (Whitham 1965, 1974) was developed for conservative constant-
coefficient nonlinear dispersive equations and is based on the averaging of appropriate con-
servation laws of the original system over the period of a single-phase periodic travelling
wave solution. The resulting system of quasi-linear equations describes the slow evolution
of the modulations (i.e. of the mean value, the wavenumber, the amplitude etc.) of the pe-
riodic travelling wave. Here, that approach is extended to the perturbed KdV equation (6)
following the general approach of Kamchatnov (2004), which extends earlier results for cer-
tain specific cases (see Gurevich and Pitaevskii (1987, 1991), Avilov, Krichever and Novikov
(1987) and Myint and Grimshaw (1995) for instance).
We suppose that the evolution of the nonlinear wave is adiabatically slow, that is, the
wave can be locally represented as a solution of the corresponding unperturbed KdV equation
(i.e. (7) with zero on the right-hand side) with its parameters slowly varying with space and
time. The one-phase periodic solution of the KdV equation can be written in the form
U(X, T ) = λ3 − λ1 − λ2 − 2(λ3 − λ2)sn2(
λ3 − λ1 θ,m) (10)
where sn(y,m) is the Jacobi elliptic sine function, λ1 ≤ λ2 ≤ λ3 are parameters and the
phase variable θ and the modulus m are given by
θ = X − V T, V = −2(λ1 + λ2 + λ3) , (11)
λ3 − λ2
λ3 − λ1
, (12)
and L =
−P (µ)
2K(m)√
λ3 − λ1
, (13)
where K(m) is the complete elliptic integral of the first kind, L is the “wavelength” along the
X-axis (which is actually a retarded time rather than a true spatial co-ordinate). Here we
have used the representation of the basic ordinary differential equation for the KdV travelling
wave solution (10) in the form (see Kamchatnov (2000) for a general motivation behind this
representation)
−P (µ), (14)
where
µ = 1
(U + s1), s1 = λ1 + λ2 + λ3 (15)
P (µ) =
(µ− λi) = µ3 − s1µ2 + s2µ− s3, (16)
that is the solution (10) is parameterized by the zeroes λ1, λ2, λ3 of the polynomial P (µ).
In a modulated wave, the parameters λ1, λ2, λ3 are allowed to be slow functions of X and
T , and their evolution is governed by the Whitham equations. For the unperturbed KdV
equation, the evolution of the modulation parameters is due to a spatial non-uniformity
of the initial distributions for λj, j = 1, 2, 3 and the typical spatio-temporal scale of the
modulation variations is determined by the scale of the initial data.
In the case of the perturbed KdV equation (7), the evolution of the parameters λ1, λ2, λ3
is caused not only by their initial spatial non-uniformity, but also by the action of the
weak perturbation, so that, generally, at least two independent spatio-temporal scales for
the modulations can be involved. However, at this point we shall not introduce any scale
separation within the modulation theory and derive general perturbed Whitham equations
assuming that the typical values of F (T ) and G(T ) are O(∂λj/∂T, ∂λj/∂X) within the
modulation theory.
It is instructive to first introduce the Whitham equations for the perturbed KdV equation
(7) using the traditional approach of averaging the (perturbed) conservation laws. To this
end, we introduce the averaging over the period (13) of the cnoidal wave (10) by
〈F〉 =
Fdθ =
−P (µ)
. (17)
In particular,
〈U〉 = 2〈µ〉 − s1 = 2(λ3 − λ1)
+ λ1 − λ2 − λ3, (18)
〈U2〉 = 8[−s1
(λ3 − λ1)
s1λ1 +
(λ21 − λ2λ3)] + s21 , (19)
where E(m) is the complete elliptic integral of the second kind. Now, one represents the
KdV equation (7) in the form of the perturbed conservation laws
= Rj , j = 1, 2, 3 , Rj ≪ 1 , (20)
where Pj and Qj are the standard expressions for the conserved densities (Kruskal integrals)
and “fluxes” of the unperturbed KdV equation. Just as in the Whitham (1965) theory for
unperturbed dispersive systems, the number of conservation laws required is equal to the
number of free parameters in the travelling wave solution, which is three in the present case.
Next, one applies the averaging (17) to the system (20) to obtain (see Dubrovin and Novikov
1989)
∂〈Pj〉
∂〈Qj〉
= 〈Rj〉 , j = 1, 2, 3 . (21)
The system (21) describes slow evolution of the parameters λj in the cnoidal wave solution
(10).
Along with these derived perturbed conservative form of the Whitham equations, we
introduce the wave conservation law which is a general condition for the existence of slowly
modulated single-phase travelling wave solutions (10) (see for instance Whitham 1974) and
must be consistent with the modulation system (21). This conservation law has the form
= 0 , (22)
where k =
, ω = kV (23)
are the “wavenumber” and the “frequency” respectively (we have put quotation marks here
because the actual wavenumber and frequency related to the physical variables x, t are
different quantities from those in (23), but are related through the transformations (3, 6) ).
The wave conservation law (22) can be introduced instead of any of three inhomogeneous
averaged conservation laws comprising the Whitham system (21).
It is known that the Whitham system for the homogeneous constant-coefficient KdV
equation can be represented in diagonal (Riemann) form (Whitham 1965, 1974) by an ap-
propriate choice of the three parameters characterising the periodic travelling wave solution.
In fact, in our solution (7) the parameters λj have already been chosen so that they coincide
with the Riemann invariants of the unperturbed KdV modulation system. Introducing them
explicitly into the perturbed system (21) we obtain (see Kamchatnov 2004)
∂L/∂λi
〈(2λi − s1 − U)R〉
j 6=i(λi − λj)
, i = 1, 2, 3, (24)
where R is the perturbation term on the right-hand side of the KdV equation (7) and
vi = −2
∂L/∂λi
, i = 1, 2, 3, (25)
are the Whitham characteristic velocities corresponding to the unperturbed KdV equation.
It should be noted that the straightforward realisation of the above lucid general algo-
rithm for obtaining perturbed modulation system in diagonal form is quite a laborious task.
In fact, to derive system (24), the so-called finite-gap integration method incorporating the
integrable structure of the unperturbed KdV equation has been used. The modulation sys-
tem (24) in a more particular form corresponding to specific choices of the perturbation term
was obtained by Myint and Grimshaw (1995) using a multiple-scale perturbation expansion.
In that latter setting, the wave conservation law (22) is an inherent part of the construction,
while in the averaging approach used here, it can be obtained as a consequence of the system
(24).
To obtain an explicit representation of the Whitham equations for the present case of
equation (7), we must substitute the perturbation R from the right-hand side of (7) and
perform the integration (17) with U given by (10). From now on, we are going to consider
only the flows where U ≥ 0 so that the perturbation term assumes the form
R(U) = G(T )U − F (T )U2 . (26)
Substituting (26) into (24) we obtain, after some detailed calculations (see Appendix),
the perturbed Whitham system in the form
= ρi = Ci[F (T )Ai −G(T )Bi], i = 1, 2, 3 (27)
where C1 =
, C2 =
E − (1−m)K
, C3 =
; (28)
(5λ1 − λ2 − λ3)E +
(λ2 − λ1)K,
(5λ2 − λ1 − λ3)E − (λ2 − λ1)
λ3 − λ1
(5λ3 − λ1 − λ2)E −
(λ2 − λ1)
(−27λ21 − 7λ22 − 7λ23 + 2λ1λ2 + 2λ1λ3 + 22λ2λ3)E
(λ2 − λ1)(3λ1 + λ2 + λ3)K,
(−7λ21 − 27λ22 − 7λ23 + 2λ1λ2 + 22λ1λ3 + 2λ2λ3)E
λ2 − λ1
λ3 − λ1
(7λ21 + 15λ
2 + 11λ
3 − 6λ1λ2 − 18λ1λ3 + 6λ2λ3)K,
(−7λ21 − 7λ22 − 27λ23 + 22λ1λ2 + 2λ1λ3 + 2λ2λ3)E
(7λ21 + 11λ
2 + 15λ
3 − 18λ1λ2 − 6λ1λ3 + 6λ2λ3)K;
and the characteristic velocities are:
v1 = −2
4(λ3 − λ1)(1−m)K
v2 = −2
4(λ3 − λ2)(1−m)K
E − (1−m)K
v3 = −2
4(λ3 − λ2)K
The equations (27) – (31) provide a general setting for studying the nonlinear modulated
wave evolution over variable topography with bottom friction. In the absence of the pertur-
bation terms (i.e. when F (T ) ≡ 0, G(T ) ≡ 0), the system (27), (31) indeed coincides with
the original Whitham equations (Whitham 1965) for the integrable KdV dynamics. In that
case the variables λ1, λ2, λ3 become Riemann invariants, so in this general (perturbed) case
we shall call them Riemann variables.
It is important to study the structure of the perturbed Whitham equations (27) – (31) in
two limiting cases when the underlying cnoidal wave degenerates into (i) a small-amplitude
sinusoidal wave (linear limit), when λ2 = λ3 (m = 0), and (ii) into a solitary wave when
λ2 = λ1 (m = 1). Since in both these limits the oscillations do not contribute to the mean
flow (they are infinitely small in the linear limit and the distance between them becomes
infinitely long in the solitary wave limit) one should expect that in both cases one of the
Whitham equations will transform into the dispersionless limit of the original perturbed
KdV equation (7) i.e.
UT + 6UUX = F (T )U −G(T )U2, (32)
Indeed, using formulae (27) – (31) we obtain for m = 0:
λ2 = λ3 ,
− 6λ1
= λ1F + λ
+ (6λ1 − 12λ3)
= λ1F + λ
Similarly, for m = 1, one has
λ2 = λ1 ,
− (4λ1 + 2λ3)
(4λ1 − λ3)F +
(7λ23 − 24λ1λ3 + 32λ21)G,
− 6λ3
= λ3F + λ
We see that, in both cases, one of the Riemann variables (taken with inverted sign) coincides
with the solution of the dispersionless equation (32) (recall that in the derivation of the
Whitham equations we assumed U ≥ 0 everywhere), namely U = 〈U〉 = −λ1 when λ2 = λ3
(m = 0) and U = 〈U〉 = −λ3 when λ2 = λ1 (m = 1).
To conclude this section, we present expressions for the physical wave parameters such as
the surface elevation wave amplitude a, mean elevation 〈A〉 speed and wavenumber in terms
of the modulation solution λj(X, T ). Using (6) and (10) we obtain for the wave amplitude
(peak to trough) and the mean elevation
(λ3 − λ2) , 〈A〉 =
〈U〉 , (35)
where the dependence of 〈U〉 on λj(X, T ), j = 1, 2, 3 is given by (18) and X = X(x, t),
T = T (x, t) by (3, 6). In order to obtain the physical wavenumber κ and the frequency Ω
we first note that the phase function θ(X, T ) defined in (11) is replaced by a more general
expression defined so that k = θX and kV = −θT are the “wavenumber” and “frequency” in
the X −T coordinate system. Then we define the physical phase function Θ(x, t) = θ(X, T )
so that we get
κ = Θx , Ω = −Θt . (36)
It now follows that
(1− hV
) , Ω = k , and
1− hV/6g
. (37)
Note that the physical frequency is the “wavenumber” in the X − T coordinate system,
and that the physical phase speed is Ω/κ. Since the validity of the KdV model (1) requires
inter alia that the wave be right-going it follows from this expression that the modulation
solution remains valid only when hV < 6g. Of course, the validity of (1) also requires that
the amplitude remains small, and this would normally also ensure that V remains small.
4 Modulation solution in the solitary wave limit
In this section, we shall integrate the perturbed modulation system (27) along the multiple
characteristic corresponding to the merging of two Riemann variables λ2 and λ1. As we shall
see later, this characteristic specifies the motion of the leading edge of the shoaling undular
bore in the case when the perturbations due to variable topography and bottom friction can
be considered as small compared with the existing spatial modulations within the bore. At
the same time, as the case λ2 = λ1 ( i.e. m = 1) corresponds to the solitary wave limit in the
travelling wave solution (10), our results here are expected to be consistent with the results
from the traditional perturbation approach to the adiabatic variation of a solitary wave due
to topography and bottom friction (see Miles 1983a,b).
In the limit m → 1 the periodic solution (10) of the KdV equation goes over to its solitary
wave solution
U(X, T ) = U0sech
λ3 − λ1(X − VsT )]− λ3, (38)
where
U0 = 2(λ3 − λ1) , Vs = −(4λ1 + 2λ3) (39)
are the solitary wave amplitude and “velocity” respectively. The solution (38) depends on two
parameters λ1 and λ3 whose adiabatic slow evolution is governed by the reduced modulation
system (34). It is important that the second equation in this system is decoupled from
the first one. Hence, evolution of the pedestal −λ3 on which the solitary wave rides, can
be found from the solution of this dispersionless equation by the method of characteristics.
When λ3(X, T ) is known, evolution of the parameter λ1 can be found from the solution of
the first equation (34). As a result, we arrive at a complete description of adiabatic slow
evolution of the solitary wave parameters taking account of its interaction with the (given)
pedestal.
However, it is important to note here that while this description of the adiabatic evolution
of a solitary wave is complete as far as the solitary wave itself is concerned, it fails to describe
the evolution of a trailing shelf, which is needed to conserve total “mass” (see, for instance,
Johnson 1973b, Grimshaw 1979 or Grimshaw 2006). This trailing shelf has a very small
amplitude, but a very large length scale, and hence can carry the same order of “mass” as
the solitary wave. But note that the “momentum” of the trailing shelf is much smaller than
that of the solitary wave, whose adiabatic deformation is in fact governed to leading order by
conservation of “momentum”, or more precisely, by conservation of wave action flux (strictly
speaking, conservation only in the absence of friction).
The situation simplifies if the solitary wave propagates into a region of still water so
that there is no pedestal ahead of the wave, that is λ3 = 0 in X > τ(T ). But then, since
λ3 = 0 is an exact solution of the degenerate Whitham system (34) for this solitary wave
configuration, we can put λ3 = 0 both in the solitary wave solution,
U(X, T ) = −2λ1sech2[
−λ1 (X − VsT )], Vs = −4λ1, (40)
and in equation (34) for the parameter λ1 to obtain,
− 4λ1
Fλ1 +
Gλ21 , (41)
As we see, the solitary wave moves with the instant velocity
= −4λ1, (42)
and the parameter λ1 changes with T along the solitary wave trajectory according to the
ordinary differential equation
F (T )λ1 +
G(T )λ21. (43)
It can be shown that equation (43) is consistent with the equation for the solitary wave half-
width γ =
−λ1 obtained by the traditional perturbation approach (see Grimshaw (1979)
for instance).
Next, we re-write equation (43) in terms the original independent x-variable. For that,
we find from (6), that
dT = (h1/2/6g3/2)dx (44)
and F = −27
)3/2 dh
, G = 4CD
. (45)
Then substituting these expressions into (43) yields the equation
= −31
λ21 (46)
which can be easily integrated to give
−C0 −
, (47)
where C0 is an integration constant and x = 0 is a reference point where h = h0. According
to (40), U0 = −2λ1 is the amplitude of the soliton expressed in terms of variable U(X, T ).
Returning to the original surface displacement A(x, t) by means of (6) and denoting C0 =
4/(3ga0h0), we find the dependence of the surface elevation soliton amplitude a = (2h
2/3g)U0
on x in the form
a = a0
CDa0h0
, (48)
where a0 is the solitary wave amplitude at x = 0. We note that for CD = 0 this reduces to
the classical Boussinesq (1872) result a ∼ h−1, while for h = h0 it reduces to the well-known
algebraic decay law a ∼ 1/(1 + constant x) due to Chezy friction. Miles (1983a,b) obtained
this expression for a linear depth variation, although we note that there is a factor of 2
difference from (48) (in Miles (1983a,b) the factor 16CD/15 is 8CD/15). The trajectory of
the soliton can be now found from (42) and (47):
− t =
dx′h−5/2(x′)
CDa0h0
h3(x)
. (49)
This expression determines implicitly the dependence of x on t along the solitary wave path
and provides the desired equation for the multiple characteristic of the modulation system
for the case m = 1.
It is instructive to derive an explicit expression for the solitary wave speed by computing
the derivative dx/dt from (49), or more simply, directly from (37),
1− a/2h
. (50)
The formula (50) yields the restriction for the relative amplitude γ = a/h < 2 which is
clearly beyond the applicability of the KdV approximation (wave breaking occurs already
at γ = 0.7 (see Whitham 1974)). In the frictionless case (CD = 0) equation (48) gives
a/h = a0h0/h
2, and so the expression (50) for the speed must fail as h → 0. It is interesting
to note that this failure of the KdV model as h → 0 due to appearance of infinite (and
further negative!) solitary wave speeds is not apparent from the expression (48) for the
solitary wave amplitude, and the implication is that the model cannot be continued as
h → 0. Curiously this restriction of the KdV model seems never to have been noticed before
in spite of numerous works on this subject. Note that taking account of bottom friction
leads to a more complicated formula for the solitary wave speed as a function of h but the
qualitative result remains the same.
It is straightforward to show from (46) or (48) that
= −hx
CDa0h0
CDa0h0
. (51)
It follows immediately that for a wave advancing into increasing depth (hx > 0), the ampli-
tude decreases due to a combination of increasing depth and bottom friction. However, for
a wave advancing into decreasing depth, there is a tendency to increase the amplitude due
to the depth decrease, but to decrease the amplitude due to bottom friction. Hence whether
or not the amplitude increases is determined by which of these effects is larger, and this in
turn is determined by the slope, the depth, and the consolidated drag parameter CDa0/h0.
To illustrate, let us consider the bottom topography in the form
h(x) = h1−α0 (h0 − δx)α , α > 0 , (52)
which satisfies the condition h(0) = h0; the parameter δ characterizes the slope of the bot-
tom. In this case the formula (48) becomes
a = a0
δ(3α− 1)h0
)(3α−1)/α
if α 6= 1/3. One can see now that if α < 1/3, then the bottom friction term is relatively
unimportant due to the smallness of CD. Of course, for this case we again recover the
Boussinesq result, now slightly modified,
a ≈ a0
δ(1− 3α)h20
, 0 < α <
, h ≪ h0. (54)
Of course, this result is impractical in the KdV context as the KdV approximation used here
requires the ratio a/h to remain small.
If α > 1/3 now obtain asymptotic formula
15(3α− 1)δ
, h ≪ h0 , (55)
which is independent of the initial amplitude a0. This expression is consistent with the small-
amplitude KdV approximation as long as (3α− 1)δ/CD is order unity. Simple inspection of
(55) shows that the solitary wave amplitude
• increases as h → 0 if 1
< α < 1
• is constant as h → 0 if α = 1
• decreases as h → 0 if α > 1
Thus for 1/3 < α < 1/2, as for the case α < 1/3, the amplitude will increase as the depth
decreases, in spite of the presence of (sufficiently small) friction. However, for α > 1/3, even
although there is usually some initial growth in the amplitude, eventually even small bottom
friction will take effect and the amplitude decreases to zero. We note that if α = 1/3 then
the integral
h−3dx in (48) diverges logarithmically as h → 0, which just slightly modifies
the result (55) for h ≪ h0 and implies growth of the amplitude ∝ ln h/h as h → 0.
Of particular interest is the case α = 1. In that case formula (53) becomes
a = a0
. (56)
and a ≈ 15
h , h ≪ h0 (57)
These expressions (56, 57) were obtained by Miles (1983a,b) using wave energy conservation
(as above, note, however, that in Miles (1983a,b) the numerical coefficient is 15/4 rather
than 15/8). Thus, these results obtained from the Whitham theory are indeed consistent, at
the leading order, with the traditional perturbation approach for a slowly-varying solitary
wave.
5 Adiabatic deformation of a cnoidal wave
Next we consider a modulated cnoidal wave (10) in the special case when the modulation does
not depend on X . While this case is, strictly speaking, impractical as it assumes there is an
infinitely long wavetrain, it can nevertheless provide some useful insights into the qualitative
effects of gradual slope and friction on undular bores which are locally represented as cnoidal
waves. In the absence of friction, the slow dependence of the cnoidal wave parameters on T
was obtained by Ostrovsky & Pelinovsky (1970, 1975) and Miles (1979) (see also Grimshaw
2006), assuming that the surface displacement had a zero mean (i.e. 〈U〉 = 0), while,
the effects of friction were taken into account by Miles (1983b) using the same zero-mean
displacement assumption. However, this assumption is inconsistent with our aim to study
undular bores where the value of 〈U〉 is essentially nonzero. Hence, we need to develop a
more general theory enabling us to take into account variations in all the parameters in the
cnoidal wave. Such a general setting is provided by the modulation system (27).
Thus we consider the case when the Riemann variables in (27) do not depend on the
variable X so that the general Whitham equations become ordinary differential equations
in T , which can be conveniently reformulated in terms of the original spatial x-coordinate
using the relationship (44):
, i = 1, 2, 3, (58)
where all variables are defined above in section 3 (see 28, 29, 30). This system can be readily
solved numerically. But it is instructive, however, to first indicate some general properties
of the solution.
First, the solution to the system (58) must have the property of conservation of “wave-
length” L (or “wavenumber” k=2π/L)
2K(m)√
λ3 − λ1
= constant (59)
Indeed, the wave conservation law (22) in absence of X-dependence assumes the form
= 0 , (60)
which yields (59). Thus the system of three equations (58) can be reduced to two equations.
Next, applying Whitham averaging directly to (7) yields
P̃ , M = 〈U〉 , P̃ = 〈|U |U〉 . (61)
P − 4CD
Q̃ , P = 〈U2〉 , Q̃ = 〈|U |3〉 . (62)
The equation set (59), (61), (62) comprise a closed modulation system for three independent
modulation parameters, say M , P̃ and m. While this system is not as convenient for further
analysis as the system (27) in Riemann variables, it does not have a restriction U > 0 inherent
in (27), and allows for some straightforward inferences regarding the possible existence of
modulation solutions with zero mean elevation, that is with M = 0. Indeed, one can see
that the solution with the zero mean is actually not generally permissible when CD 6= 0, a
situation overlooked in Miles (1983b). Indeed, M = 0 immediately then implies that P̃ = 0
by (61). But then due to (59) we have all three modulation parameters fixed which is clearly
inconsistent with the remaining equation (62) (except for the trivial case M = 0, P = 0,
Q̃ = 0). However, in the absence of friction, when CD = 0, equation (61) uncouples and
permits a nontrivial solution with a zero mean. In general, when CD = 0 equations (61),
(62) can be easily integrated to give
d = Mh9/4 = constant; σ = Ph9/2 = constant. (63)
Then, using (18, 19, 59) one readily gets the formula for the variation of the modulus m,
and hence of all the other wave parameters, as a function of h
K2[2(2−m)EK − 3E2 − (1−m)K2] =
(σ − d2)L4
. (64)
200
 400
 600
 800
0.
2
0.
4
0.
6
0.
8
C  = 0
C  = 0.01
Figure 2: Dependence of the modulus m on the physical space coordinate x in the cases
without and with bottom friction in the X-independent modulation solution.
Formula (64) generalises to the case M 6= 0 (i.e. d 6= 0) the expressions of Ostrovsky &
Pelinovsky (1970, 1975), Miles (1979) and Grimshaw (2006) (note that in Grimshaw (2006)
the zero mean restriction in actually not necessary). We note here that, again with CD = 0,
equation (5) implies conservation of 〈B〉 and 〈B2〉 (the averaged wave action flux), which,
together with (59), also yield (64).
The physical frequency Ω and wavenumber κ in the modulated periodic wave under study
are given by the formula (37), and we recall here that k = 2π/L is constant (see (59)). As
discussed before at the end of Section 3 we must require that the phase speed stays positive as
the wave evolves, and here that requires that the physical wavenumber κ > 0. Since a/h (and
hence hV/6g) is supposed to be small within the range of applicability of the KdV equation
(2) the expression (37) implies the behaviour κ ≃ Ω/
gh which of course agrees with the
well known result for linear waves on a sloping beach (see Johnson 1997 for instance). This
effect will be slightly attenuated for the nonlinear cnoidal wave, since V h/6g > 0, but the
overall effect will be a “squeezing” of the cnoidal wave, a result important for our further
study of undular bores. Next we study numerically the combined effect of slope and friction
on a cnoidal wave.
As we have shown, in the presence of Chezy friction M 6= 0, and we have also assumed
that U > 0, which is necessary when we come to study undular bores. Now we use the
stationary modulation system (58) in Riemann variables, which was derived using this as-
sumption. We solve the coupled ordinary differential equation system (58) for the case of a
linear slope
h(x) = h0 − δx (65)
with h0 = 10, δ = 0.01, and with the initial conditions
λ1 = −0.441, λ2 = 0.147, λ3 = 0.294 at x = 0, (66)
which corresponds to a nearly harmonic wave with m = 0.2, a/h0 = 0.2, 〈A〉/h0 ≈ 0.3
at x = 0 (see (35)). Also we note that for the chosen parameters we have V = 0, so at
x = 0 we have κ = Ω/
gh0 as in linear theory. It is instructive to compare solutions with
(CD = 0.01) and without (CD = 0) friction. In Fig. 2 the dependence of the modulus m
100 200 300 400 500
C  = 0
C  = 0.01
100 200 300 400 500
h  <A>
1/4 C  = 0
C  = 0.01
Figure 3: Left: Dependence of the mean value 〈A〉 in theX-independent modulation solution
on the physical space coordinate x without (dashed line) and with (solid line) bottom friction;
Right: Same but multiplied by the Green’s law factor, h1/4
100
 200
 300
 400
 500
1.
4
1.
6
1.
8
2.
2
2.
4
C   = 0
C   = 0.01
Figure 4: Dependence of the surface elevation amplitude a on the space coordinate x. Dashed
line corresponds to the frictionless case and solid line to the case with bottom friction.
on x is shown for both cases. We see that for the frictionless case m → 1 with decrease
of depth, i.e. the wave crests assume the shape of solitary waves when one approaches the
shoreline. When CD 6= 0 the modulus also grows with decrease of depth but never reaches
unity. The dependence on x of the mean surface elevation 〈A〉 for the cases without and
with friction is shown in Fig. 3. We have checked that the “wavelength” L (59) is constant
for both solutions. Also, one can see from Fig. 3 (right) that the value h1/4〈A〉 ∝ d is indeed
conserved in the frictionless case but is not constant if friction is present (the same holds
true for the value h1/2〈A2〉 ∝ σ but we do not present the graph here). Finally, in Fig. 4 the
dependence of the physical elevation wave amplitude a on the spatial coordinate x is shown.
One can see that the amplitude adiabatically grows with distance in the frictionless case
due to the effect of the slope (without friction) but, not unexpectedly, gradually decreases
in the case when bottom friction is present, where the decrease for these parameter settings
is comparable in magnitude to the effect of the slope. In both cases the main qualitative
changes occur in the wave shape and the wavelength.
Overall, we can infer from these results that the main local effect of a slope and bottom
friction on a cnoidal wave, along with the adiabatic amplitude variations, is twofold: a wave
with a m < 1 at x = 0 tends to transform into a sequence of solitary waves as x decreases,
and at the same time the distance between subsequent wave crests tends to decrease. This
is in sharp contrast with the behaviour of modulated cnoidal waves in problems described
by the unperturbed KdV equation, where growth of the modulus m is accompanied by an
increase of the distance between the wave crests. Generally, in the study of behaviour of
unsteady undular bores in the presence of a slope and bottom friction we will have to deal
with the combination of these two opposite tendencies.
6 Undular bore propagation over variable topography
with bottom friction
6.1 Gurevich-Pitaevskii problem for flat-bottom zero-friction case
We now turn to the problem (b) outlined in Section 2. We study the evolution of an undular
bore developing from an initial surface elevation jump ∆ > 0, located at some point x0 < 0.
As discussed below, the undular bore will expand with time so that at some t = t0 its lead
solitary wave enters the gradual slope region, which begins at x = 0 (see Fig. 1b). We assume
that for x < 0 one has h = h0 = constant and CD ≡ 0. We shall first present a formulation
of the Gurevich-Pitaevskii problem for the perturbation-free KdV equation and reproduce
the well-known similarity modulation solution describing the evolution of the undular bore
until the moment it enters the slope. We emphasize that, although this formulation and,
especially, this similarity solution are known very well and have been used by many authors,
some of the inferences important for the present application to fluid dynamics have not been
widely appreciated, as far as we can discern. Pertinent to our main objective in this paper,
we undertake a detailed study of the characteristics of the Whitham modulation system in
the vicinity of the leading edge of the undular bore solution, and show that the boundary con-
ditions of Gurevich-Pitaevskii type permit only two possible characteristics configurations,
implying two qualitatively different types of the leading solitary wave behaviour. Next, we
shall show how this Gurevich-Pitaevskii formulation of the problem applies to the perturbed
modulation system in the form (27) and finally we will study the effects of the perturbation
on the modulations in the vicinity of the leading edge of the undular bore.
In the case of a flat, frictionless bottom the original equation (1) becomes the constant-
coefficient KdV equation which can be cast into the standard form
ηζ + 6ηηξ + ηξξξ = 0 (67)
by introducing the new variables
A , ξ =
(x+ x0 −
gh0t) , ζ =
t , (68)
where x0 < 0 is an arbitrary constant. In the Gurevich-Pitaevskii (GP) approach, one
considers a large-scale initial disturbance η(ξ, 0) = f(ξ), in the form of a decreasing profile,
f ′(ξ) < 0 (e.g. a smooth step: f(ξ) → 0 as ξ → +∞; f(ξ) → η0 > 0 as ξ → −∞), whose
initial evolution until some critical (breaking) time ζb can be described by the dispersionless
limit of the KdV equation, i.e. by the Hopf equation,
ζ < ζb : η ≈ r(ξ, ζ), rζ + 6rrξ = 0 , r(ξ, 0) = f(ξ) . (69)
The evolution (69) leads to wave-breaking of the r(ξ)-profile at some ζ = ζb, with the
consequence that the dispersive term in the KdV equation then comes into play, and an
undular bore forms, which can be locally represented as a single-phase travelling wave. This
travelling wave is modulated in such a way that it acquires the form of a solitary wave at the
leading edge ξ = ξ+(ζ) and gradually degenerates, via the nonlinear cnoidal-wave regime, to
a linear wave packet at the trailing edge ξ = ξ−(ζ). It is important that this undular bore
is essentially unsteady, i.e. the region ξ−(ζ) < ξ < ξ+(ζ) expands with time ζ .
The single-phase travelling wave solution of the KdV equation (67) has the form (cf.
(10))
η(ξ, ζ) = r3 − r1 − r2 − 2(r3 − r2)sn2(
r3 − r1θ,m) (70)
θ = ξ + 2(r1 + r2 + r3)ζ , m =
r3 − r2
r3 − r1
. (71)
The parameters r1 ≤ r2 ≤ r3 ≤ 0 in the undular bore are slowly varying functions of ξ, ζ ,
whose evolution is governed by the Whitham equations
+ vj(r1, r2, r3)
= 0 , j = 1, 2, 3. (72)
The characteristic velocities in (72) are given by (31). We stress that, although analytical
expressions (70) and (10) (as well as (72) and the homogeneous version of (27)) are identical,
they are written for completely different sets of variables, both dependent and independent.
The Riemann invariants rj(ξ, ζ) are subject to special matching conditions at the free
boundaries, ξ = ξ±(ζ) defined by the conditions m = 0 (trailing edge) and m = 1 (leading
edge), formulated in Gurevich and Pitaevskii (1974) (see also Kamchatnov (2000) or El
(2005) for a detailed description).
At the trailing (harmonic) edge, where the wave amplitude a = 2(r3 − r2) vanishes and
m = 0, one has
ξ = ξ−(ζ) : r2 = r3 , −r1 = r . (73)
At the leading (soliton) edge, where m = 1 one has
ξ = ξ+(ζ) : r2 = r1 , −r3 = r . (74)
In both (73) and (74), r(ξ, ζ) is the solution of the Hopf equation (69).
The curves ξ = ξ±(ζ) are defined for the solution of the GP problem (72), (73), (74) by
the ordinary differential equations
= v−(ξ−, ζ) ,
= v+(ξ+, ζ) , (75)
where v± are calculated as the values of double characteristic velocities of the modulation
system at the undular bore edges,
v− = v2(r1, r3, r3)|ξ=ξ−(ζ) = v3(r1, r3, r3)|ξ=ξ−(ζ), (76)
v+ = v2(r1, r1, r3)|ξ=ξ+(ζ) = v1(r1, r1, r3)|ξ=ξ+(ζ) (77)
These equations (75) essentially represent kinematic boundary conditions for the undular
bore (see El 2005). Indeed, the double characteristic velocity v2(r1, r3, r3) = v3(r1, r3, r3) can
be shown to coincide with the linear group velocity of the small-amplitude KdV wavepacket
while the double characteristic velocity v2(r1, r1, r3) = v1(r1, r1, r3) is the soliton speed.
One might infer from this GP formulation of the problem that, since the leading edge of
the undular bore specified by (75), (77) is a characteristic of the modulation system, then
the value of the double Riemann invariant r+ ≡ r2 = r1 is constant. Then, on considering an
undular bore propagating into still water, where r = 0, one would obtain from the matching
condition (74) at the leading edge that r3|ξ=ξ+ = 0 and thus, the amplitude of the lead solitary
wave a+ = 2(r3−r1)|ξ=ξ+ = −r+ would always be constant as well. However, this contradicts
the general physical reasoning that the amplitude of the lead solitary wave should be allowed
to change in the case of general initial data. The apparent contradiction is resolved by noting
that the leading edge specified by (75), (77) can be an envelope of the characteristic family,
i.e. a caustic, rather than necessarily a regular characteristic, and hence there is no necessity
for the double Riemann invariant r+ to be constant along the curve ξ = ξ+(ζ) in general case.
On the other hand, since the leading edge is defined by the condition m = 1, the wave form
at the leading edge will coincide with the spatial profile of the standard KdV soliton. Thus
we arrive at the conclusion that, in general, the amplitude of the leading KdV solitary wave
will vary, even in the absence of the perturbation terms. Of course, in the unperturbed KdV
equation, such varying solitary waves cannot not exist on their own, and require the presence
of the rest of the undular bore. We also stress that these variations of the leading solitary
wave in the undular bore, as described here, have a completely different physical nature to
the variations of the parameters of an individual solitary wave due to small perturbations as
described in Section 4. They are caused by nonlinear wave interactions within the undular
bore rather than by a local adiabatic response of the solitary wave to a perturbation induced
by topography and friction. Importantly for our study, however, it will transpire that the
action of these same perturbation terms on the undular bore can lead to both a local and a
nonlocal response of the leading solitary wave.
6.2 Undular bore developing from an initial jump
Next we consider the simplest solution of the modulation system, which describes an undular
bore developing from an initial discontinuity placed at the point x = −x0. In (η; ξ, ζ) -
variables we have the initial conditions
η(ξ, 0) = ∆ for ξ < 0 ; η(ξ, 0) = 0 for ξ > 0 , (78)
where ∆ > 0 is a constant. Then, on using (69), the initial conditions (78) are readily
translated into the free-boundary matching conditions (73), (74) for the Riemann invariants.
Because of the absence of a length scale in this problem, the corresponding solution of the
modulation system must depend on the self-similar variable τ = ξ/ζ alone, which reduces
the modulation system to the ordinary differential equations
(vi − τ)
= 0 , i = 1, 2, 3. (79)
-4
0
 -3
0
 -2
0
 -1
0
 10
 20
-0.8
-0.6
-0.4
-0.2
-30 -20 -10 10
η(ξ, ζ = 5)
Figure 5: Left: Riemann invariants behaviour in the similarity modulation solution for the
flat-bottom zero-friction case ; Right: corresponding undular bore profile η(ξ).
The boundary conditions for (79) follow from the matching conditions (73), (74) using the
initial condition (78):
τ = τ− : r2 = r3 , r1 = −∆
τ = τ+ : r2 = r1 , r3 = 0 .
where τ± are self-similar coordinates (speeds) of the leading and trailing edges, ξ± = τ±ζ .
Taking into account the inequality r1 ≤ r2 ≤ r3 one obtains the well-known modulation
solution of Gurevich and Pitaevskii (1974) (see also Fornberg and Whitham 1978) in the
r1 = −∆ , r3 = 0 , r2 = −m∆ , (81)
= v2(−∆,−m∆, 0) = 2∆[(1 +m)−
2m(1−m)K(m)
E(m)− (1−m)K(m)
] . (82)
This modulation solution (81), (82) (see Fig. 5a) represents the replacement, due to averag-
ing over the oscillations, of the unphysical formal three-valued solution of the dispersionless
KdV equation (i.e. of the Hopf equation) which would have taken place in the absence of
the dispersive regularisation by the undular bore. We see that (82) describes an expansion
fan in the characteristic (ξ, ζ)-plane and thus is a global solution. Substituting (81), (82)
into the travelling wave solution (70) one obtains the asymptotic wave form of the undular
bore (see Fig. 5b), which then can be readily represented in terms of the original physical
variables using the relationships (68).
The equations of the trailing and leading edges of the undular bore are determined from
(82) by putting m = 0 and m = 1 respectively
= τ− = v2(−∆, 0, 0) = −6∆ ,
= τ+ = v2(−∆,−∆, 0) = 4∆ . (83)
The leading solitary wave amplitude is η0 = 2(r3−r1) = 2∆, which is exactly twice the height
of the initial jump. This corresponds to the amplitude of the surface elevation a = 3h0∆ (see
(68)). Note that, to get the leading solitary wave of the same initial amplitude a0 as for the
separate solitary wave considered in Section 4, one should use the jump value ∆0 = a0/3h0,
which of course is just 2∆̃, where ∆̃ = 3h0∆/2 is the initial discontinuity in the surface
elevation.
6.3 Structure of the undular bore front
We are especially interested in the behaviour of the modulation solution (81), (82) in the
vicinity of the leading edge ξ = ξ+(ζ). This behaviour is essentially determined by the
manner in which the pair of characteristics corresponding to the velocities v2 and v1 merge
into a multiple eigenvalue v+ of the modulation system at ξ = ξ+(ζ).
First, one can readily infer from the modulation solution (81), (82) that the phase velocity
c = −2(r1 + r2 + r3) = 2∆(1 +m) > v2(−∆,−m, 0) for m < 1 and c = v2 for m = 1. Thus,
any individual wave crest generated at the trailing edge of the undular bore moves towards
the leading edge, i.e. for any crest m → 1 as ζ → ∞. Thus, for any particular wave crest,
except for the very first one, the solitary wave ‘status’ is achieved only asymptotically as
ζ → ∞.
Without loss of generality we assume in this section that ∆ = 1 in (81), (82). First, as
we have already mentioned, the characteristic family Γ2 : dξ/dζ = v2 is an expansion fan in
the ξ, ζ - plane,
Γ2 : ξ = C2ζ , (84)
parameterised by a constant C2, −6 ≤ C2 ≤ 4 . Next, in (82) we make an asymptotic
expansion of v2(−1,−m, 0) for small (1−m) ≪ 1, to get
2(1−m) ln(16/(1−m)) ≃ τ+ − ξ/ζ (85)
or, with logarithmic accuracy,
(τ+ − ξ/ζ) ≪ 1 : 1−m ≃
τ+ − ξ/ζ
2 ln[1/(τ+ − ξ/ζ)]
. (86)
Next, expanding v1(−1,−m, 0) for (1 − m) ≪ 1 and using (86) we get the asymptotic
equation for the characteristics family Γ1,
= v1 = τ
+ + (τ+ − ξ/ζ) +O(1−m) , (87)
which is readily integrated to leading order to give
Γ1 : ξ ≃ τ+ζ −
, (88)
where C1 ≥ 0 is an arbitrary constant ‘labeling’ the characteristics; C1 = 0 corresponds to
the leading edge of the undular bore. This asymptotic formula (88) is valid as long as ζ ≫ 1.
The behaviour of the characteristics belonging to the families Γ1 and Γ2 near the leading
edge is shown in Fig. 6a.
Next, expanding the equation for the third characteristic family, Γ3: dξ/dζ = v3(−1,−m, 0)
for (1−m) ≪ 1, we get on using (86)
τ+ − ξ/ζ
ln(1/(τ+ − ξ/ζ))
+O(τ+ − ξ/ζ) . (89)
Figure 6: Characteristics behaviour for the similarity modulation solution near the leading
edge ξ+(ζ): (a) families Γ1: dξ/dζ = v1 and Γ2 : ξ = C2ζ , (b) family Γ3: dξ/dζ = v3.
Integrating (89) we obtain to first order
Γ3 : ξ ≃ C3 − g(ζ) , (90)
where g(ζ) =
τ+ζ − C3
ln |τ+ζ − C3| − ln ζ
dζ , g(C3/τ
+) = 0 , (91)
C3 being an arbitrary constant. The asymptotic formula (90) is valid as long as g(ζ)/C3 ≪
1. Since the characteristics Γ3 intersect the leading edge ξ = τ
+ζ we must indicate their
behaviour outside the undular bore. It follows from the matching condition (74) and the
limiting structure (34) of the characteristic velocities of the Whitham system, that the
characteristics from the family Γ3 match with the Hopf equation characteristics dξ/dζ = 6r
carrying the value of the Riemann invariant r = 0 corresponding to still water upstream the
undular bore. Therefore, the sought external characteristics are simply vertical lines ξ = C3.
The qualitative behaviour of the characteristics from the family Γ3 is shown in Fig. 6b.
It is clear from the asymptotic behaviour of the characteristics that the edge characteristic
ξ = τ+ζ corresponding to the motion of the leading solitary wave intersects only with
characteristics of the family Γ3 carrying the Riemann invariant value r3 = 0 into the undular
bore domain. Since, according to the matching condition (80), r3 ≡ 0 everywhere along the
edge characteristic one can infer that the leading solitary wave motion is completely specified
by its amplitude at ζ = 0. Indeed, in this case, the leading edge represents a genuine multiple
characteristic of the modulation system, along which the Riemann invariant r+ = r2 = r1 is
a constant. Given the constant value of r1 = −1 for the solution (82), one infers that the
amplitude of the lead soliton of the self-similar undular bore, η0 = 2(r3 − r+) = 2 is also a
constant value. Thus, in the undular bore evolving from an initial jump, the leading solitary
wave represents an independent soliton of the KdV equation. Of course, this fact follows
directly from the modulation solution (82) but now we have established its meaning in the
context of the characteristics, which will play an important role below.
Next we discuss the structure of the undular bore front in the case when the initial
profile η(ξ, 0) is not a simple jump discontinuity, and instead has the form of a monotonically
decreasing function, for instance, (−ξ)1/2 when ξ ≤ 0 and η(ξ, 0) = 0 for ξ > 0. In that case,
the modulation solution for the undular bore no longer possesses x/t-similarity as in the
Figure 7: a) Leading edge ξ+(ζ) of non-self-similar undular bore as an envelope of pairwise
merging characteristics from the families dξ/dζ = v1 and dξ/dζ = v2; b) behaviour of the
Riemann invariants in non-self-similar modulation solution with r3 ≡ 0.
jump resolution case and, as a result, the speed (and therefore, the amplitude) of the lead
solitary wave is not constant. For instance, for the afore-mentioned square-root initial profile
the amplitude of the lead solitary wave grows as ζ2 (see Gurevich, Krylov and Mazur 1989,
or Kamchatnov 2000). Clearly, such an amplitude variation is impossible if the leading edge
ξ+(ζ) was a regular characteristic carrying a constant value of the Riemann invariant r+. As
discussed above, however, the GP matching conditions (73) -(77) admit another possibility;
the leading edge curve is the envelope of the characteristic families Γ1: dξ/dζ = v1 and Γ2:
dξ/dζ = v2 merging when m = 1. This configuration is shown in Fig. 7a. In this case, the
behaviour of the modulus m in the vicinity of the leading edge is given by the asymptotic
formula found in Gurevich & Pitaevskii (1974):
(1−m)2
(r+)2
(ξ+ − ξ) (92)
where the function r+(ζ) 6= constant is assumed to be known. Another specific feature of
this (general) configuration is that dr1,2/dξ → ±∞ as ξ → ξ+ (see Fig. 7b - also found in
Gurevich & Pitaevskii 1974, see also Kamchatnov 2000), which is in drastic contrast with
similarity solution (see Fig. 6a). This particular difference was discussed in relation with
undular bores in the KdV-Burgers equation in Gurevich and Pitaevskii (1987).
In summary, we see from (92) that the structure of the modulation solution in the vicin-
ity of the leading edge of an undular bore defined as a characteristic envelope is qualitatively
different compared to that for the similarity case (see (85)). The more general (but qual-
itatively similar to (92)) asymptotic formula which takes into account small perturbations
due to a variable topography and bottom friction will be derived later. At the moment,
it is important for us that in this configuration, when the leading edge is a characteristic
envelope rather than just a characteristic, the value r+, and thus, the leading solitary wave
amplitude are allowed to vary.
The analysis of the corresponding modulation solution in Gurevich, Krylov and Mazur
(1989) showed that, while in the case of an initial jump the wave crests generated at the
trailing edge reach the leading edge (and therefore, transform into solitary waves) only
asymptotically as t → ∞, for the more general case of decreasing initial data each wave
crest generated at the trailing edge reaches the leading edge in finite time and replaces
(overtakes) the existing leading solitary wave. This process is manifested as a continuous
amplitude growth of the (apparent) leading solitary wave. As in classical soliton theory,
an alternative explanation of the leading solitary wave amplitude growth can be made in
terms of the momentum exchange between the “instantaneous” leading solitary wave and
solitary waves of greater amplitude coming from the left. Indeed, as the rigorous analysis of
Lax, Levermore and Venakides showed (see Lax, Levermore and Venakides (1994) and the
references therein), the whole modulated structure of the undular bore can be asymptotically
described in terms of the interactions of a large number of KdV solitons initially ‘packed’
into a non-oscillating large-scale initial profile.
This latter interpretation is especially instructive for our purposes. Our point is that
the specific cause of the enhanced soliton interactions resulting in amplitude growth at the
leading edge is not essential; it can be large-scale spatial variations of the initial profile as
just described, but it could also equally well be an effect of small perturbations in the KdV
equation itself. Indeed, in the weakly perturbed KdV equation, the local wave structure
of the undular bore must be described to leading order by the periodic solution (70) of
the unperturbed KdV equation, so if one assumes the GP boundary conditions analogous to
(73) – (77) for the perturbed modulation system (27), one invariably will have to deal with
one of the two possible types of the characteristics behaviour (shown in Figs. 7a and 8a)
in the vicinity of the leading edge of the undular bore, because this qualitative behaviour
is determined only by the structure of the GP boundary conditions and by the associated
asymptotic structure of the characteristic velocities of the Whitham system for (1−m) ≪ 1,
which are the same for both unperturbed and perturbed modulation systems. Next, we will
show that, by using the knowledge of this qualitative behaviour of the characteristics, one
is able to construct the asymptotic modulation solution for the undular bore front in the
presence of variable topography and bottom friction even if the full solution of the perturbed
modulation system is not available.
6.4 Gurevich-Pitaevskii problem for perturbed modulation sys-
We investigate now how the GP matching problem applies to the perturbed modulation
system (27). As in the original GP problem, we postulate the natural physical requirement
that the mean value 〈U〉 is continuous across the undular bore edges, which represent free
boundaries and are defined by the conditions m = 0 (trailing edge X = X−(T )) and m = 1
(leading edge X = X+(T )). Also, we consider propagation of the undular bore into still
water, hence 〈U〉|X=X+(T ) = 0. Now, using the explicit expression (18) for 〈U〉 in terms of
complete elliptic integrals and calculating its limits as m → 0 and m → 1 one has
X = X−(T ) : λ2 = λ3 , 〈U〉 = −λ1 = u ,
X = X+(T ) : λ2 = λ1 , 〈U〉 = −λ3 = 0 ,
where u(X, T ) is solution of the dispersionless perturbed KdV equation (7), i.e.
uT + 6uuX = F (T )u−G(T )u2, (94)
with the boundary conditions
∆0 if τ < τ0; u
= 0 if τ > τ0 , (95)
where τ0 = −x0/
gh0. The boundary conditions (95) correspond to a discontinuous initial
surface elevation A(x, t) at x = −x0, obtained by using transformations (3) and (6) where
one sets t = 0. As earlier, ∆0 = a0/(3h0) is the value of the discontinuity in A, chosen in
such a way that the amplitude of the lead solitary wave in the undular bore was exactly a0
in the flat-bottom zero-friction region (see Section 6.2).
This free-boundary matching problem is then complemented by the kinematic conditions
explicitly defining the boundaries X = X±(T ). These are formulated using the multiple
characteristic directions of the perturbed modulation system (27) in the limits as m → 0
and m → 1 (cf. (75) - (77)),
= V −(X−, T ) ,
= V +(X+, T ) , (96)
where V − = v2(u, λ
−, λ−) = v3(u, λ
−, λ−), (97)
V + = v2(λ
+, λ+, 0) = v1(λ
+, λ+, 0) , (98)
and λ− = λ2(X
−, T ) = λ3(X
−, T ) , λ+ = λ2(X
+, T ) = λ1(X
+, T ). (99)
Thus, for the perturbed KdV equation the leading and trailing edges of the undular bore are
defined mathematically in the same way as for the unperturbed one, albeit for a different
set of variables.
6.5 Deformation of the undular bore front due to variable topog-
raphy and bottom friction
Finally we study the effects of gradual slope and bottom friction on the leading front of the
self-similar expanding undular bore described in Sections 6.2, 6.3. The result will essentially
depend on the relative values of the small parameters appearing in the problem. We note
that in general there are three distinct relevant small parameters,
≪ 1 , δ = max(hx) ≪ 1, CD ≪ 1 (100)
The first small parameter is determined by the ratio of the equilibrium depth in the flat
bottom region, to the distance from the beginning of the slope region to the location of the
initial jump discontinuity in the surface displacement. This measures the typical relative
spatial variations of the modulation parameters in the undular bore when it reaches the
beginning of the slope. The second and third parameters are contained in the KdV equation
(1) itself and measure the values of the slope and bottom friction respectively. In terms of
the transformed variables appearing in (7), |F (T )| ∼ δ, |G(T )| ∼ CD (see (8)). Generally
we assume δ ∼ CD (the possible orderings δ ≪ CD or CD ≪ δ can be then considered as
particular cases).
To obtain a quantitative description of the vicinity of the leading edge of the undular
bore we perform an expansion of the Whitham modulation system (27) for (1 − m) ≪ 1.
We first introduce the substitutions
λi(X, T ) = λ
+(T ) + li(X̃, T ) , vi = V
+ + v′i , ρi = ρ
+ + ρ′i, i = 1, 2. (101)
where X̃ = X+ −X , V + = −4λ+ , ρ+ =
F (T )λ+ +
G(T )(λ+)2. (102)
Since λ2 ≥ λ1, v2 ≥ v1 one always has l2 ≥ l1, v′2 ≥ v′1. Assuming X̃/X+ ≪ 1 ⇔ 1−m ≪ 1
and using that λ3 = 0 to leading order in the vicinity of the leading edge (see the matching
condition (93)), we have from asymptotic expansions of (28) – (31) as (1−m) ≪ 1
v′1 = M1(l2 − l1) ≡ −2
ln(16/(1−m))
1 + 1
(1−m) ln(16/(1−m))
(l2 − l1),
v′2 = M2(l2 − l1) ≡ −2
1− ln(16/(1−m))
(1−m) ln(16/(1−m))
(l2 − l1),
(103)
ρ′1 = N1(l2 − l1) ≡
1 + ln
l2 − l1
−16λ+
2λ+ ln
l2 − l1
−16λ+
− 3λ+
(l2 − l1)
ρ′2 = N2(l2 − l1) ≡
5 + ln
l2 − l1
−16λ+
2λ+ ln
l2 − l1
−16λ+
+ 13λ+
(l2 − l1).
(104)
Naturally, v′i and ρ
i vanish when l2 = l1. Now, substituting (101), (102) into the modulation
system (27) we obtain
− (V + + v′i)
= ρ+ + ρ′i, i = 1, 2. (105)
On using the kinematic condition (96) at the leading edge, this reduces to
− v′i
= ρ+ + ρ′i, i = 1, 2. (106)
There are two qualitatively different cases to consider:
(i) limX̃→0 |dli/dX̃| < ∞, i = 1, 2 (Fig. 8a)
(ii) limX̃→0 |dli/dX̃| = ∞, i = 1, 2 (Fig. 8b)
The case (i) implies that to leading order (106) reduces to
= ρ+ , (107)
which, together with the kinematic condition dX+/dT = −4λ+, defines the leading edge
curve X+(T ). One can observe that this system coincides with (43), (42) defining the
Figure 8: Riemann variables behaviour in the vicinity of the leading edge of the undular
bore propagating over gradual slope with bottom friction (a) Adiabatic variations of the
similarity GP regime, δ ≪ ǫ, CD ≪ ǫ; (b) General case, δ ∼ CD ∼ ǫ.
motion of a separate solitary wave over a gradual slope with bottom friction. Its integral
expressed in terms of original physical x, t-variables is given by (49). Therefore, in the case
(i) the lead solitary wave in the undular bore to leading order is not restrained by interactions
with the remaining part of the bore and behaves as a separate solitary wave. Physically this
case corresponds to adiabatic deformation of the similarity modulation solution (81), (82)
and implies the following small parameter ordering : δ ≪ ǫ, CD ≪ ǫ.
Next, we study the structure of this weakly perturbed similarity modulation solution in
the vicinity of the leading edge. The next leading order of the system (106) yields
− v′i
= ρ′i, i = 1, 2, (108)
that is
= −N1
= −N2
. (109)
Subtraction of one equation (109) from another with account of the relationship l2 − l1 ∼=
−λ+(1−m) leads consistently to leading order to the differential equation for 1−m
∂(1 −m)
F (T )
16G(T )
, (110)
This equation should be solved with the initial condition
1−m = 0 at X̃ = 0 . (111)
Elementary integration gives with the accuracy O(1−m) (cf. (85))
(1−m) ln 16
F (T )− 16
λ+G(T )
X+ −X
. (112)
This formula determines the dependence of the modulus m on T and X (as long as 1−m ≪
Now, we make use of the solution λ+ of equation (107) given by (47) with C0 =
4/(3ga0h0) (see (48)). Under supposition that the integral
h−3dx diverges as h → 0,
so that the turbulent bottom friction plays an essential role in the undular bore front be-
haviour (see Section 4 for a similar approximation for an isolated solitary wave), we obtain
for h ≪ h0
(1−m) ln
2 + 3h2
(X+ −X). (113)
At last, if the bottom topography is approximated by the dependence (52), we get with the
same accuracy
(1−m) ln 16
(3α− 1)δ
(X+ −X) , (114)
where α > 1/3. The second term in square brackets tends to zero as h → 0. However, the
region where it can be neglected may be very narrow because of smallness of the parameter
δ. We recall that in this formula X+ is given by (49) and X is defined by (3) in terms of
the original physical independent variables x and t.
Summarising, if the conditions δ, CD ≪ ǫ are satisfied, the lead solitary wave of the
undular bore behaves as an individual (noninteracting) solitary wave adiabatically varying
under small perturbation due to variable topography and bottom friction. The modulation
solution in the vicinity of the leading edge also varies adiabatically, however, its qualitative
structure considered in Section 6.4 (see Figs 5,6) remains unchanged.
In a sharp contrast with the described case of adiabatic deformation of an undular bore
front is case (ii) when the second term in the left-hand side of (106) contributes to the
leading order, i.e. to the motion of the leading edge itself. Namely, we have
= ρ+ + v′i
, i = 1, 2. (115)
Now dλ+/dT 6= ρ+ which means that the amplitude of the lead solitary wave a = −2λ+ varies
essentially differently compared to the case of an isolated solitary wave. Indeed, the term
ρ+ in the right-hand side of (115) is responsible for local adiabatic variations of the solitary
wave while the term v′i∂li/∂X̃ describes nonlocal parts of the variations associated with
the wave interactions within the undular bore. Using asymptotic formulae (103) implying
v′2 ≥ 0, v′1 ≤ 0, and the condition limX̃→0 |dl1,2/dX̃| = ∞ along with l2 ≥ l1, it is not
difficult to show that this nonlocal term is always nonnegative , i.e. the lead solitary wave
in the undular bore propagating over a gradual slope with bottom friction always moves
faster (and, therefore, has greater amplitude) than an isolated solitary wave of the same
initial amplitude in the beginning of the slope. Indeed, as we have shown in Section 5, the
presence of a slope and bottom friction always result in “squeezing” the cnoidal wave, hence
increasing momentum exchange between solitary waves in the vicinity of the leading edge
of the undular bore and acceleration of the lead solitary wave itself. The situation here is
qualitatively analogous to that described in Section 6.4 where the general global modulation
solution for the unperturbed KdV equation was discussed. Similarly to that case, the leading
edge now represents a characteristic envelope – a caustic (otherwise we are back in the case
(i) implying dλ+/dT = ρ+) (see Fig. 6a).
Unlike the case of adiabatic variations of the leading edge, determination of the function
λ+(T ) requires now knowledge of the full solution of the perturbed modulation system (27)
with the matching conditions (93). While the analytic methods to construct such a solution
for inhomogeneous quasilinear systems are not available presently, it is instructive to assume
that dλ+/dT − ρ+ is a known function of T and to study the structure of the solution in
close vicinity of the leading edge. With an account of the explicit form (103) of the velocity
corrections, equations (115) assume the form
= −dλ
+/dT − ρ+
2(l2 − l1)
ln[16/(1−m)]
(1−m)
, (116)
= −dλ
+/dT − ρ+
2(l2 − l1)
ln[16/(1−m)]
(1−m)
. (117)
Taking the difference of (116) and (117) we transform it to the form
∂(1 −m)
dλ+/dT − ρ+
(λ+)2
(1−m) ln[16/(1−m)]
. (118)
This equation can be readily integrated with the initial condition (111) to give
(1−m)2
2(dλ+/dT − ρ+)
(λ+)2
(X+ −X). (119)
This solution coincides with the asymptotic formula (92) for the behaviour of the modulus
in the vicinity of the leading edge of the undular bore in general unperturbed GP problem
[16] but instead of the derivative dλ+/dT in (92) we have the difference dλ+/dT −ρ+ (which
is always positive as we have established).
7 Conclusions
We have studied the effects of a gradual slope and turbulent (Chezy) bottom friction on the
propagation of solitary waves, nonlinear periodic waves and undular bores in shallow-water
flows in the framework of the variable-coefficient perturbed KdV equation. The analysis has
been performed in the most general setting provided by the associated Whitham equations
describing slow modulations of a periodic travelling wave due to the slope, bottom friction
and spatial nonuniformity of initial data. This modulation theory, developed in general form
for perturbed integrable equations in Kamchatnov (2004) was applied here to the perturbed
KdV equation and allowed us to take into account slow variations of all three parameters
in the cnoidal wave solution. The particular time-independent solutions of the perturbed
modulation equations were shown to be consistent with the adiabatically varying solutions
for a single solitary wave and for a periodic wave propagating over a slope without bottom
friction obtained in Ostrovsky & Pelinovsky (1970, 1975) and Miles (1979, 1983a). It was
shown, however, that the assumption of zero mean elevation used in these papers for the
description of slow variations of a cnoidal wave, ceases to be valid in the case when the
turbulent bottom friction is present. In this case, a more general solution was obtained
numerically improving the results of Miles (1983b).
Further, the derived full time-dependent modulation system was used for the descrip-
tion of the effects of variable topography and bottom friction on the propagation of undular
bores, in particular on the variations of the undular bore front representing a system of
weakly interacting solitary waves. By the analysis of the characteristics of the Whitham
system in the vicinity of the leading edge of the undular bore, two possible configurations
have been identified depending on whether the leading edge of the undular bore represents a
regular characteristic of the modulation system or its singular characteristic, i.e. a caustic.
The first case was shown to correspond to adiabatically slow deformations of the classi-
cal Gurevich-Pitaevskii modulation solution and is realised when the perturbations due to
variable topography and bottom friction are small compared with the existing spatial non-
uniformity of modulations in the undular bore (which is supposed to be formed outside the
region of variable topography/bottom friction). In the case when modulations due to the
external perturbations are comparable in magnitude with the existing modulations in the
undular bore, the leading edge becomes a caustic, and this situation was shown to corre-
spond to enhanced solitary wave interactions within the undular bore front. These enhanced
interactions have been shown to lead to a “nonlocal” leading solitary wave amplitude growth,
which cannot be predicted in the frame of the traditional local adiabatic approach to prop-
agation of an isolated solitary wave in a variable environment. As we mentioned in the
Introduction, one of our original motivations for this study was the possibility to model a
shoreward propagating tsunami as an undular bore. In this context, we would suggest that
the second scenario described above is the more relevant, which has the implication that
the growth, and eventual breaking of the leading waves in a tsunami wavetrain, cannot be
modeled as a local effect for that particular wave, but is determined instead by the whole
structure of the wavetrain.
Acknowledgements
This work was started during the visit of A.M.K. at the Department of Mathematical Sci-
ences, Loughborough University, UK. A.M.K. is grateful to EPSRC for financial support.
Appendix A: Derivation of the perturbed modulation system
We express the integrand function in the right-hand side of (24) in terms of the µ-variable
(15):
(2λi − s1 − U)R = 8Gµ3 − [8Gλi + 4(F + 2s1G)]µ2
+ [4(F + 2s1G)λi + 2s1(s1G+ F )]µ− 2s1(s1G+ F )λi.
(120)
Then we obtain with the use of (13), (14), and (16) the following expressions:
〈µ〉 =
µdθ =
−P (µ)
〈µ2〉 = 1
µ2dθ =
〈µ3〉 =
µ3dθ = −
+ s1〈µ2〉 − s2〈µ〉+ s3,
(121)
where I is a known integral
(λ3 − µ)(µ− λ2)(µ− λ1) dµ
(λ3 − λ1)5/2[(1−m+m2)E(m)− (1−m)(1−m/2)K(m)],
(122)
K(m) and E(m) being the complete elliptic integrals of the first and second kind, respec-
tively. The derivatives of I with respect to λi are also known table integrals (Gradshtein &
Ryzhik 1980):
(λ3 − µ)(µ− λ2)
µ− λ1
λ3 − λ1[(λ2 + λ3 − 2λ1)E − 2(λ2 − λ1)K],
(λ3 − µ)(µ− λ1)
µ− λ2
λ3 − λ1[(λ3 − λ1)K + (λ1 + λ3 − 2λ2)E],
(µλ2)(µ− λ1)
λ3 − µ
λ3 − λ1[(2λ3 − λ1 − λ2)E − (λ2 − λ1)K].
(123)
We can easily express the si-derivatives in terms of λi derivatives by differentiation of the
formulae (see (16))
s1 = λ1 + λ2 + λ3, s2 = λ1λ2 + λ1λ3 + λ2λ3, s3 = λ1λ2λ3 (124)
and solving the linear system for differentials. Simple calculation gives
(−1)3−k
j 6=i(λi − λj)
. (125)
Then, combining (123) and (125), we obtain the derivatives ∂I/∂si and hence the expressions
(λ3 − λ1)
(s21 − 3s2)
(λ2 − λ1)(λ2 + λ3 − 2λ1)
+ s1λ1 + λ
1 − λ2λ3
(λ3 − λ1)
(126)
To complete the calculation of the right-hand side of (24), we need also expressions
∂L/∂λ1
= 2(λ2 − λ1)
∂L/∂λ2
2(λ3 − λ2)(1−m)K
E − (1−m)K
∂L/∂λ3
2(λ3 − λ2)K
(127)
Collecting all contributions into perturbations terms, we obtain the Whitham equations in
the form
= ρi = Ci[F (T )Ai −G(T )Bi], (128)
where Cj , Aj , Bj and vj , j = 1, 2, 3 are specified by formulae (28) - (30).
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	Introduction
	Problem formulation
	Modulation equations
	Modulation solution in the solitary wave limit
	Adiabatic deformation of a cnoidal wave
	Undular bore propagation over variable topography with bottom friction
	Gurevich-Pitaevskii problem for flat-bottom zero-friction case
	Undular bore developing from an initial jump
	Structure of the undular bore front
	Gurevich-Pitaevskii problem for perturbed modulation system
	Deformation of the undular bore front due to variable topography and bottom friction
	Conclusions

  This paper considers the propagation of shallow-water solitary and nonlinear
periodic waves over a gradual slope with bottom friction in the framework of a
variable-coefficient Korteweg-de Vries equation. We use the Whitham averaging
method, using a recent development of this theory for perturbed integrable
equations. This general approach enables us not only to improve known results
on the adiabatic evolution of isolated solitary waves and periodic wave trains
in the presence of variable topography and bottom friction, modeled by the
Chezy law, but also importantly, to study the effects of these factors on the
propagation of undular bores, which are essentially unsteady in the system
under consideration. In particular, it is shown that the combined action of
variable topography and bottom friction generally imposes certain global
restrictions on the undular bore propagation so that the evolution of the
leading solitary wave can be substantially different from that of an isolated
solitary wave with the same initial amplitude. This non-local effect is due to
nonlinear wave interactions within the undular bore and can lead to an
additional solitary wave amplitude growth, which cannot be predicted in the
framework of the traditional adiabatic approach to the propagation of solitary
waves in slowly varying media.

Introduction
There have been many studies of the propagation of water waves over a slope, sometimes
also subject to the effects of bottom friction. Many of these works have considered linear
waves, or have been numerical simulations in the framework of various nonlinear long-wave
model equations. Our interest here is in the propagation of weakly nonlinear long water
http://arxiv.org/abs/0704.0045v1
waves over a slope, simultaneously subject to bottom friction, a combination apparently
first considered by Miles (1983a,b) albeit for the special case of a single solitary wave, or a
periodic wavetrain. An appropriate model equation for this scenario is the variable-coefficient
perturbed Korteweg-de Vries (KdV) equation (see Grimshaw 1981, Johnson 1973a,b),
At + cAx +
AAx +
Axxx = −CD
|A|A. (1)
Here A(x, t) is the free surface elevation above the undisturbed depth h(x) and c(x) =
gh(x) is the linear long wave phase speed. The bottom friction term on the right-hand
side is represented by the Chezy law, modelling a turbulent boundary layer. Here CD is a
non-dimensional drag coefficient, often assumed to have a value around 0.01 (Miles 1983a,b).
Other forms of friction could be used (see, for instance Grimshaw et al 2003) but the Chezy
law seems to be the most appropriate for water waves in a shallow depth. In (1) the first
two terms on the left-hand side are the dominant terms, and by themselves describe the
propagation of a linear long wave with speed c. The remaining terms on the left-hand
side represent, respectively, the effect of varying depth, weakly nonlinear effects and weak
linear dispersion. The equation is derived using the usual KdV balance in which the linear
dispersion, represented by ∂2/∂x2, is balanced by nonlinearity, represented by A. Here we
have added to this balance weak inhomogeneity so that cx/c scales as h
2∂3/∂x3, and weak
friction so that CD scales with h∂/∂x. Within this basic balance of terms, we can cast (1)
into the asymptotically equivalent form
AAX +
AXXX = −CD
|A|A, (2)
where τ =
c(x′)
, X = τ − t. (3)
Here we have h = h(x(τ)), explicitly dependent on the variable τ which describes evolution
along the path of the wave.
The governing equation (2) can be cast into several equivalent forms. That most com-
monly used is the variable-coefficient KdV equation, obtained here by putting
B = (gh)1/4A (4)
so that Bτ +
2g1/4h5/4
BBX +
BXXX = −CD
|B|B . (5)
This form shows that, in the absence of friction term, i.e. when CD ≡ 0, equation (2)
has two integrals of motion with the densities proportional to h1/4A and h1/2A2. These
are often referred to as laws for the conservation of “mass” and “momentum”. However,
these densities do not necessarily correspond to the corresponding physical entities. Indeed,
to leading order, the “momentum” density is proportional to the wave action flux, while
the “mass” density differs slightly from the actual mass density. This latter issue has been
explored by Miles (1979), where it was shown that the difference is smaller than the error
incurred in the derivation of equation (4), and is due to reflected waves.
Our main concern in this paper is with the behaviour of an undular bore over a slope
in the presence of bottom friction, using the perturbed KdV equation (2), where we were
originally motivated by the possibility that the behaviour of a tsunami approaching the
shore might be modeled in this way. The undular bore solution to the unperturbed KdV
equation can be constructed using the well-known Gurevich-Pitaevskii (GP) (1974) approach
(see also Fornberg and Whitham 1978). In this approach, the undular bore is represented
as a modulated nonlinear periodic wave train. The main feature of this unsteady undular
bore is the presence of a solitary wave (which is the limiting wave form of the periodic
cnoidal wave) at its leading edge. The original initial-value problem for the KdV equation is
then replaced by a certain boundary-value problem for the associated modulation Whitham
equations. We note, however, that so far, the simplest, “(x/t)”-similarity solutions of the
modulation equations have been used for the modelling of undular bores in various contexts
(see Grimshaw and Smyth 1986, Smyth 1987 or Apel 2003 for instance). These solutions,
while effectively describing many features of undular bores, are degenerate and fail to cap-
ture, even qualitatively, some important effects associated with non-self-similar modulation
dynamics. In particular, in the classical GP solution for the resolution of an initial jump
in the unperturbed KdV equation, the amplitude of the lead solitary wave in the undular
bore is constant (twice the value of the initial jump). On the other hand, the modulation
solution for the undular bore evolving from a general monotonically decreasing initial profile
shows that the lead solitary wave amplitude in fact grows with time (Gurevich, Krylov and
Mazur 1989; Gurevich, Krylov and El 1992; Kamchatnov 2000). As we shall see, the very
possibility of such variations in the modulated solutions of the unperturbed KdV equation
has a very important fluid dynamics implication: in a general setting, the undular bore lead
solitary wave cannot be treated as an individual KdV solitary wave but rather represents a
part of the global nonlinear wave structure. In other words, while at every particular moment
of time the lead solitary wave has the spatial profile of the familiar KdV soliton, generally,
the temporal dependence of its amplitude cannot be obtained in the framework of single
solitary wave perturbation theory.
In the unperturbed KdV equation, the growth of the lead solitary wave amplitude is
caused by the spatial inhomogeneity of the initial data. Here, however, the presence of a
perturbation due to topography and/or friction serves as an alternative and/or additional
cause for variation of the lead solitary wave amplitude. Thus, in the present case, the
variation in the amplitude will have two components (which generally, of course, cannot be
separated because of the nonlinear nature of the problem); one is local, described by the
adiabatic perturbation theory for a single solitary wave, and the other one is nonlocal, which
in principle requires the study of the full modulation solution. Depending on the relative
values of the small parameters associated with the slope, friction and spatial non-uniformity
of the initial modulations, we can take into account only one of these components, or a
combination of them.
The structure of the paper is as follows. First, in Section 2, we reformulate the basic model
(1) as a constant-coefficient KdV equation perturbed by terms representing topography
and friction. Then we derive in Section 3 the associated perturbed Whitham modulation
equations using methods recently developed by Kamchatnov (2004). Next, in Section 4, this
Whitham system is integrated in the solitary-wave limit. Our purpose here is primarily to
obtain the equation of a multiple characteristic, which defines the leading edge of a shoaling
undular bore in the case when the modulations due to the combined action of the slope and
bottom friction are small compared to the existing spatial modulations due to non-uniformity
of the initial data. As a by-product of this integration, we reproduce and extend the known
results on the adiabatic variation of a single solitary wave (Miles 1983a,b). Then, in Section
5, we carry out an analogous study of a cnoidal wave, propagating over a gradual slope and
subject to friction, a case studied previously by Miles (1983 b) but under the restriction of
zero mean flow, which is removed here. Finally, in Section 6 we study effects of a gradual
slope and bottom friction on the front of an undular bore which represents a modulated
cnoidal wave transforming into a system of weakly interacting solitons near its leading edge.
2 Problem formulation
For the purpose of the present paper it is convenient to recast (2) into the standard KdV
equation form with constant coefficients, modified by certain perturbation terms. Thus we
introduce the new variables
A, T =
hdτ =
6g3/2
h(x)dx. (6)
so that UT + 6UUX + UXXX = R = F (T )U −G(T )|U |U, (7)
where F (T ) = −9hT
, G(T ) = 4CD
. (8)
In this form, the governing equation (7) has the structure of the integrable KdV equation
on the left-hand side, while the separate effects of the varying depth and the bottom friction
are represented by the two terms on the right-hand side. This structure enables us to use
the general theory developed in Kamchatnov (2004) for perturbed integrable systems.
For much of the subsequent discussion, it is useful to assume that h(x) = constant,
CD = 0 for x < 0 in the original equation (1), which corresponds to F (T ) = G(T ) = 0 for
T < 0 in (7). We shall also assume that A = 0 for x > 0 at t = 0, which corresponds to
U = 0 for X > 0 on X = τ(T ) (see (6)). Then we shall propose two types of initial-value
problem for (1), and correspondingly for (7).
(a) Let a solitary wave of a given amplitude a0 initially propagating over a flat bottom
without friction (i.e a soliton described by an unperturbed KdV equation), enter the variable
topography and bottom friction region at t = 0, x = 0 (Fig. 1 a).
(b) Let an undular bore of a given intensity propagate over a flat bottom without friction
(the corresponding solution of the unperturbed KdV equation will be discussed in Section
5). Let the lead solitary wave of this undular bore have the same amplitude a0 and enter
the variable topography and bottom friction region at t = 0, x = 0 (Fig. 1b).
In particular, we shall be interested in the comparison of the slow evolution of these
two, initially identical, solitary waves in the two different problems described above. The
expected essential difference in the evolution is due to the fact that the lead solitary wave
in the undular bore is generally not independent of the remaining part of the bore and can
exhibit features that cannot be captured by a local perturbation analysis. The well-known
example of such a behaviour, when a solitary wave is constrained by the condition of being
a part of a global nonlinear wave structure, is provided by the undular bore solution of the
KdV-Burgers (KdV-B) equation
ut + 6uux + uxxx = µuxx, µ ≪ 1 . (9)
( )h x
  a) 
( )h x
Figure 1: Isolated solitary wave (a) and undular bore (b) entering the variable topogra-
phy/bottom friction region.
Indeed, the undular bore solution of the KdV-B equation (9) is known to have a solitary
wave at its leading edge (see Johnson 1970; Gurevich & Pitaevskii 1987; Avilov, Krichever
& Novikov 1987) and this solitary wave: (a) is asymptotically close to a soliton solution
of the unperturbed KdV equation; and (b) has the amplitude, say a0, that is constant in
time. At the same time, it is clear that if one takes an isolated KdV soliton of the same
amplitude a0 as initial data for the KdV-Burgers equation it would damp with time due
to dissipation. The physical explanation of such a drastic difference in the behaviour of an
isolated soliton and a lead solitary wave in the undular bore for the same weakly dissipative
KdV-B equation is that the action of weak dissipation on an expanding undular bore is
twofold: on the one hand, the dissipation tends to decrease the amplitude of the wave
locally but, on the other hand, it “squeezes” the undular bore so that the interaction (i.e.
momentum exchange) between separate solitons within the bore becomes stronger than in
the absence of dissipation and this acts as the amplitude increasing factor. The additional
momentum is extracted from the upstream flow with a greater depth (see Benjamin and
Lighthill 1954). As a result, in the case of the KdV-B equation, an equilibrium non-zero
value for the lead solitary wave amplitude in the undular bore is established. Of course,
for other types of dissipation, a stationary value of the lead soliton amplitude would not
necessarily exist, but in general, due to the expected increase of the soliton interactions near
the leading edge, the amplitude of the lead soliton of the undular bore would decay slower
than that of an isolated soliton. Indeed, the presence here of variable topography as well
can result in an additional “nonlocal” amplitude growth.
While the problem (a) can be solved using traditional perturbation analysis for a single
solitary wave, which leads to an ordinary differential equation along the solitary wave path
(see Miles 1983a,b), the undular bore evolution problem (b) requires a more general approach
which can be developed on the basis of Whitham’s modulation theory leading to a system of
three nonlinear hyperbolic partial differential equations of the first order. Since the Whitham
method, being equivalent to a nonlinear multiple scale perturbation procedure, contains the
adiabatic theory of slow evolution of a single solitary wave as a particular (albeit singular)
limit, it is instructive for the purposes of this paper to treat both problems (a) and (b) using
the general Whitham theory.
3 Modulation equations
The original Whithammethod (Whitham 1965, 1974) was developed for conservative constant-
coefficient nonlinear dispersive equations and is based on the averaging of appropriate con-
servation laws of the original system over the period of a single-phase periodic travelling
wave solution. The resulting system of quasi-linear equations describes the slow evolution
of the modulations (i.e. of the mean value, the wavenumber, the amplitude etc.) of the pe-
riodic travelling wave. Here, that approach is extended to the perturbed KdV equation (6)
following the general approach of Kamchatnov (2004), which extends earlier results for cer-
tain specific cases (see Gurevich and Pitaevskii (1987, 1991), Avilov, Krichever and Novikov
(1987) and Myint and Grimshaw (1995) for instance).
We suppose that the evolution of the nonlinear wave is adiabatically slow, that is, the
wave can be locally represented as a solution of the corresponding unperturbed KdV equation
(i.e. (7) with zero on the right-hand side) with its parameters slowly varying with space and
time. The one-phase periodic solution of the KdV equation can be written in the form
U(X, T ) = λ3 − λ1 − λ2 − 2(λ3 − λ2)sn2(
λ3 − λ1 θ,m) (10)
where sn(y,m) is the Jacobi elliptic sine function, λ1 ≤ λ2 ≤ λ3 are parameters and the
phase variable θ and the modulus m are given by
θ = X − V T, V = −2(λ1 + λ2 + λ3) , (11)
λ3 − λ2
λ3 − λ1
, (12)
and L =
−P (µ)
2K(m)√
λ3 − λ1
, (13)
where K(m) is the complete elliptic integral of the first kind, L is the “wavelength” along the
X-axis (which is actually a retarded time rather than a true spatial co-ordinate). Here we
have used the representation of the basic ordinary differential equation for the KdV travelling
wave solution (10) in the form (see Kamchatnov (2000) for a general motivation behind this
representation)
−P (µ), (14)
where
µ = 1
(U + s1), s1 = λ1 + λ2 + λ3 (15)
P (µ) =
(µ− λi) = µ3 − s1µ2 + s2µ− s3, (16)
that is the solution (10) is parameterized by the zeroes λ1, λ2, λ3 of the polynomial P (µ).
In a modulated wave, the parameters λ1, λ2, λ3 are allowed to be slow functions of X and
T , and their evolution is governed by the Whitham equations. For the unperturbed KdV
equation, the evolution of the modulation parameters is due to a spatial non-uniformity
of the initial distributions for λj, j = 1, 2, 3 and the typical spatio-temporal scale of the
modulation variations is determined by the scale of the initial data.
In the case of the perturbed KdV equation (7), the evolution of the parameters λ1, λ2, λ3
is caused not only by their initial spatial non-uniformity, but also by the action of the
weak perturbation, so that, generally, at least two independent spatio-temporal scales for
the modulations can be involved. However, at this point we shall not introduce any scale
separation within the modulation theory and derive general perturbed Whitham equations
assuming that the typical values of F (T ) and G(T ) are O(∂λj/∂T, ∂λj/∂X) within the
modulation theory.
It is instructive to first introduce the Whitham equations for the perturbed KdV equation
(7) using the traditional approach of averaging the (perturbed) conservation laws. To this
end, we introduce the averaging over the period (13) of the cnoidal wave (10) by
〈F〉 =
Fdθ =
−P (µ)
. (17)
In particular,
〈U〉 = 2〈µ〉 − s1 = 2(λ3 − λ1)
+ λ1 − λ2 − λ3, (18)
〈U2〉 = 8[−s1
(λ3 − λ1)
s1λ1 +
(λ21 − λ2λ3)] + s21 , (19)
where E(m) is the complete elliptic integral of the second kind. Now, one represents the
KdV equation (7) in the form of the perturbed conservation laws
= Rj , j = 1, 2, 3 , Rj ≪ 1 , (20)
where Pj and Qj are the standard expressions for the conserved densities (Kruskal integrals)
and “fluxes” of the unperturbed KdV equation. Just as in the Whitham (1965) theory for
unperturbed dispersive systems, the number of conservation laws required is equal to the
number of free parameters in the travelling wave solution, which is three in the present case.
Next, one applies the averaging (17) to the system (20) to obtain (see Dubrovin and Novikov
1989)
∂〈Pj〉
∂〈Qj〉
= 〈Rj〉 , j = 1, 2, 3 . (21)
The system (21) describes slow evolution of the parameters λj in the cnoidal wave solution
(10).
Along with these derived perturbed conservative form of the Whitham equations, we
introduce the wave conservation law which is a general condition for the existence of slowly
modulated single-phase travelling wave solutions (10) (see for instance Whitham 1974) and
must be consistent with the modulation system (21). This conservation law has the form
= 0 , (22)
where k =
, ω = kV (23)
are the “wavenumber” and the “frequency” respectively (we have put quotation marks here
because the actual wavenumber and frequency related to the physical variables x, t are
different quantities from those in (23), but are related through the transformations (3, 6) ).
The wave conservation law (22) can be introduced instead of any of three inhomogeneous
averaged conservation laws comprising the Whitham system (21).
It is known that the Whitham system for the homogeneous constant-coefficient KdV
equation can be represented in diagonal (Riemann) form (Whitham 1965, 1974) by an ap-
propriate choice of the three parameters characterising the periodic travelling wave solution.
In fact, in our solution (7) the parameters λj have already been chosen so that they coincide
with the Riemann invariants of the unperturbed KdV modulation system. Introducing them
explicitly into the perturbed system (21) we obtain (see Kamchatnov 2004)
∂L/∂λi
〈(2λi − s1 − U)R〉
j 6=i(λi − λj)
, i = 1, 2, 3, (24)
where R is the perturbation term on the right-hand side of the KdV equation (7) and
vi = −2
∂L/∂λi
, i = 1, 2, 3, (25)
are the Whitham characteristic velocities corresponding to the unperturbed KdV equation.
It should be noted that the straightforward realisation of the above lucid general algo-
rithm for obtaining perturbed modulation system in diagonal form is quite a laborious task.
In fact, to derive system (24), the so-called finite-gap integration method incorporating the
integrable structure of the unperturbed KdV equation has been used. The modulation sys-
tem (24) in a more particular form corresponding to specific choices of the perturbation term
was obtained by Myint and Grimshaw (1995) using a multiple-scale perturbation expansion.
In that latter setting, the wave conservation law (22) is an inherent part of the construction,
while in the averaging approach used here, it can be obtained as a consequence of the system
(24).
To obtain an explicit representation of the Whitham equations for the present case of
equation (7), we must substitute the perturbation R from the right-hand side of (7) and
perform the integration (17) with U given by (10). From now on, we are going to consider
only the flows where U ≥ 0 so that the perturbation term assumes the form
R(U) = G(T )U − F (T )U2 . (26)
Substituting (26) into (24) we obtain, after some detailed calculations (see Appendix),
the perturbed Whitham system in the form
= ρi = Ci[F (T )Ai −G(T )Bi], i = 1, 2, 3 (27)
where C1 =
, C2 =
E − (1−m)K
, C3 =
; (28)
(5λ1 − λ2 − λ3)E +
(λ2 − λ1)K,
(5λ2 − λ1 − λ3)E − (λ2 − λ1)
λ3 − λ1
(5λ3 − λ1 − λ2)E −
(λ2 − λ1)
(−27λ21 − 7λ22 − 7λ23 + 2λ1λ2 + 2λ1λ3 + 22λ2λ3)E
(λ2 − λ1)(3λ1 + λ2 + λ3)K,
(−7λ21 − 27λ22 − 7λ23 + 2λ1λ2 + 22λ1λ3 + 2λ2λ3)E
λ2 − λ1
λ3 − λ1
(7λ21 + 15λ
2 + 11λ
3 − 6λ1λ2 − 18λ1λ3 + 6λ2λ3)K,
(−7λ21 − 7λ22 − 27λ23 + 22λ1λ2 + 2λ1λ3 + 2λ2λ3)E
(7λ21 + 11λ
2 + 15λ
3 − 18λ1λ2 − 6λ1λ3 + 6λ2λ3)K;
and the characteristic velocities are:
v1 = −2
4(λ3 − λ1)(1−m)K
v2 = −2
4(λ3 − λ2)(1−m)K
E − (1−m)K
v3 = −2
4(λ3 − λ2)K
The equations (27) – (31) provide a general setting for studying the nonlinear modulated
wave evolution over variable topography with bottom friction. In the absence of the pertur-
bation terms (i.e. when F (T ) ≡ 0, G(T ) ≡ 0), the system (27), (31) indeed coincides with
the original Whitham equations (Whitham 1965) for the integrable KdV dynamics. In that
case the variables λ1, λ2, λ3 become Riemann invariants, so in this general (perturbed) case
we shall call them Riemann variables.
It is important to study the structure of the perturbed Whitham equations (27) – (31) in
two limiting cases when the underlying cnoidal wave degenerates into (i) a small-amplitude
sinusoidal wave (linear limit), when λ2 = λ3 (m = 0), and (ii) into a solitary wave when
λ2 = λ1 (m = 1). Since in both these limits the oscillations do not contribute to the mean
flow (they are infinitely small in the linear limit and the distance between them becomes
infinitely long in the solitary wave limit) one should expect that in both cases one of the
Whitham equations will transform into the dispersionless limit of the original perturbed
KdV equation (7) i.e.
UT + 6UUX = F (T )U −G(T )U2, (32)
Indeed, using formulae (27) – (31) we obtain for m = 0:
λ2 = λ3 ,
− 6λ1
= λ1F + λ
+ (6λ1 − 12λ3)
= λ1F + λ
Similarly, for m = 1, one has
λ2 = λ1 ,
− (4λ1 + 2λ3)
(4λ1 − λ3)F +
(7λ23 − 24λ1λ3 + 32λ21)G,
− 6λ3
= λ3F + λ
We see that, in both cases, one of the Riemann variables (taken with inverted sign) coincides
with the solution of the dispersionless equation (32) (recall that in the derivation of the
Whitham equations we assumed U ≥ 0 everywhere), namely U = 〈U〉 = −λ1 when λ2 = λ3
(m = 0) and U = 〈U〉 = −λ3 when λ2 = λ1 (m = 1).
To conclude this section, we present expressions for the physical wave parameters such as
the surface elevation wave amplitude a, mean elevation 〈A〉 speed and wavenumber in terms
of the modulation solution λj(X, T ). Using (6) and (10) we obtain for the wave amplitude
(peak to trough) and the mean elevation
(λ3 − λ2) , 〈A〉 =
〈U〉 , (35)
where the dependence of 〈U〉 on λj(X, T ), j = 1, 2, 3 is given by (18) and X = X(x, t),
T = T (x, t) by (3, 6). In order to obtain the physical wavenumber κ and the frequency Ω
we first note that the phase function θ(X, T ) defined in (11) is replaced by a more general
expression defined so that k = θX and kV = −θT are the “wavenumber” and “frequency” in
the X −T coordinate system. Then we define the physical phase function Θ(x, t) = θ(X, T )
so that we get
κ = Θx , Ω = −Θt . (36)
It now follows that
(1− hV
) , Ω = k , and
1− hV/6g
. (37)
Note that the physical frequency is the “wavenumber” in the X − T coordinate system,
and that the physical phase speed is Ω/κ. Since the validity of the KdV model (1) requires
inter alia that the wave be right-going it follows from this expression that the modulation
solution remains valid only when hV < 6g. Of course, the validity of (1) also requires that
the amplitude remains small, and this would normally also ensure that V remains small.
4 Modulation solution in the solitary wave limit
In this section, we shall integrate the perturbed modulation system (27) along the multiple
characteristic corresponding to the merging of two Riemann variables λ2 and λ1. As we shall
see later, this characteristic specifies the motion of the leading edge of the shoaling undular
bore in the case when the perturbations due to variable topography and bottom friction can
be considered as small compared with the existing spatial modulations within the bore. At
the same time, as the case λ2 = λ1 ( i.e. m = 1) corresponds to the solitary wave limit in the
travelling wave solution (10), our results here are expected to be consistent with the results
from the traditional perturbation approach to the adiabatic variation of a solitary wave due
to topography and bottom friction (see Miles 1983a,b).
In the limit m → 1 the periodic solution (10) of the KdV equation goes over to its solitary
wave solution
U(X, T ) = U0sech
λ3 − λ1(X − VsT )]− λ3, (38)
where
U0 = 2(λ3 − λ1) , Vs = −(4λ1 + 2λ3) (39)
are the solitary wave amplitude and “velocity” respectively. The solution (38) depends on two
parameters λ1 and λ3 whose adiabatic slow evolution is governed by the reduced modulation
system (34). It is important that the second equation in this system is decoupled from
the first one. Hence, evolution of the pedestal −λ3 on which the solitary wave rides, can
be found from the solution of this dispersionless equation by the method of characteristics.
When λ3(X, T ) is known, evolution of the parameter λ1 can be found from the solution of
the first equation (34). As a result, we arrive at a complete description of adiabatic slow
evolution of the solitary wave parameters taking account of its interaction with the (given)
pedestal.
However, it is important to note here that while this description of the adiabatic evolution
of a solitary wave is complete as far as the solitary wave itself is concerned, it fails to describe
the evolution of a trailing shelf, which is needed to conserve total “mass” (see, for instance,
Johnson 1973b, Grimshaw 1979 or Grimshaw 2006). This trailing shelf has a very small
amplitude, but a very large length scale, and hence can carry the same order of “mass” as
the solitary wave. But note that the “momentum” of the trailing shelf is much smaller than
that of the solitary wave, whose adiabatic deformation is in fact governed to leading order by
conservation of “momentum”, or more precisely, by conservation of wave action flux (strictly
speaking, conservation only in the absence of friction).
The situation simplifies if the solitary wave propagates into a region of still water so
that there is no pedestal ahead of the wave, that is λ3 = 0 in X > τ(T ). But then, since
λ3 = 0 is an exact solution of the degenerate Whitham system (34) for this solitary wave
configuration, we can put λ3 = 0 both in the solitary wave solution,
U(X, T ) = −2λ1sech2[
−λ1 (X − VsT )], Vs = −4λ1, (40)
and in equation (34) for the parameter λ1 to obtain,
− 4λ1
Fλ1 +
Gλ21 , (41)
As we see, the solitary wave moves with the instant velocity
= −4λ1, (42)
and the parameter λ1 changes with T along the solitary wave trajectory according to the
ordinary differential equation
F (T )λ1 +
G(T )λ21. (43)
It can be shown that equation (43) is consistent with the equation for the solitary wave half-
width γ =
−λ1 obtained by the traditional perturbation approach (see Grimshaw (1979)
for instance).
Next, we re-write equation (43) in terms the original independent x-variable. For that,
we find from (6), that
dT = (h1/2/6g3/2)dx (44)
and F = −27
)3/2 dh
, G = 4CD
. (45)
Then substituting these expressions into (43) yields the equation
= −31
λ21 (46)
which can be easily integrated to give
−C0 −
, (47)
where C0 is an integration constant and x = 0 is a reference point where h = h0. According
to (40), U0 = −2λ1 is the amplitude of the soliton expressed in terms of variable U(X, T ).
Returning to the original surface displacement A(x, t) by means of (6) and denoting C0 =
4/(3ga0h0), we find the dependence of the surface elevation soliton amplitude a = (2h
2/3g)U0
on x in the form
a = a0
CDa0h0
, (48)
where a0 is the solitary wave amplitude at x = 0. We note that for CD = 0 this reduces to
the classical Boussinesq (1872) result a ∼ h−1, while for h = h0 it reduces to the well-known
algebraic decay law a ∼ 1/(1 + constant x) due to Chezy friction. Miles (1983a,b) obtained
this expression for a linear depth variation, although we note that there is a factor of 2
difference from (48) (in Miles (1983a,b) the factor 16CD/15 is 8CD/15). The trajectory of
the soliton can be now found from (42) and (47):
− t =
dx′h−5/2(x′)
CDa0h0
h3(x)
. (49)
This expression determines implicitly the dependence of x on t along the solitary wave path
and provides the desired equation for the multiple characteristic of the modulation system
for the case m = 1.
It is instructive to derive an explicit expression for the solitary wave speed by computing
the derivative dx/dt from (49), or more simply, directly from (37),
1− a/2h
. (50)
The formula (50) yields the restriction for the relative amplitude γ = a/h < 2 which is
clearly beyond the applicability of the KdV approximation (wave breaking occurs already
at γ = 0.7 (see Whitham 1974)). In the frictionless case (CD = 0) equation (48) gives
a/h = a0h0/h
2, and so the expression (50) for the speed must fail as h → 0. It is interesting
to note that this failure of the KdV model as h → 0 due to appearance of infinite (and
further negative!) solitary wave speeds is not apparent from the expression (48) for the
solitary wave amplitude, and the implication is that the model cannot be continued as
h → 0. Curiously this restriction of the KdV model seems never to have been noticed before
in spite of numerous works on this subject. Note that taking account of bottom friction
leads to a more complicated formula for the solitary wave speed as a function of h but the
qualitative result remains the same.
It is straightforward to show from (46) or (48) that
= −hx
CDa0h0
CDa0h0
. (51)
It follows immediately that for a wave advancing into increasing depth (hx > 0), the ampli-
tude decreases due to a combination of increasing depth and bottom friction. However, for
a wave advancing into decreasing depth, there is a tendency to increase the amplitude due
to the depth decrease, but to decrease the amplitude due to bottom friction. Hence whether
or not the amplitude increases is determined by which of these effects is larger, and this in
turn is determined by the slope, the depth, and the consolidated drag parameter CDa0/h0.
To illustrate, let us consider the bottom topography in the form
h(x) = h1−α0 (h0 − δx)α , α > 0 , (52)
which satisfies the condition h(0) = h0; the parameter δ characterizes the slope of the bot-
tom. In this case the formula (48) becomes
a = a0
δ(3α− 1)h0
)(3α−1)/α
if α 6= 1/3. One can see now that if α < 1/3, then the bottom friction term is relatively
unimportant due to the smallness of CD. Of course, for this case we again recover the
Boussinesq result, now slightly modified,
a ≈ a0
δ(1− 3α)h20
, 0 < α <
, h ≪ h0. (54)
Of course, this result is impractical in the KdV context as the KdV approximation used here
requires the ratio a/h to remain small.
If α > 1/3 now obtain asymptotic formula
15(3α− 1)δ
, h ≪ h0 , (55)
which is independent of the initial amplitude a0. This expression is consistent with the small-
amplitude KdV approximation as long as (3α− 1)δ/CD is order unity. Simple inspection of
(55) shows that the solitary wave amplitude
• increases as h → 0 if 1
< α < 1
• is constant as h → 0 if α = 1
• decreases as h → 0 if α > 1
Thus for 1/3 < α < 1/2, as for the case α < 1/3, the amplitude will increase as the depth
decreases, in spite of the presence of (sufficiently small) friction. However, for α > 1/3, even
although there is usually some initial growth in the amplitude, eventually even small bottom
friction will take effect and the amplitude decreases to zero. We note that if α = 1/3 then
the integral
h−3dx in (48) diverges logarithmically as h → 0, which just slightly modifies
the result (55) for h ≪ h0 and implies growth of the amplitude ∝ ln h/h as h → 0.
Of particular interest is the case α = 1. In that case formula (53) becomes
a = a0
. (56)
and a ≈ 15
h , h ≪ h0 (57)
These expressions (56, 57) were obtained by Miles (1983a,b) using wave energy conservation
(as above, note, however, that in Miles (1983a,b) the numerical coefficient is 15/4 rather
than 15/8). Thus, these results obtained from the Whitham theory are indeed consistent, at
the leading order, with the traditional perturbation approach for a slowly-varying solitary
wave.
5 Adiabatic deformation of a cnoidal wave
Next we consider a modulated cnoidal wave (10) in the special case when the modulation does
not depend on X . While this case is, strictly speaking, impractical as it assumes there is an
infinitely long wavetrain, it can nevertheless provide some useful insights into the qualitative
effects of gradual slope and friction on undular bores which are locally represented as cnoidal
waves. In the absence of friction, the slow dependence of the cnoidal wave parameters on T
was obtained by Ostrovsky & Pelinovsky (1970, 1975) and Miles (1979) (see also Grimshaw
2006), assuming that the surface displacement had a zero mean (i.e. 〈U〉 = 0), while,
the effects of friction were taken into account by Miles (1983b) using the same zero-mean
displacement assumption. However, this assumption is inconsistent with our aim to study
undular bores where the value of 〈U〉 is essentially nonzero. Hence, we need to develop a
more general theory enabling us to take into account variations in all the parameters in the
cnoidal wave. Such a general setting is provided by the modulation system (27).
Thus we consider the case when the Riemann variables in (27) do not depend on the
variable X so that the general Whitham equations become ordinary differential equations
in T , which can be conveniently reformulated in terms of the original spatial x-coordinate
using the relationship (44):
, i = 1, 2, 3, (58)
where all variables are defined above in section 3 (see 28, 29, 30). This system can be readily
solved numerically. But it is instructive, however, to first indicate some general properties
of the solution.
First, the solution to the system (58) must have the property of conservation of “wave-
length” L (or “wavenumber” k=2π/L)
2K(m)√
λ3 − λ1
= constant (59)
Indeed, the wave conservation law (22) in absence of X-dependence assumes the form
= 0 , (60)
which yields (59). Thus the system of three equations (58) can be reduced to two equations.
Next, applying Whitham averaging directly to (7) yields
P̃ , M = 〈U〉 , P̃ = 〈|U |U〉 . (61)
P − 4CD
Q̃ , P = 〈U2〉 , Q̃ = 〈|U |3〉 . (62)
The equation set (59), (61), (62) comprise a closed modulation system for three independent
modulation parameters, say M , P̃ and m. While this system is not as convenient for further
analysis as the system (27) in Riemann variables, it does not have a restriction U > 0 inherent
in (27), and allows for some straightforward inferences regarding the possible existence of
modulation solutions with zero mean elevation, that is with M = 0. Indeed, one can see
that the solution with the zero mean is actually not generally permissible when CD 6= 0, a
situation overlooked in Miles (1983b). Indeed, M = 0 immediately then implies that P̃ = 0
by (61). But then due to (59) we have all three modulation parameters fixed which is clearly
inconsistent with the remaining equation (62) (except for the trivial case M = 0, P = 0,
Q̃ = 0). However, in the absence of friction, when CD = 0, equation (61) uncouples and
permits a nontrivial solution with a zero mean. In general, when CD = 0 equations (61),
(62) can be easily integrated to give
d = Mh9/4 = constant; σ = Ph9/2 = constant. (63)
Then, using (18, 19, 59) one readily gets the formula for the variation of the modulus m,
and hence of all the other wave parameters, as a function of h
K2[2(2−m)EK − 3E2 − (1−m)K2] =
(σ − d2)L4
. (64)
200
 400
 600
 800
0.
2
0.
4
0.
6
0.
8
C  = 0
C  = 0.01
Figure 2: Dependence of the modulus m on the physical space coordinate x in the cases
without and with bottom friction in the X-independent modulation solution.
Formula (64) generalises to the case M 6= 0 (i.e. d 6= 0) the expressions of Ostrovsky &
Pelinovsky (1970, 1975), Miles (1979) and Grimshaw (2006) (note that in Grimshaw (2006)
the zero mean restriction in actually not necessary). We note here that, again with CD = 0,
equation (5) implies conservation of 〈B〉 and 〈B2〉 (the averaged wave action flux), which,
together with (59), also yield (64).
The physical frequency Ω and wavenumber κ in the modulated periodic wave under study
are given by the formula (37), and we recall here that k = 2π/L is constant (see (59)). As
discussed before at the end of Section 3 we must require that the phase speed stays positive as
the wave evolves, and here that requires that the physical wavenumber κ > 0. Since a/h (and
hence hV/6g) is supposed to be small within the range of applicability of the KdV equation
(2) the expression (37) implies the behaviour κ ≃ Ω/
gh which of course agrees with the
well known result for linear waves on a sloping beach (see Johnson 1997 for instance). This
effect will be slightly attenuated for the nonlinear cnoidal wave, since V h/6g > 0, but the
overall effect will be a “squeezing” of the cnoidal wave, a result important for our further
study of undular bores. Next we study numerically the combined effect of slope and friction
on a cnoidal wave.
As we have shown, in the presence of Chezy friction M 6= 0, and we have also assumed
that U > 0, which is necessary when we come to study undular bores. Now we use the
stationary modulation system (58) in Riemann variables, which was derived using this as-
sumption. We solve the coupled ordinary differential equation system (58) for the case of a
linear slope
h(x) = h0 − δx (65)
with h0 = 10, δ = 0.01, and with the initial conditions
λ1 = −0.441, λ2 = 0.147, λ3 = 0.294 at x = 0, (66)
which corresponds to a nearly harmonic wave with m = 0.2, a/h0 = 0.2, 〈A〉/h0 ≈ 0.3
at x = 0 (see (35)). Also we note that for the chosen parameters we have V = 0, so at
x = 0 we have κ = Ω/
gh0 as in linear theory. It is instructive to compare solutions with
(CD = 0.01) and without (CD = 0) friction. In Fig. 2 the dependence of the modulus m
100 200 300 400 500
C  = 0
C  = 0.01
100 200 300 400 500
h  <A>
1/4 C  = 0
C  = 0.01
Figure 3: Left: Dependence of the mean value 〈A〉 in theX-independent modulation solution
on the physical space coordinate x without (dashed line) and with (solid line) bottom friction;
Right: Same but multiplied by the Green’s law factor, h1/4
100
 200
 300
 400
 500
1.
4
1.
6
1.
8
2.
2
2.
4
C   = 0
C   = 0.01
Figure 4: Dependence of the surface elevation amplitude a on the space coordinate x. Dashed
line corresponds to the frictionless case and solid line to the case with bottom friction.
on x is shown for both cases. We see that for the frictionless case m → 1 with decrease
of depth, i.e. the wave crests assume the shape of solitary waves when one approaches the
shoreline. When CD 6= 0 the modulus also grows with decrease of depth but never reaches
unity. The dependence on x of the mean surface elevation 〈A〉 for the cases without and
with friction is shown in Fig. 3. We have checked that the “wavelength” L (59) is constant
for both solutions. Also, one can see from Fig. 3 (right) that the value h1/4〈A〉 ∝ d is indeed
conserved in the frictionless case but is not constant if friction is present (the same holds
true for the value h1/2〈A2〉 ∝ σ but we do not present the graph here). Finally, in Fig. 4 the
dependence of the physical elevation wave amplitude a on the spatial coordinate x is shown.
One can see that the amplitude adiabatically grows with distance in the frictionless case
due to the effect of the slope (without friction) but, not unexpectedly, gradually decreases
in the case when bottom friction is present, where the decrease for these parameter settings
is comparable in magnitude to the effect of the slope. In both cases the main qualitative
changes occur in the wave shape and the wavelength.
Overall, we can infer from these results that the main local effect of a slope and bottom
friction on a cnoidal wave, along with the adiabatic amplitude variations, is twofold: a wave
with a m < 1 at x = 0 tends to transform into a sequence of solitary waves as x decreases,
and at the same time the distance between subsequent wave crests tends to decrease. This
is in sharp contrast with the behaviour of modulated cnoidal waves in problems described
by the unperturbed KdV equation, where growth of the modulus m is accompanied by an
increase of the distance between the wave crests. Generally, in the study of behaviour of
unsteady undular bores in the presence of a slope and bottom friction we will have to deal
with the combination of these two opposite tendencies.
6 Undular bore propagation over variable topography
with bottom friction
6.1 Gurevich-Pitaevskii problem for flat-bottom zero-friction case
We now turn to the problem (b) outlined in Section 2. We study the evolution of an undular
bore developing from an initial surface elevation jump ∆ > 0, located at some point x0 < 0.
As discussed below, the undular bore will expand with time so that at some t = t0 its lead
solitary wave enters the gradual slope region, which begins at x = 0 (see Fig. 1b). We assume
that for x < 0 one has h = h0 = constant and CD ≡ 0. We shall first present a formulation
of the Gurevich-Pitaevskii problem for the perturbation-free KdV equation and reproduce
the well-known similarity modulation solution describing the evolution of the undular bore
until the moment it enters the slope. We emphasize that, although this formulation and,
especially, this similarity solution are known very well and have been used by many authors,
some of the inferences important for the present application to fluid dynamics have not been
widely appreciated, as far as we can discern. Pertinent to our main objective in this paper,
we undertake a detailed study of the characteristics of the Whitham modulation system in
the vicinity of the leading edge of the undular bore solution, and show that the boundary con-
ditions of Gurevich-Pitaevskii type permit only two possible characteristics configurations,
implying two qualitatively different types of the leading solitary wave behaviour. Next, we
shall show how this Gurevich-Pitaevskii formulation of the problem applies to the perturbed
modulation system in the form (27) and finally we will study the effects of the perturbation
on the modulations in the vicinity of the leading edge of the undular bore.
In the case of a flat, frictionless bottom the original equation (1) becomes the constant-
coefficient KdV equation which can be cast into the standard form
ηζ + 6ηηξ + ηξξξ = 0 (67)
by introducing the new variables
A , ξ =
(x+ x0 −
gh0t) , ζ =
t , (68)
where x0 < 0 is an arbitrary constant. In the Gurevich-Pitaevskii (GP) approach, one
considers a large-scale initial disturbance η(ξ, 0) = f(ξ), in the form of a decreasing profile,
f ′(ξ) < 0 (e.g. a smooth step: f(ξ) → 0 as ξ → +∞; f(ξ) → η0 > 0 as ξ → −∞), whose
initial evolution until some critical (breaking) time ζb can be described by the dispersionless
limit of the KdV equation, i.e. by the Hopf equation,
ζ < ζb : η ≈ r(ξ, ζ), rζ + 6rrξ = 0 , r(ξ, 0) = f(ξ) . (69)
The evolution (69) leads to wave-breaking of the r(ξ)-profile at some ζ = ζb, with the
consequence that the dispersive term in the KdV equation then comes into play, and an
undular bore forms, which can be locally represented as a single-phase travelling wave. This
travelling wave is modulated in such a way that it acquires the form of a solitary wave at the
leading edge ξ = ξ+(ζ) and gradually degenerates, via the nonlinear cnoidal-wave regime, to
a linear wave packet at the trailing edge ξ = ξ−(ζ). It is important that this undular bore
is essentially unsteady, i.e. the region ξ−(ζ) < ξ < ξ+(ζ) expands with time ζ .
The single-phase travelling wave solution of the KdV equation (67) has the form (cf.
(10))
η(ξ, ζ) = r3 − r1 − r2 − 2(r3 − r2)sn2(
r3 − r1θ,m) (70)
θ = ξ + 2(r1 + r2 + r3)ζ , m =
r3 − r2
r3 − r1
. (71)
The parameters r1 ≤ r2 ≤ r3 ≤ 0 in the undular bore are slowly varying functions of ξ, ζ ,
whose evolution is governed by the Whitham equations
+ vj(r1, r2, r3)
= 0 , j = 1, 2, 3. (72)
The characteristic velocities in (72) are given by (31). We stress that, although analytical
expressions (70) and (10) (as well as (72) and the homogeneous version of (27)) are identical,
they are written for completely different sets of variables, both dependent and independent.
The Riemann invariants rj(ξ, ζ) are subject to special matching conditions at the free
boundaries, ξ = ξ±(ζ) defined by the conditions m = 0 (trailing edge) and m = 1 (leading
edge), formulated in Gurevich and Pitaevskii (1974) (see also Kamchatnov (2000) or El
(2005) for a detailed description).
At the trailing (harmonic) edge, where the wave amplitude a = 2(r3 − r2) vanishes and
m = 0, one has
ξ = ξ−(ζ) : r2 = r3 , −r1 = r . (73)
At the leading (soliton) edge, where m = 1 one has
ξ = ξ+(ζ) : r2 = r1 , −r3 = r . (74)
In both (73) and (74), r(ξ, ζ) is the solution of the Hopf equation (69).
The curves ξ = ξ±(ζ) are defined for the solution of the GP problem (72), (73), (74) by
the ordinary differential equations
= v−(ξ−, ζ) ,
= v+(ξ+, ζ) , (75)
where v± are calculated as the values of double characteristic velocities of the modulation
system at the undular bore edges,
v− = v2(r1, r3, r3)|ξ=ξ−(ζ) = v3(r1, r3, r3)|ξ=ξ−(ζ), (76)
v+ = v2(r1, r1, r3)|ξ=ξ+(ζ) = v1(r1, r1, r3)|ξ=ξ+(ζ) (77)
These equations (75) essentially represent kinematic boundary conditions for the undular
bore (see El 2005). Indeed, the double characteristic velocity v2(r1, r3, r3) = v3(r1, r3, r3) can
be shown to coincide with the linear group velocity of the small-amplitude KdV wavepacket
while the double characteristic velocity v2(r1, r1, r3) = v1(r1, r1, r3) is the soliton speed.
One might infer from this GP formulation of the problem that, since the leading edge of
the undular bore specified by (75), (77) is a characteristic of the modulation system, then
the value of the double Riemann invariant r+ ≡ r2 = r1 is constant. Then, on considering an
undular bore propagating into still water, where r = 0, one would obtain from the matching
condition (74) at the leading edge that r3|ξ=ξ+ = 0 and thus, the amplitude of the lead solitary
wave a+ = 2(r3−r1)|ξ=ξ+ = −r+ would always be constant as well. However, this contradicts
the general physical reasoning that the amplitude of the lead solitary wave should be allowed
to change in the case of general initial data. The apparent contradiction is resolved by noting
that the leading edge specified by (75), (77) can be an envelope of the characteristic family,
i.e. a caustic, rather than necessarily a regular characteristic, and hence there is no necessity
for the double Riemann invariant r+ to be constant along the curve ξ = ξ+(ζ) in general case.
On the other hand, since the leading edge is defined by the condition m = 1, the wave form
at the leading edge will coincide with the spatial profile of the standard KdV soliton. Thus
we arrive at the conclusion that, in general, the amplitude of the leading KdV solitary wave
will vary, even in the absence of the perturbation terms. Of course, in the unperturbed KdV
equation, such varying solitary waves cannot not exist on their own, and require the presence
of the rest of the undular bore. We also stress that these variations of the leading solitary
wave in the undular bore, as described here, have a completely different physical nature to
the variations of the parameters of an individual solitary wave due to small perturbations as
described in Section 4. They are caused by nonlinear wave interactions within the undular
bore rather than by a local adiabatic response of the solitary wave to a perturbation induced
by topography and friction. Importantly for our study, however, it will transpire that the
action of these same perturbation terms on the undular bore can lead to both a local and a
nonlocal response of the leading solitary wave.
6.2 Undular bore developing from an initial jump
Next we consider the simplest solution of the modulation system, which describes an undular
bore developing from an initial discontinuity placed at the point x = −x0. In (η; ξ, ζ) -
variables we have the initial conditions
η(ξ, 0) = ∆ for ξ < 0 ; η(ξ, 0) = 0 for ξ > 0 , (78)
where ∆ > 0 is a constant. Then, on using (69), the initial conditions (78) are readily
translated into the free-boundary matching conditions (73), (74) for the Riemann invariants.
Because of the absence of a length scale in this problem, the corresponding solution of the
modulation system must depend on the self-similar variable τ = ξ/ζ alone, which reduces
the modulation system to the ordinary differential equations
(vi − τ)
= 0 , i = 1, 2, 3. (79)
-4
0
 -3
0
 -2
0
 -1
0
 10
 20
-0.8
-0.6
-0.4
-0.2
-30 -20 -10 10
η(ξ, ζ = 5)
Figure 5: Left: Riemann invariants behaviour in the similarity modulation solution for the
flat-bottom zero-friction case ; Right: corresponding undular bore profile η(ξ).
The boundary conditions for (79) follow from the matching conditions (73), (74) using the
initial condition (78):
τ = τ− : r2 = r3 , r1 = −∆
τ = τ+ : r2 = r1 , r3 = 0 .
where τ± are self-similar coordinates (speeds) of the leading and trailing edges, ξ± = τ±ζ .
Taking into account the inequality r1 ≤ r2 ≤ r3 one obtains the well-known modulation
solution of Gurevich and Pitaevskii (1974) (see also Fornberg and Whitham 1978) in the
r1 = −∆ , r3 = 0 , r2 = −m∆ , (81)
= v2(−∆,−m∆, 0) = 2∆[(1 +m)−
2m(1−m)K(m)
E(m)− (1−m)K(m)
] . (82)
This modulation solution (81), (82) (see Fig. 5a) represents the replacement, due to averag-
ing over the oscillations, of the unphysical formal three-valued solution of the dispersionless
KdV equation (i.e. of the Hopf equation) which would have taken place in the absence of
the dispersive regularisation by the undular bore. We see that (82) describes an expansion
fan in the characteristic (ξ, ζ)-plane and thus is a global solution. Substituting (81), (82)
into the travelling wave solution (70) one obtains the asymptotic wave form of the undular
bore (see Fig. 5b), which then can be readily represented in terms of the original physical
variables using the relationships (68).
The equations of the trailing and leading edges of the undular bore are determined from
(82) by putting m = 0 and m = 1 respectively
= τ− = v2(−∆, 0, 0) = −6∆ ,
= τ+ = v2(−∆,−∆, 0) = 4∆ . (83)
The leading solitary wave amplitude is η0 = 2(r3−r1) = 2∆, which is exactly twice the height
of the initial jump. This corresponds to the amplitude of the surface elevation a = 3h0∆ (see
(68)). Note that, to get the leading solitary wave of the same initial amplitude a0 as for the
separate solitary wave considered in Section 4, one should use the jump value ∆0 = a0/3h0,
which of course is just 2∆̃, where ∆̃ = 3h0∆/2 is the initial discontinuity in the surface
elevation.
6.3 Structure of the undular bore front
We are especially interested in the behaviour of the modulation solution (81), (82) in the
vicinity of the leading edge ξ = ξ+(ζ). This behaviour is essentially determined by the
manner in which the pair of characteristics corresponding to the velocities v2 and v1 merge
into a multiple eigenvalue v+ of the modulation system at ξ = ξ+(ζ).
First, one can readily infer from the modulation solution (81), (82) that the phase velocity
c = −2(r1 + r2 + r3) = 2∆(1 +m) > v2(−∆,−m, 0) for m < 1 and c = v2 for m = 1. Thus,
any individual wave crest generated at the trailing edge of the undular bore moves towards
the leading edge, i.e. for any crest m → 1 as ζ → ∞. Thus, for any particular wave crest,
except for the very first one, the solitary wave ‘status’ is achieved only asymptotically as
ζ → ∞.
Without loss of generality we assume in this section that ∆ = 1 in (81), (82). First, as
we have already mentioned, the characteristic family Γ2 : dξ/dζ = v2 is an expansion fan in
the ξ, ζ - plane,
Γ2 : ξ = C2ζ , (84)
parameterised by a constant C2, −6 ≤ C2 ≤ 4 . Next, in (82) we make an asymptotic
expansion of v2(−1,−m, 0) for small (1−m) ≪ 1, to get
2(1−m) ln(16/(1−m)) ≃ τ+ − ξ/ζ (85)
or, with logarithmic accuracy,
(τ+ − ξ/ζ) ≪ 1 : 1−m ≃
τ+ − ξ/ζ
2 ln[1/(τ+ − ξ/ζ)]
. (86)
Next, expanding v1(−1,−m, 0) for (1 − m) ≪ 1 and using (86) we get the asymptotic
equation for the characteristics family Γ1,
= v1 = τ
+ + (τ+ − ξ/ζ) +O(1−m) , (87)
which is readily integrated to leading order to give
Γ1 : ξ ≃ τ+ζ −
, (88)
where C1 ≥ 0 is an arbitrary constant ‘labeling’ the characteristics; C1 = 0 corresponds to
the leading edge of the undular bore. This asymptotic formula (88) is valid as long as ζ ≫ 1.
The behaviour of the characteristics belonging to the families Γ1 and Γ2 near the leading
edge is shown in Fig. 6a.
Next, expanding the equation for the third characteristic family, Γ3: dξ/dζ = v3(−1,−m, 0)
for (1−m) ≪ 1, we get on using (86)
τ+ − ξ/ζ
ln(1/(τ+ − ξ/ζ))
+O(τ+ − ξ/ζ) . (89)
Figure 6: Characteristics behaviour for the similarity modulation solution near the leading
edge ξ+(ζ): (a) families Γ1: dξ/dζ = v1 and Γ2 : ξ = C2ζ , (b) family Γ3: dξ/dζ = v3.
Integrating (89) we obtain to first order
Γ3 : ξ ≃ C3 − g(ζ) , (90)
where g(ζ) =
τ+ζ − C3
ln |τ+ζ − C3| − ln ζ
dζ , g(C3/τ
+) = 0 , (91)
C3 being an arbitrary constant. The asymptotic formula (90) is valid as long as g(ζ)/C3 ≪
1. Since the characteristics Γ3 intersect the leading edge ξ = τ
+ζ we must indicate their
behaviour outside the undular bore. It follows from the matching condition (74) and the
limiting structure (34) of the characteristic velocities of the Whitham system, that the
characteristics from the family Γ3 match with the Hopf equation characteristics dξ/dζ = 6r
carrying the value of the Riemann invariant r = 0 corresponding to still water upstream the
undular bore. Therefore, the sought external characteristics are simply vertical lines ξ = C3.
The qualitative behaviour of the characteristics from the family Γ3 is shown in Fig. 6b.
It is clear from the asymptotic behaviour of the characteristics that the edge characteristic
ξ = τ+ζ corresponding to the motion of the leading solitary wave intersects only with
characteristics of the family Γ3 carrying the Riemann invariant value r3 = 0 into the undular
bore domain. Since, according to the matching condition (80), r3 ≡ 0 everywhere along the
edge characteristic one can infer that the leading solitary wave motion is completely specified
by its amplitude at ζ = 0. Indeed, in this case, the leading edge represents a genuine multiple
characteristic of the modulation system, along which the Riemann invariant r+ = r2 = r1 is
a constant. Given the constant value of r1 = −1 for the solution (82), one infers that the
amplitude of the lead soliton of the self-similar undular bore, η0 = 2(r3 − r+) = 2 is also a
constant value. Thus, in the undular bore evolving from an initial jump, the leading solitary
wave represents an independent soliton of the KdV equation. Of course, this fact follows
directly from the modulation solution (82) but now we have established its meaning in the
context of the characteristics, which will play an important role below.
Next we discuss the structure of the undular bore front in the case when the initial
profile η(ξ, 0) is not a simple jump discontinuity, and instead has the form of a monotonically
decreasing function, for instance, (−ξ)1/2 when ξ ≤ 0 and η(ξ, 0) = 0 for ξ > 0. In that case,
the modulation solution for the undular bore no longer possesses x/t-similarity as in the
Figure 7: a) Leading edge ξ+(ζ) of non-self-similar undular bore as an envelope of pairwise
merging characteristics from the families dξ/dζ = v1 and dξ/dζ = v2; b) behaviour of the
Riemann invariants in non-self-similar modulation solution with r3 ≡ 0.
jump resolution case and, as a result, the speed (and therefore, the amplitude) of the lead
solitary wave is not constant. For instance, for the afore-mentioned square-root initial profile
the amplitude of the lead solitary wave grows as ζ2 (see Gurevich, Krylov and Mazur 1989,
or Kamchatnov 2000). Clearly, such an amplitude variation is impossible if the leading edge
ξ+(ζ) was a regular characteristic carrying a constant value of the Riemann invariant r+. As
discussed above, however, the GP matching conditions (73) -(77) admit another possibility;
the leading edge curve is the envelope of the characteristic families Γ1: dξ/dζ = v1 and Γ2:
dξ/dζ = v2 merging when m = 1. This configuration is shown in Fig. 7a. In this case, the
behaviour of the modulus m in the vicinity of the leading edge is given by the asymptotic
formula found in Gurevich & Pitaevskii (1974):
(1−m)2
(r+)2
(ξ+ − ξ) (92)
where the function r+(ζ) 6= constant is assumed to be known. Another specific feature of
this (general) configuration is that dr1,2/dξ → ±∞ as ξ → ξ+ (see Fig. 7b - also found in
Gurevich & Pitaevskii 1974, see also Kamchatnov 2000), which is in drastic contrast with
similarity solution (see Fig. 6a). This particular difference was discussed in relation with
undular bores in the KdV-Burgers equation in Gurevich and Pitaevskii (1987).
In summary, we see from (92) that the structure of the modulation solution in the vicin-
ity of the leading edge of an undular bore defined as a characteristic envelope is qualitatively
different compared to that for the similarity case (see (85)). The more general (but qual-
itatively similar to (92)) asymptotic formula which takes into account small perturbations
due to a variable topography and bottom friction will be derived later. At the moment,
it is important for us that in this configuration, when the leading edge is a characteristic
envelope rather than just a characteristic, the value r+, and thus, the leading solitary wave
amplitude are allowed to vary.
The analysis of the corresponding modulation solution in Gurevich, Krylov and Mazur
(1989) showed that, while in the case of an initial jump the wave crests generated at the
trailing edge reach the leading edge (and therefore, transform into solitary waves) only
asymptotically as t → ∞, for the more general case of decreasing initial data each wave
crest generated at the trailing edge reaches the leading edge in finite time and replaces
(overtakes) the existing leading solitary wave. This process is manifested as a continuous
amplitude growth of the (apparent) leading solitary wave. As in classical soliton theory,
an alternative explanation of the leading solitary wave amplitude growth can be made in
terms of the momentum exchange between the “instantaneous” leading solitary wave and
solitary waves of greater amplitude coming from the left. Indeed, as the rigorous analysis of
Lax, Levermore and Venakides showed (see Lax, Levermore and Venakides (1994) and the
references therein), the whole modulated structure of the undular bore can be asymptotically
described in terms of the interactions of a large number of KdV solitons initially ‘packed’
into a non-oscillating large-scale initial profile.
This latter interpretation is especially instructive for our purposes. Our point is that
the specific cause of the enhanced soliton interactions resulting in amplitude growth at the
leading edge is not essential; it can be large-scale spatial variations of the initial profile as
just described, but it could also equally well be an effect of small perturbations in the KdV
equation itself. Indeed, in the weakly perturbed KdV equation, the local wave structure
of the undular bore must be described to leading order by the periodic solution (70) of
the unperturbed KdV equation, so if one assumes the GP boundary conditions analogous to
(73) – (77) for the perturbed modulation system (27), one invariably will have to deal with
one of the two possible types of the characteristics behaviour (shown in Figs. 7a and 8a)
in the vicinity of the leading edge of the undular bore, because this qualitative behaviour
is determined only by the structure of the GP boundary conditions and by the associated
asymptotic structure of the characteristic velocities of the Whitham system for (1−m) ≪ 1,
which are the same for both unperturbed and perturbed modulation systems. Next, we will
show that, by using the knowledge of this qualitative behaviour of the characteristics, one
is able to construct the asymptotic modulation solution for the undular bore front in the
presence of variable topography and bottom friction even if the full solution of the perturbed
modulation system is not available.
6.4 Gurevich-Pitaevskii problem for perturbed modulation sys-
We investigate now how the GP matching problem applies to the perturbed modulation
system (27). As in the original GP problem, we postulate the natural physical requirement
that the mean value 〈U〉 is continuous across the undular bore edges, which represent free
boundaries and are defined by the conditions m = 0 (trailing edge X = X−(T )) and m = 1
(leading edge X = X+(T )). Also, we consider propagation of the undular bore into still
water, hence 〈U〉|X=X+(T ) = 0. Now, using the explicit expression (18) for 〈U〉 in terms of
complete elliptic integrals and calculating its limits as m → 0 and m → 1 one has
X = X−(T ) : λ2 = λ3 , 〈U〉 = −λ1 = u ,
X = X+(T ) : λ2 = λ1 , 〈U〉 = −λ3 = 0 ,
where u(X, T ) is solution of the dispersionless perturbed KdV equation (7), i.e.
uT + 6uuX = F (T )u−G(T )u2, (94)
with the boundary conditions
∆0 if τ < τ0; u
= 0 if τ > τ0 , (95)
where τ0 = −x0/
gh0. The boundary conditions (95) correspond to a discontinuous initial
surface elevation A(x, t) at x = −x0, obtained by using transformations (3) and (6) where
one sets t = 0. As earlier, ∆0 = a0/(3h0) is the value of the discontinuity in A, chosen in
such a way that the amplitude of the lead solitary wave in the undular bore was exactly a0
in the flat-bottom zero-friction region (see Section 6.2).
This free-boundary matching problem is then complemented by the kinematic conditions
explicitly defining the boundaries X = X±(T ). These are formulated using the multiple
characteristic directions of the perturbed modulation system (27) in the limits as m → 0
and m → 1 (cf. (75) - (77)),
= V −(X−, T ) ,
= V +(X+, T ) , (96)
where V − = v2(u, λ
−, λ−) = v3(u, λ
−, λ−), (97)
V + = v2(λ
+, λ+, 0) = v1(λ
+, λ+, 0) , (98)
and λ− = λ2(X
−, T ) = λ3(X
−, T ) , λ+ = λ2(X
+, T ) = λ1(X
+, T ). (99)
Thus, for the perturbed KdV equation the leading and trailing edges of the undular bore are
defined mathematically in the same way as for the unperturbed one, albeit for a different
set of variables.
6.5 Deformation of the undular bore front due to variable topog-
raphy and bottom friction
Finally we study the effects of gradual slope and bottom friction on the leading front of the
self-similar expanding undular bore described in Sections 6.2, 6.3. The result will essentially
depend on the relative values of the small parameters appearing in the problem. We note
that in general there are three distinct relevant small parameters,
≪ 1 , δ = max(hx) ≪ 1, CD ≪ 1 (100)
The first small parameter is determined by the ratio of the equilibrium depth in the flat
bottom region, to the distance from the beginning of the slope region to the location of the
initial jump discontinuity in the surface displacement. This measures the typical relative
spatial variations of the modulation parameters in the undular bore when it reaches the
beginning of the slope. The second and third parameters are contained in the KdV equation
(1) itself and measure the values of the slope and bottom friction respectively. In terms of
the transformed variables appearing in (7), |F (T )| ∼ δ, |G(T )| ∼ CD (see (8)). Generally
we assume δ ∼ CD (the possible orderings δ ≪ CD or CD ≪ δ can be then considered as
particular cases).
To obtain a quantitative description of the vicinity of the leading edge of the undular
bore we perform an expansion of the Whitham modulation system (27) for (1 − m) ≪ 1.
We first introduce the substitutions
λi(X, T ) = λ
+(T ) + li(X̃, T ) , vi = V
+ + v′i , ρi = ρ
+ + ρ′i, i = 1, 2. (101)
where X̃ = X+ −X , V + = −4λ+ , ρ+ =
F (T )λ+ +
G(T )(λ+)2. (102)
Since λ2 ≥ λ1, v2 ≥ v1 one always has l2 ≥ l1, v′2 ≥ v′1. Assuming X̃/X+ ≪ 1 ⇔ 1−m ≪ 1
and using that λ3 = 0 to leading order in the vicinity of the leading edge (see the matching
condition (93)), we have from asymptotic expansions of (28) – (31) as (1−m) ≪ 1
v′1 = M1(l2 − l1) ≡ −2
ln(16/(1−m))
1 + 1
(1−m) ln(16/(1−m))
(l2 − l1),
v′2 = M2(l2 − l1) ≡ −2
1− ln(16/(1−m))
(1−m) ln(16/(1−m))
(l2 − l1),
(103)
ρ′1 = N1(l2 − l1) ≡
1 + ln
l2 − l1
−16λ+
2λ+ ln
l2 − l1
−16λ+
− 3λ+
(l2 − l1)
ρ′2 = N2(l2 − l1) ≡
5 + ln
l2 − l1
−16λ+
2λ+ ln
l2 − l1
−16λ+
+ 13λ+
(l2 − l1).
(104)
Naturally, v′i and ρ
i vanish when l2 = l1. Now, substituting (101), (102) into the modulation
system (27) we obtain
− (V + + v′i)
= ρ+ + ρ′i, i = 1, 2. (105)
On using the kinematic condition (96) at the leading edge, this reduces to
− v′i
= ρ+ + ρ′i, i = 1, 2. (106)
There are two qualitatively different cases to consider:
(i) limX̃→0 |dli/dX̃| < ∞, i = 1, 2 (Fig. 8a)
(ii) limX̃→0 |dli/dX̃| = ∞, i = 1, 2 (Fig. 8b)
The case (i) implies that to leading order (106) reduces to
= ρ+ , (107)
which, together with the kinematic condition dX+/dT = −4λ+, defines the leading edge
curve X+(T ). One can observe that this system coincides with (43), (42) defining the
Figure 8: Riemann variables behaviour in the vicinity of the leading edge of the undular
bore propagating over gradual slope with bottom friction (a) Adiabatic variations of the
similarity GP regime, δ ≪ ǫ, CD ≪ ǫ; (b) General case, δ ∼ CD ∼ ǫ.
motion of a separate solitary wave over a gradual slope with bottom friction. Its integral
expressed in terms of original physical x, t-variables is given by (49). Therefore, in the case
(i) the lead solitary wave in the undular bore to leading order is not restrained by interactions
with the remaining part of the bore and behaves as a separate solitary wave. Physically this
case corresponds to adiabatic deformation of the similarity modulation solution (81), (82)
and implies the following small parameter ordering : δ ≪ ǫ, CD ≪ ǫ.
Next, we study the structure of this weakly perturbed similarity modulation solution in
the vicinity of the leading edge. The next leading order of the system (106) yields
− v′i
= ρ′i, i = 1, 2, (108)
that is
= −N1
= −N2
. (109)
Subtraction of one equation (109) from another with account of the relationship l2 − l1 ∼=
−λ+(1−m) leads consistently to leading order to the differential equation for 1−m
∂(1 −m)
F (T )
16G(T )
, (110)
This equation should be solved with the initial condition
1−m = 0 at X̃ = 0 . (111)
Elementary integration gives with the accuracy O(1−m) (cf. (85))
(1−m) ln 16
F (T )− 16
λ+G(T )
X+ −X
. (112)
This formula determines the dependence of the modulus m on T and X (as long as 1−m ≪
Now, we make use of the solution λ+ of equation (107) given by (47) with C0 =
4/(3ga0h0) (see (48)). Under supposition that the integral
h−3dx diverges as h → 0,
so that the turbulent bottom friction plays an essential role in the undular bore front be-
haviour (see Section 4 for a similar approximation for an isolated solitary wave), we obtain
for h ≪ h0
(1−m) ln
2 + 3h2
(X+ −X). (113)
At last, if the bottom topography is approximated by the dependence (52), we get with the
same accuracy
(1−m) ln 16
(3α− 1)δ
(X+ −X) , (114)
where α > 1/3. The second term in square brackets tends to zero as h → 0. However, the
region where it can be neglected may be very narrow because of smallness of the parameter
δ. We recall that in this formula X+ is given by (49) and X is defined by (3) in terms of
the original physical independent variables x and t.
Summarising, if the conditions δ, CD ≪ ǫ are satisfied, the lead solitary wave of the
undular bore behaves as an individual (noninteracting) solitary wave adiabatically varying
under small perturbation due to variable topography and bottom friction. The modulation
solution in the vicinity of the leading edge also varies adiabatically, however, its qualitative
structure considered in Section 6.4 (see Figs 5,6) remains unchanged.
In a sharp contrast with the described case of adiabatic deformation of an undular bore
front is case (ii) when the second term in the left-hand side of (106) contributes to the
leading order, i.e. to the motion of the leading edge itself. Namely, we have
= ρ+ + v′i
, i = 1, 2. (115)
Now dλ+/dT 6= ρ+ which means that the amplitude of the lead solitary wave a = −2λ+ varies
essentially differently compared to the case of an isolated solitary wave. Indeed, the term
ρ+ in the right-hand side of (115) is responsible for local adiabatic variations of the solitary
wave while the term v′i∂li/∂X̃ describes nonlocal parts of the variations associated with
the wave interactions within the undular bore. Using asymptotic formulae (103) implying
v′2 ≥ 0, v′1 ≤ 0, and the condition limX̃→0 |dl1,2/dX̃| = ∞ along with l2 ≥ l1, it is not
difficult to show that this nonlocal term is always nonnegative , i.e. the lead solitary wave
in the undular bore propagating over a gradual slope with bottom friction always moves
faster (and, therefore, has greater amplitude) than an isolated solitary wave of the same
initial amplitude in the beginning of the slope. Indeed, as we have shown in Section 5, the
presence of a slope and bottom friction always result in “squeezing” the cnoidal wave, hence
increasing momentum exchange between solitary waves in the vicinity of the leading edge
of the undular bore and acceleration of the lead solitary wave itself. The situation here is
qualitatively analogous to that described in Section 6.4 where the general global modulation
solution for the unperturbed KdV equation was discussed. Similarly to that case, the leading
edge now represents a characteristic envelope – a caustic (otherwise we are back in the case
(i) implying dλ+/dT = ρ+) (see Fig. 6a).
Unlike the case of adiabatic variations of the leading edge, determination of the function
λ+(T ) requires now knowledge of the full solution of the perturbed modulation system (27)
with the matching conditions (93). While the analytic methods to construct such a solution
for inhomogeneous quasilinear systems are not available presently, it is instructive to assume
that dλ+/dT − ρ+ is a known function of T and to study the structure of the solution in
close vicinity of the leading edge. With an account of the explicit form (103) of the velocity
corrections, equations (115) assume the form
= −dλ
+/dT − ρ+
2(l2 − l1)
ln[16/(1−m)]
(1−m)
, (116)
= −dλ
+/dT − ρ+
2(l2 − l1)
ln[16/(1−m)]
(1−m)
. (117)
Taking the difference of (116) and (117) we transform it to the form
∂(1 −m)
dλ+/dT − ρ+
(λ+)2
(1−m) ln[16/(1−m)]
. (118)
This equation can be readily integrated with the initial condition (111) to give
(1−m)2
2(dλ+/dT − ρ+)
(λ+)2
(X+ −X). (119)
This solution coincides with the asymptotic formula (92) for the behaviour of the modulus
in the vicinity of the leading edge of the undular bore in general unperturbed GP problem
[16] but instead of the derivative dλ+/dT in (92) we have the difference dλ+/dT −ρ+ (which
is always positive as we have established).
7 Conclusions
We have studied the effects of a gradual slope and turbulent (Chezy) bottom friction on the
propagation of solitary waves, nonlinear periodic waves and undular bores in shallow-water
flows in the framework of the variable-coefficient perturbed KdV equation. The analysis has
been performed in the most general setting provided by the associated Whitham equations
describing slow modulations of a periodic travelling wave due to the slope, bottom friction
and spatial nonuniformity of initial data. This modulation theory, developed in general form
for perturbed integrable equations in Kamchatnov (2004) was applied here to the perturbed
KdV equation and allowed us to take into account slow variations of all three parameters
in the cnoidal wave solution. The particular time-independent solutions of the perturbed
modulation equations were shown to be consistent with the adiabatically varying solutions
for a single solitary wave and for a periodic wave propagating over a slope without bottom
friction obtained in Ostrovsky & Pelinovsky (1970, 1975) and Miles (1979, 1983a). It was
shown, however, that the assumption of zero mean elevation used in these papers for the
description of slow variations of a cnoidal wave, ceases to be valid in the case when the
turbulent bottom friction is present. In this case, a more general solution was obtained
numerically improving the results of Miles (1983b).
Further, the derived full time-dependent modulation system was used for the descrip-
tion of the effects of variable topography and bottom friction on the propagation of undular
bores, in particular on the variations of the undular bore front representing a system of
weakly interacting solitary waves. By the analysis of the characteristics of the Whitham
system in the vicinity of the leading edge of the undular bore, two possible configurations
have been identified depending on whether the leading edge of the undular bore represents a
regular characteristic of the modulation system or its singular characteristic, i.e. a caustic.
The first case was shown to correspond to adiabatically slow deformations of the classi-
cal Gurevich-Pitaevskii modulation solution and is realised when the perturbations due to
variable topography and bottom friction are small compared with the existing spatial non-
uniformity of modulations in the undular bore (which is supposed to be formed outside the
region of variable topography/bottom friction). In the case when modulations due to the
external perturbations are comparable in magnitude with the existing modulations in the
undular bore, the leading edge becomes a caustic, and this situation was shown to corre-
spond to enhanced solitary wave interactions within the undular bore front. These enhanced
interactions have been shown to lead to a “nonlocal” leading solitary wave amplitude growth,
which cannot be predicted in the frame of the traditional local adiabatic approach to prop-
agation of an isolated solitary wave in a variable environment. As we mentioned in the
Introduction, one of our original motivations for this study was the possibility to model a
shoreward propagating tsunami as an undular bore. In this context, we would suggest that
the second scenario described above is the more relevant, which has the implication that
the growth, and eventual breaking of the leading waves in a tsunami wavetrain, cannot be
modeled as a local effect for that particular wave, but is determined instead by the whole
structure of the wavetrain.
Acknowledgements
This work was started during the visit of A.M.K. at the Department of Mathematical Sci-
ences, Loughborough University, UK. A.M.K. is grateful to EPSRC for financial support.
Appendix A: Derivation of the perturbed modulation system
We express the integrand function in the right-hand side of (24) in terms of the µ-variable
(15):
(2λi − s1 − U)R = 8Gµ3 − [8Gλi + 4(F + 2s1G)]µ2
+ [4(F + 2s1G)λi + 2s1(s1G+ F )]µ− 2s1(s1G+ F )λi.
(120)
Then we obtain with the use of (13), (14), and (16) the following expressions:
〈µ〉 =
µdθ =
−P (µ)
〈µ2〉 = 1
µ2dθ =
〈µ3〉 =
µ3dθ = −
+ s1〈µ2〉 − s2〈µ〉+ s3,
(121)
where I is a known integral
(λ3 − µ)(µ− λ2)(µ− λ1) dµ
(λ3 − λ1)5/2[(1−m+m2)E(m)− (1−m)(1−m/2)K(m)],
(122)
K(m) and E(m) being the complete elliptic integrals of the first and second kind, respec-
tively. The derivatives of I with respect to λi are also known table integrals (Gradshtein &
Ryzhik 1980):
(λ3 − µ)(µ− λ2)
µ− λ1
λ3 − λ1[(λ2 + λ3 − 2λ1)E − 2(λ2 − λ1)K],
(λ3 − µ)(µ− λ1)
µ− λ2
λ3 − λ1[(λ3 − λ1)K + (λ1 + λ3 − 2λ2)E],
(µλ2)(µ− λ1)
λ3 − µ
λ3 − λ1[(2λ3 − λ1 − λ2)E − (λ2 − λ1)K].
(123)
We can easily express the si-derivatives in terms of λi derivatives by differentiation of the
formulae (see (16))
s1 = λ1 + λ2 + λ3, s2 = λ1λ2 + λ1λ3 + λ2λ3, s3 = λ1λ2λ3 (124)
and solving the linear system for differentials. Simple calculation gives
(−1)3−k
j 6=i(λi − λj)
. (125)
Then, combining (123) and (125), we obtain the derivatives ∂I/∂si and hence the expressions
(λ3 − λ1)
(s21 − 3s2)
(λ2 − λ1)(λ2 + λ3 − 2λ1)
+ s1λ1 + λ
1 − λ2λ3
(λ3 − λ1)
(126)
To complete the calculation of the right-hand side of (24), we need also expressions
∂L/∂λ1
= 2(λ2 − λ1)
∂L/∂λ2
2(λ3 − λ2)(1−m)K
E − (1−m)K
∂L/∂λ3
2(λ3 − λ2)K
(127)
Collecting all contributions into perturbations terms, we obtain the Whitham equations in
the form
= ρi = Ci[F (T )Ai −G(T )Bi], (128)
where Cj , Aj , Bj and vj , j = 1, 2, 3 are specified by formulae (28) - (30).
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	Introduction
	Problem formulation
	Modulation equations
	Modulation solution in the solitary wave limit
	Adiabatic deformation of a cnoidal wave
	Undular bore propagation over variable topography with bottom friction
	Gurevich-Pitaevskii problem for flat-bottom zero-friction case
	Undular bore developing from an initial jump
	Structure of the undular bore front
	Gurevich-Pitaevskii problem for perturbed modulation system
	Deformation of the undular bore front due to variable topography and bottom friction
	Conclusions
