数理解析研究所講究録922 巻 1995 年 36-54

36Interaction of an algebraic soliton with uneven bottom topography
in a two-layer fluid system
好雅 (Yoshimasa Matsuno)
Department of Physics, Faculty of Liberal Arts,
Yamaguchi University, Yamaguchi 753, Japan
松野

Abstract
The propagation of long interfacial waves of finite amplitude is investigated ina two-layer fluid system where a layer of a light fluid overlies a layer of heavier one
resting on an uneven bottom. The upstream condition is imposed such that theconstant flow is in the same direction in both layers. Under appropriate balanceamong nonlinearity, dispersion and topographic effect, the long term evolution of
the interfacial elevation is shown to be governed by a forced Benjamin-Ono
equation to study
equation. A direct soliton perturbation theory is applied to thethe interaction of an algebraic soliton with a bottom topography when the Froudenumber is nearly equal to unity. A system of ordinary differential equations describing
the change of the amplitude and position of the soliton is derived for a simple bottom
profile. The solutions of these equations exhibit a variety of phenomena like thecapture and repulsion of the soliton by topography and the occurrence of soliton-like
phase shifts due to the interaction of the soliton with topography. We also examine
the effect of small dissipation on the dynamics of the soliton. Special emphasis isgiven to the appearance of new branch of stationary states of the soliton which has
never been observed in the absence of dissipation.
$(\mathrm{f}\mathrm{B}\mathrm{O})$$\mathrm{f}\mathrm{B}\mathrm{O}$I. INTRODUCTION
There has been much interest in studying the effect of external forces on the propagation of nonlinear waves. In the context of fluid dynamics, the forcing is mainly due
a bottom topography. Many authors have been
to a moving pressure distribution
concerned with a variety of phenomena arising from the forced systems such as the generation of solitary waves by a moving pressure force [1-4], or by uniform flow over a localizedbottom topography [5-7]. These problems have been elucidated extensively from bothanalytical and numerical points of view on the basis of the Korteweg-de Vries
equation. It is now
equation with forcing terms which is called the forced
equation is derived from the underlying system of equations on
well-known that thethe assumptions of shallow water and weak nonlinearity together with weak topographic
effect. At this stage, a natural question arises: what kind of nonlinear evolution equations are obtained when we introduce deep-water approximation in place of shallow-water
approximation. This is a motivation of the present study.
To answer this question, we shall consider a simple two-layer fluid system where alayer of a light fluid overlies a layer of heavier one resting on an uneven bottom. The depth$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$$(\mathrm{K}\mathrm{d}\mathrm{V})$$\mathrm{K}\mathrm{d}\mathrm{V}(\mathrm{f}\mathrm{l}<\mathrm{d}\mathrm{V})$$\mathrm{f}\mathrm{l}<\mathrm{d}\mathrm{V}$37of the upper layer is taken to be infinite corresponding to a deep-water approximation.
The upstream condition is imposed such that the constant flow is in the same direction inboth layers. We consider the two-dimensional irrotational flow of an incompressible andinviscid fluid. Applying a singular perturbation method [8-10] developed recently to thissystem, we show that the long term evolution of the interfacial elevation is governed by aforced Benjamin-Ono (fflO) equation. In the present case, the forcing term comes froman uneven bottom topography. We assume that the effect of topography is weak so thatit can be regarded as a small perturbation to the BO equation. It is noteworthy that aequation is derived similarly from the same system if we make the assumption ofshallow water in the upper layer [7].
While a number of problems to be resolved by the fflO equation are at hand, weshall be concerned here with an analytical study of the interaction process of an algebraic
soliton with the bottom topography. For the purpose, we employ a direct soliton perturbation theory $[11,12]$ developed quite recently to examine the dynamics of interacting
algebraic solitons under small dispersive or dissipative perturbations. The present analysis predicts the phenomena like the capture and repulsion of a soliton by topography
and the occurrence of soliton-like phase shifts. We also discuss the effect of small dissipation on the propagation of the soliton. Although similar problems have been studied
equations with various types of forcing terms [13-16], the analysis
extensively using
equation has not been performed as yet.
based on a
In Sec. II, we first derive a finite-depth analog of the fflO equation starting from theequation is then obtained simply bybasic system of hydrodynamic equations. The
taking the depth of the upper layer infinite. In Sec. III, the interaction of an algebraic
soliton with topography is investigated on the basis of the BO equation. We focus ourattention on the situation when the Froude number is nearly equal to unity. In thisproblem, the forcing term is regarded as a small perturbation to the BO equation sothat the amplitude and initial position of the soliton are no more constant and become
slowly varying functions of time. A system of ordinary differential equations describing thechange of these soliton parameters is derived by using the soliton perturbation theory. Fora simple bottom profile, the system is shown to be completely integrable. The analyticalsolutions thus obtained are investigated in detail in phase plane to show that they exhibit
the interesting phenomena already mentioned above. Also we briefly discuss the effectof small dissipation on the dynamics of the soliton. To simplify the analysis, we add the
Burgers term to the fflO equation which is proportional to the second derivative of the
wave profile. It is found that the characterist.ics of the stationary states of the soliton
are changed drastically owing to the dissipation. Section IV is devoted to summary andoutlook.
$\mathrm{f}\mathrm{K}\mathrm{d}\mathrm{V}$$\mathrm{f}\mathrm{K}\mathrm{d}\mathrm{V}$$\mathrm{f}\mathrm{B}\mathrm{O}$$\mathrm{f}\mathrm{B}\mathrm{O}$II. DERIVATION OF FORCED BENJAMIN-ONO EQUATION
A. Basic hydrodynamic equations
In this section, we shall derive the fflO equation starting from the hydrodynamic
equations subject to appropriate boundary conditions. The configuration of the two-38

layer fluid system under consideration is shown in Fig. 1.

FIG.1. Configuration of a two-layer fluid system

All the physical quantities are wnitten in terms of the dimensionless variables. The
dimensional quantities, with tildes, are related to the corresponding dimensionless onesby the relations$\overline{x}=lx,\overline{y}=h_{1}y,$$t\sim=(l/c_{0})t,\overline{\phi}_{j}=(gla/c0)\phi j(j=1,2),\overline{\eta}=a\eta,\overline{h}=h_{1}h$

(2.1)

,

and the dimensionless parameters are defined by$\epsilon=\frac{a}{l},$$\alpha_{j}=\frac{a}{h_{j}}(\dot{j}’=1,2),$$\delta_{j}=\frac{h_{j}}{1_{}}(j=1,2),$$F= \frac{U}{c_{0}},$(2.2)$\triangle=\frac{\rho_{1}}{\rho_{2}}$are characteristic scales of length, amplitude and velocity of the wave,
where
and
and measure theis the thickness of the upper layer. The parametersrespectively,
is themagnitude of nonlinearity and dispersion of the wave, respectively andsteepness parameter. We assume that $\alpha_{j}(j=1,2)$ and are small compared to unity
is very large, which is equivalent to introducing shallow-water approximation
whilein the lower fluid and deep-water approximation in the upper fluid in addition to weak
nonlinearity of the wave. The lower boundary is represented by the equation$l_{\text{ノ}},$$a$$c_{0}$$h_{1}$$\alpha_{j}$$\delta_{j}$$\epsilon(=\alpha_{j}\delta_{j})$$\delta_{2}$$\epsilon,$$\delta_{1}$$y=-h(x)=- \frac{\delta_{2}}{\delta_{1}}[1-\gamma\alpha_{2}^{2}B(x)]$

(2.3)

,

where $B(x)$ characterizes the unevenness of the bottom topography and is a positive
constant. In what follows, we assume that its profile is sufficiently smooth, localized andis the, implying thathence it vanishes at infinity. When $|x|arrow\infty,$ tends tothickness of the lower layer at infinity because it has been scaled by the thickness of the$\gamma$$h$$\delta_{2}/\delta_{1}$$h_{2}$

39

upper layer. The coefficientmeasures the magnitude of $B(x)$ . While this canbe taken arbitrary, we must impose for it appropriate smallness to lead correctly to the
fflO equation, as will be shown below. The basic flow is specified at $x=-\infty$ and it
is supposed to be in the positive -direction with a constant velocity $U$ in both layers.
The external force is only due to the gravitational acceleration and the effect of surface
tension is neglected. A characteristic velocity of the linear wave is given by $c_{0}=\sqrt{gl/\kappa}$
where is a parameter depending on
and $\delta_{j}(j=1,2)$ . Hereis the density ratio
which is assumed to be less than unity to assure a stable equilibrium state of fluids. The
Froude number $F$ is a very important parameter characterizing the property of the flow.
Its magnitude will be specified in the analysis developed in Sec. III.
Now, the dimensionless governing equations for inviscid incompressible and irrotational flow can be written as follows:
equations$h_{1}$$\gamma\alpha_{2}^{2}$

$x$

$g$

$\triangle$$\triangle$$\kappa$$(\mathrm{i})_{\mathrm{C}O}ntinuity$$\delta_{1}^{2}\phi_{1},xx+\phi 1,yy=0(-\infty<x<\infty, \alpha_{1}\eta<y<1)$,$\delta_{1}^{2}\phi 2,xx+\phi_{2,yy}=0(-\infty<x<\infty, -h(x)<y<\alpha_{1}\eta)$$(\mathrm{i}\mathrm{i})kinematic$.

$(2.4b)$boundary conditions at the fluid interface$\eta_{t}+F\eta_{x}+\kappa\in\phi_{1,\eta_{x}=}x\frac{\kappa}{\delta_{1}}\phi_{1,y}(y=\alpha_{1\eta))}$$(2.5a)$.

$(2.5b)$$\eta_{t}+F\eta_{x}+\kappa\epsilon\phi 2_{\mathcal{I}},\eta x=\frac{\kappa}{\delta_{1}}\phi_{2,y}(y=\alpha_{1}\eta)$$(\mathrm{i}\mathrm{i}\mathrm{i})dynamic$$(2.4a)$boundary condition at the fluid interface$\triangle[\phi_{1,t}+\frac{\kappa\epsilon}{2\delta_{1}^{2}}\{\delta_{1}^{2}(\frac{F}{\kappa\epsilon}+\phi 1,x)2+\phi^{2}1,y\}+\eta-\eta 0]$$= \phi_{2,t}+\frac{\kappa\epsilon}{2\delta_{1}^{2}}\{\delta_{1}^{2}(\frac{F}{\kappa\epsilon}+\phi_{2,x})^{2}+\phi_{2,y}^{2}\}+\eta-\eta_{0}(y=\alpha_{1\eta})$$(\mathrm{i}\mathrm{v})upper$

.

(2.6)

and lower boundary conditions$\phi_{1,y}=0(y=1)$ ,$\phi_{2,y}=-\delta_{1}^{2}(\frac{F}{\kappa\epsilon}+\phi 2,x)h_{x}(y=-h(x))$$(2.7a)$.

$(2.7b)$Here, $\phi_{1}=\phi_{1}(x, y, t)$ and $\phi_{2}=\phi_{2}(x, y, t)$ are the velocity potentials for the upperand lower fiuids, respectively about the perturbations of the upstream constant flow,$\eta=\eta(x, t)$ is the interfacial elevation andis a constant determined by the undisturbed
uniform state at infinity. The subscripts
and appended toand denote partial
differentiations. This notation is used throughout the paper.
$\eta_{0}$$t,$ $x$$y$$\phi,$$\eta$

B. Solutions for the velocity potentials

$h$

40

The solutions of the Laplace equations (2.4) for the velocity potentials which satisfy the boundary conditions (2.7) are constructed by following the procedure developed
recently [8-10]. For the upper fluid, it can be written in a closed form as$\phi_{1}(x, y, t)=-i[f_{1}^{+}(x+i\delta_{1y}, t)-f_{1}^{-}(x-i\delta 1y, t)]$with the complex functions$f_{1}^{\pm}$,$(2.8a)$given by$f_{1}^{\pm}(z, t)= \pm\frac{1}{4i\delta_{1}}P\int_{-\infty}^{\infty}\coth[\pi(y-Z)/2\delta_{1}]f_{1}(y, t)dy$$(2.8b)$where is a real function defined appropriately on the real axis and the symbol $P$ denotes
the Cauchy principal value integral.
For the lower fluid, on the other hand, the solution is expressed in a form of infinite
series as$f_{1}$$\phi_{2}(x, y, t)=\sum_{n=0}[y+h(X)]^{n}\phi_{2}^{()}(nXt\infty,)$,$(2.9a)$where the functions $\phi_{2}^{(n)}(n=0,1, \ldots)$ are determined by the recurrence formula$\delta_{1}^{2}[(n+1)h_{xx}\phi_{2}^{(}n+1)+(n+1)(n+2)h_{x}2\phi_{2}(n+2)+2(n+1)h_{x}\phi_{2,x}(n+1)+\phi_{2,xx}^{(n)}]$$+(n+1)(n+2)\phi_{2}^{(n+2)}=0,$ $(n=0,1, \ldots)$ ,$(2.9b)$subject to the boundary condition (2.7b). This can be solved easily and the first two of
which are written in the form$\phi_{2}^{(1)}=-\frac{\delta_{1}^{2}(\frac{F}{\kappa\in}+\phi 2,x(0))h_{x}}{1+\delta_{1}^{2}h_{x}^{2}}$,$(2.10a)$$\phi_{2}^{(2)}=-\frac{\delta_{1}^{2}(h_{xx}\phi 2+2h_{x}\phi_{2}^{()(0)}x+\phi 2,xx(1)1)}{2(1+\delta_{1}^{2}h_{x}^{2}’)}$

Here$\phi_{2}^{(0)}$.

$(2.10b)$

is an unknown function to be determined later.

C. Perturbation analysis
To perform the perturbation analysis, we must specify the magnitude of the paramand . We assume the weak nonlinearity of the wave, . $\alpha_{j}<<1(j=1,2)$ as
eters
well as the shallow-water approximation for the lower layer, i.e. $\delta_{2}<<1$ . For the upper
layer, however, we tentatively suppose $\delta_{1}=O(1)$ for the sake of generality. Furthermore,
it is necessary to balance the magnitude of nonlinearity, dispersion and topography appropriately to obtain forced soliton equations. In the present case, this can be accomplished
and $\gamma=O(1)$ .
by introducing the scalings$\mathrm{i}.\mathrm{e}$$\delta_{j}$$\alpha_{j}$$\alpha_{2}=O(\delta_{2})$41

These assumptions enable us to expand all the physical quantities in powers of theand . In particular, the derivatives of the velocity potentials are
small parameters
easily evaluated at the fluid interface. To that end, it is convenient to introduce thehorizontal components of the interfacial velocities by$\delta_{2}$$\alpha_{2}$$u_{j}=\phi_{j,x}|_{y=\alpha_{1}\eta},$(2.11)$(j=1,2)$ .

In addition, we take $\kappa=[(1-\triangle)\delta 2]-1$ to normalize the phase velocity of the linear wave
to unity since the linear dispersion relation relevant to the present problem is given by. If we retain the first two terms of the expansions, we obtain for theupper fluid$\omega^{2}=\kappa(1-\triangle)\delta_{2}k^{2}$$\phi_{1,y}|_{y=\alpha_{1}\eta}=-\delta_{1}[\overline{T}_{1}u_{1}+\epsilon\{\eta u_{1,x}+\overline{T}_{1}(\eta\overline{\tau}_{1}u1,x)\}+O(\epsilon^{2})]$$(\phi_{1,t}|_{^{-}}-\alpha 1\eta)_{x}=u_{1,t}+\epsilon(\eta_{t}\overline{\tau}_{1}u_{1,x}-\eta x\overline{T}1u1,t)+O(\epsilon^{2})$where$\overline{T}_{1}$,$(2.12a)$,$(2.12b)$is an integral operator defined by$\overline{T}_{1}f(X, t)=-\frac{1}{2\delta_{1}}P\int_{-\infty}^{\infty}\frac{f(y,t)}{\sinh[\pi(y-x)/2\delta_{1}]}dy$.

$(2.12_{C})$For the lower fluid, on the other hand, the corresponding expressions take the form$\phi_{2,y}|_{27}=\alpha_{1}\eta^{=}-\delta_{1}\delta_{2}[u_{2,x}+\alpha_{2}\{\eta u_{2,x}-\gamma(1-\triangle)FB_{x}\}+O(\alpha_{2}^{2})]$$(\phi_{2,t}|_{y=\alpha_{1\eta}})_{x}=u_{2,t}+O(\alpha_{2}^{2})$,$(2.13a)$.

$(2.13b)$

Finally, substituting (2.12) and (2.13) into (2.5) and (2.6) and neglecting the terms of theand . We quote, we obtain a closed system of equations fororder ofonly the final result:$(\alpha_{2}^{2}, \alpha_{22,2}\delta\delta 2)$$\eta,$$u_{2}$$u_{1}$$\eta_{t}+F\eta_{x}+(1-\triangle)-1u_{2,x}+(1-\triangle)^{-1}\alpha_{2}(\eta u2)_{x}=\gamma\alpha_{2}FB_{x}$,(2.14)$\eta_{t}+F\eta_{x}+\frac{1}{(1-\triangle)\delta_{2}}\overline{T}_{1}u_{1}+\frac{\alpha_{2}}{(1-\triangle)}[(u_{1}\eta)_{x}+\overline{T}_{1}(\eta\tilde{\tau}_{1}u1,x)]=0$,(2.15)$\triangle[u_{1,t}+Fu_{1,x}+\eta_{x}+\frac{\alpha_{2}}{2(1-\triangle)}[u_{1}^{2}+(\overline{T}_{1}u_{1})^{2}]_{x]}$$=u_{2,t}+Fu_{2,x}+\eta_{x}+(1-\triangle)^{-1}\alpha 2u2u_{2,x}$.

(2.16)In the above equations, the forcing term appears only on the right-hand side of (2.14) because of the present scaling hypothesis. Although these equations incorporate the lowestorder effect of nonlinearity and dispersion, we can proceed the perturbation analysis toinclude higher-order terms. However, the resulting equations become too complicated to
extract useful informations within the framework of analytical treatment. Nevertheless,42

it is nothing to say that higher-order equations are necessary particularly in dealing withlarge amplitude waves.
It is also possible to derive a single equation for . This can be performed by
andfrom $(2.14)-(2.16)$ . In the same approximation used above, theeliminating
final equation thus obtained can be written as follows:$\eta$$u_{1}$$u_{2}$$\eta u-(1-F2)\eta_{xx}+2F\eta xt-\triangle\delta_{2}T1\eta xxx$,$(2.17a)$$+\alpha_{2}[\eta_{t}s\eta_{t}-(1+2F^{2})\eta\eta_{x}-2F\eta\eta t+F\eta_{x}S\eta_{t}]_{x}=\gamma\alpha_{2}F^{2}B_{xx}$where the integral operators$T_{1}$and$S$

are defined by,$T_{1}u(X, t)= \tilde{T}_{1}^{-1}u(X, t)=\frac{1}{2\delta_{1}}P\int_{-\infty}^{\infty}\coth[\pi(y-X)/2\delta_{1}]u(y, t)dy$$Su(x, t)= \int_{-\infty}^{\infty}\mathrm{s}\mathrm{g}\mathrm{n}(y-x)u(y, t_{\text{ノ}})dy$,$(2.17b)$$(2.17_{C})$While Eq. (2.17) describes interfacial waves propagating to both right and left directions,
we can separate it into two equations, each describing a unidirectional motion. Following
the standard procedure [17], we find the evolution equations$\eta_{t}+(F-1)\eta x-\frac{3\alpha_{2}}{2}\eta\eta_{x}-\frac{\triangle\delta_{2}}{2}T_{1}\eta xx=\frac{\gamma\alpha_{2}F^{2}}{2}B_{x}$,(2.18)$\eta_{t}+(F+1)\eta_{x}+\frac{3\alpha_{2}}{2}\eta\eta_{x}+\frac{\triangle\delta_{2}}{2}T_{1}\eta_{xx}=-\frac{\gamma\alpha_{2}F^{2}}{2}B_{x}$.

(2.19)In the absence of the forcing term, the above equations reduce to the intermediate longwave (ILW) equations [10,18-20]. Therefore, they may be termed the forced ILW equations. The
equations are simply obtained from (2.18) and (2.19) by taking the
deep-water limit. Since in this limit the operator
reduces to the following
Hilbert transform$\mathrm{f}\mathrm{B}\mathrm{O}$$\delta_{1}arrow\infty$$T_{1}$$Hu(x, t)= \frac{1}{\pi}P\int_{-\infty}^{\infty}\frac{u(y,t)}{y-x}dy$,(2.20)

Eqs. (2.18) and (2.19) become$\eta_{t}+(F-1)\eta_{x}-\frac{3\alpha_{2}}{2}\eta\eta_{x}-\frac{\triangle\delta_{2}}{2}H\eta xx--\frac{\gamma\alpha_{2}F^{2}}{2}B_{x}$,(2.21).

$\eta_{t}+(F+1)\eta_{x}+\frac{3\alpha_{2}}{2}\eta\eta_{x}+\frac{\triangle\delta_{2}}{2}H\eta_{xx}=-\frac{\gamma\alpha_{2}F^{2}}{2}B_{x}$(2.22)

It is obvious that Eq. (2.22) describes unidirectional motion of the wave moving to theright. However, the situation is quite different for Eq. (2.21). In fact, the direction of
propagation will change according to the magnitude of $F$ as well as the amplitude of
the wave. Furthermore, the forcing term may affect the propagation characteristics even43

if it is small when compared to other terms. In view of this observation, Eq. (2.21) is
supposed to have a richer structure than Eq. (2.22) has in describing underlying physicalphenomena. This is a reason why we use Eq. (2.21) extensively in the next section.
III. INTERACTION OF ALGEBRAIC SOLITON WITH TOPOGRAPHY
A. Statement of the problem
equation (2.21) has wide applications in real physical phenomena,
While thewe shall study the interaction of an algebraic soliton with topography in the presence ofnearly critical flow, namely$F=1+\alpha_{2}\Gamma$ ,(3.1)$\mathrm{f}\mathrm{B}\mathrm{O}$where is a parameter which measures the departure of the Froude number from unity.
Furthermore, we assume that the effect of topography is weak in comparison with otherterms. More specifically, we introduce the scaling $\alpha_{2}<<\gamma<<1$ . Under these conditions,the second to fourth terms on the left-hand side of Eq. (2.21) enter at the same orderwhereas the forcing term becomes higher order and hence it can be treated as a small
ensures
perturbation to the BO equation. It should be noted that the inequality
in derivingthat the forcing term is still larger than neglected terms of order
Eq. (2.21), which is consistent with the perturbation analysis developed in Sec. II.
To proceed the analysis further, it is convenient to rescale the variables according to$\etaarrow(8/3)u$ and $Barrow(16/3F^{2})B$ . Then, Eq. (2.21) takes
the following form$\prime u_{t}+\Gamma u_{x}-4uu_{x}-Hu_{xx}=\gamma B_{x}$ .
(3.2)$\Gamma$$\alpha_{2}<<\gamma$$(\alpha_{2}^{2}, \alpha_{2}\delta_{2}, \delta_{2}2)$$tarrow(\triangle\delta_{2}/\alpha_{2}^{2})t,$$xarrow(\delta_{2}/\alpha_{2})x,$

It is well-known that if the forcing term is absent, Eq. (3.2) exhibits soliton solutionsof algebraic type [21-23]. The main concern here is to investigate the effect of small
perturbation which originates from localized topography on the dynamics of an algebraic
soliton. For the purpose, we shall consider a symmetric isolated bottom topography ofthe form$B(x)= \frac{\lambda b}{(bx)2+1},$ $(\lambda=\pm 1)$ ,(3.3)
is a typical length scale of topography and characterizes the nature of forcing.
Although the functional form of $B$ adopted here is rather specific, an advantage is thatthe resulting motion of a soliton is described by an integrable Hamiltonian system, as will
be seen below. The problem under consideration can now be dealt with by means of thesoliton perturbation theory, which we shall describe briefly.

where$b^{-1}$$\lambda$

B. Soliton perturbation theory
A direct multisoliton perturbation theory for the BO equation has been developed
quite recently using multiple time scale expansion $[11,12]$ . We shall summarize the main
ingredient of the theory.
We write the perturbed BO equation in the form
$u_{t}+4uu_{x}+Hu_{x}x=\epsilon R[u]$

,

(3.4)

44

represents a perturbation. We first introduce the different time scales
where$t_{j}=\epsilon^{j}t(j=0,1,\ldots)$ so that the time derivative is replaced by the relation
by. If we expand into an asymptotic series of the form
$u_{j}=$( $x,$ to, $t1,$ ), substitute these expressions into (3.4) and equate the coefficient of like
powers of , we obtain the hierarchy of equations for , the first two members of which
read in the form
(3.5)
$u0,t_{00}+4u0u,x+Hu0_{x},x=0$ ,$\epsilon R[u]$$t_{j}$$\partial/\partial t=$$u_{j}$$u= \sum_{j=0}^{\infty}\epsilon^{j}uj,$$u$$\sum_{j=0^{\epsilon^{j}}}^{\infty}\partial/\partial tj$$\ldots$$\epsilon$$u_{j}$$u_{1,t_{0}}+4(u_{0}u1)_{x}+Hu_{1,xx}=R[u_{0}]-u0,t_{1}$

(3.6)

.

Equation (3.5) is just the BO equation. We solve the above system of equations successively starting with a specific solution of Eq. (3.5). Since we are interested in the soliton
problem, we take it the $N$-soliton solution which includes the $N$ amplitude parameters
and the $N$ position parameters $\xi_{j}(j=1,2, \ldots)$ . In the absence of the perturbation,
being the initial position ofare constant independent of and $\xi_{j}=a_{j}t_{0}+\xi_{j0}$ withthesoliton. Under the action of the perturbation, these parameters would be modu, so we may assume their time dependence aslated slowly on the time scale of order$(j=1,2, \ldots, N)$ .
andandis determined by requiring that the correction term
The time evolution ofis no more singular than the leading-order solution . This turns out to be equivalent
to imposing the following compatibility conditions$a_{j}$$a_{j}$$t_{0}$$\xi_{j0}$$j\mathrm{t}\mathrm{h}$$\epsilon^{-1}$$\xi_{j0}=\xi_{j0}(t_{1}, t2, \ldots),$$a_{j}=a_{j}(t_{1}, t_{2}, \ldots)$$\xi_{j0}$$a_{j}$$u_{0}$$u_{1}$$(g_{j}, R[u_{0}]-u0,t_{1}) \equiv\int_{-\infty}^{\infty}g_{j}(R[u0]-u0,t_{1})dx=0,$

Here$g_{\dot{j}}$$(j=1,2, \ldots)$ .

(3.7)

are solutions of the adjoint equation for the homogeneous part of (3.6)$g_{j,t_{0}}+4u0gj,x+Hg_{j,xx}=0,$$(j=1,2, ..)$ .

(3.8)

The $2N$ independent bounded solutions for (3.8) are found to be as$g_{j}= \int_{-\infty}^{x}\frac{\partial u_{0}}{\partial a_{j}}dx,$$g_{j+N}= \int_{-\infty}^{x}\frac{\partial u_{0}}{\partial\xi_{j0}}dx,$$(j=1,2, \ldots, N)$ .

$(3.9a, b)$

The final step of our perturbation scheme is to calculate the inner products in (3.7) with
and
where
the aid of certain orthogonality relations which hold between. Theseor
while noting the relation
stands fororthogonality relations are derived directly from (3.5) and (3.9) and they are expressed as$g_{j}$$a_{j}$$\partial u_{0}/\partial p_{j}$$p_{j}$$u_{0,t_{1}}= \sum_{j=1}^{N}(a,u_{0}J^{t_{1}},a_{j}+\xi_{j,t_{1}}u_{0,\xi j})$$\xi_{j0}$$(g_{i}, u_{0},\xi_{j}0)=-(g_{i+N}, u0,aj)=(\pi/4)\delta_{ij},$ $(g_{i}+N, u0,\xi_{j}0)=(g_{i}, u_{0},)a_{j}=0(i, j=1,2, \ldots, N)$

is Kronecker’s delta. The evolution equations for the soliton parameters thus
where
obtained are written in terms of the original time variables as follows:$\delta_{ij}$$\frac{da_{j}}{dt}=-\frac{4\epsilon}{\pi}(g_{j+N}, R[u\mathrm{o}]),$$\frac{d\xi_{j}}{dt}=a_{j}+\frac{4\epsilon}{\pi}(g_{j}, R[u\mathrm{o}]),$$(j=1,2, \ldots, N)$ ,$(3.10a, b)$soliton has been used instead offor pracwhere the position parameter of thetical convenience. It should be emphasized that for the purpose of calculating the inner$\xi_{j}$$j\mathrm{t}\mathrm{h}$$\xi_{j0}$45products in (3.10) we only need the information of the $N$-soliton solution which can beusually obtained without recourse to inverse scattering transform method [21-23]. In thisrespect, it is worth remarking that the soliton perturbation theory based on the inverse
scattering transform method has not been applied as yet to a certain class of integrodifferential evolution equations like the BO and ILW equations.
C. Perturbation equations for soliton parameters
Now let us derive the perturbation equations for the soliton parameters on the basisof Eq. (3.10). In the present one-soliton problem, one can take an algebraic soliton$u_{0}= \frac{a}{a^{2}(x-\xi)^{2}+1}$(3.11),

as a lowest-order solution of Eq. (3.2) where and are the amplitude and position ofaction of small perturbation, and evolve according
the soliton, respectively. Under
.
$a$

th.

to the equations

$\xi$

$a$

$\mathrm{e}$$\frac{da}{dt}=-\frac{4\gamma}{\pi}(g_{2}, B_{x}),$$\xi$$\frac{d\xi}{dt}=\Gamma-a+\frac{4\gamma}{\pi}(g_{1}, B_{x})$,$(3.12a, b)$where$g_{1}= \int_{-\infty}^{x}\frac{\partial u_{0}}{\partial a}dx=\frac{x-\xi}{a^{2}(x-\xi)^{2}+1},$$g_{2}= \int_{-\infty}^{x}\frac{\partial u_{0}}{\partial\xi}dx=-\frac{a}{a^{2}(x-\xi)^{2}+1}$.

$(3.13a, b)$

It is pointed out that the amplitude equation (3.12a) coincides with the equation derived
with given by (3.11). However,from the energy balance equation$\partial(1,u)/\partial
t=\gamma(1,B_{x})=0$
gives rise to a trivial result
the momentum balance equation$0=0$ sincein the leading order of the expansion. As for the latter equation,
equation where
the situation is quite different from the corresponding one for the
depends on a soliton amplitude.
the integral
The inner products in (3.12) are now evaluated by using the.
$\partial(u, u)/\partial t=2\gamma(u, B_{x})$$u$$\int_{-\infty^{u}}^{\infty}dx=\pi$$\mathrm{f}\mathrm{K}\mathrm{d}\mathrm{V}$$\int_{-\infty^{u_{0}d}}^{\infty}X$$\mathrm{f}_{\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{u}}1\mathrm{a}\mathrm{s}.$$\int_{-\infty}^{\infty}\frac{x}{[a^{2}(x-\xi)2+1](b2+X^{2}1)^{2}}d_{X}=\frac{\pi}{ab^{3}}\frac{(\frac{1}{a}+\frac{1}{b})\xi}{[\xi^{2}+(\frac{1}{\mathrm{c}\iota}+\frac{1}{b})^{2}]^{2}}$,$(3.14a)$.

$\int_{-\infty}^{\infty}\frac{x^{2}}{[a^{2}(x-\xi)2+1](b2+X^{2}1)^{2}}dX=\frac{\pi}{2ab^{3}}\frac{(\frac{1}{a}+\frac{2}{b})\xi^{2}+\frac{1}{a}(\frac{1}{(\iota}+\frac{1}{b})^{2}}{[\xi^{2}+(\frac{1}{a}+\frac{1}{b})^{2}]^{2}}$$(3.\mathrm{i}4b)$

After some algebras, we obtain the final evolution equations for the soliton parameters
and as follows:

$a$

$\xi$

$\frac{da}{dt}=-\frac{8\lambda\gamma(\frac{1}{a}+\frac{1}{b})\xi}{[\xi^{2}+(\frac{1}{a}+\frac{1}{b})^{2}]^{2}},$$\frac{d\xi}{dt}=\Gamma-a+\frac{4\lambda\gamma}{a^{2}}\frac{\xi^{2}-(\frac{1}{a}+\frac{1}{b})^{2}}{[\xi^{2}+(\frac{1}{a}+\frac{1}{b})^{2}]^{2}}$.

$(3.15a, b)$

In the absence of the perturbation, i.e. $\gamma=0,$ remains its initial value while $\xi=$$(\Gamma-a)t+\xi_{0}$ withbeing the initial position. Furthermore, if we introduce a new$a$$\xi_{0}$46dependent variable bytransparent forms as$\theta$$\theta=(\frac{1}{a}+\frac{1}{b})^{-1}\xi$$\frac{da}{dt}=-\frac{8\lambda\gamma}{(\frac{1}{a}+\frac{1}{b})^{2}}\frac{\theta}{(\theta^{2}+1)^{2}},$, we can transform Eqs. (3.15) into the more.

$\frac{d\theta}{dt}=\frac{\Gamma-a}{\frac{1}{a}+\frac{1}{b}}-\frac{4\lambda\gamma}{a^{2}(\frac{1}{a}+\frac{1}{b})^{3}}\frac{1}{\theta^{2}+1}$$(3.16a, b)$

Remarkably, we find that the above system of equations is completely integrable. To
show this, we define a new variable by $A=a/b+\ln$ and a Hamiltonian $H$ given by
$A$

$a$

$H=H(A, \theta)=\Gamma a-\frac{a^{2}}{2}-\frac{1}{\frac{1}{a}+\frac{1}{b}}\frac{4\lambda\gamma}{\theta^{2}+1},$

Note that is a single-valued function of
to Hamilton’s equations
$a$

$A$

$(a=a(A))$ .

(3.17). Then, Eqs. (3.16) are seen to be equivalent.
$\frac{dA}{dt}=-\frac{\partial H}{\partial\theta},$$\frac{d\theta}{dt}=\frac{\partial H}{\partial A}$.

$(3.18a, b)$

Since the Hamiltonian (3.17) does not depend on time explicitly, it becomes a constantof motion. For later use, we wnite it in the form$[(a- \Gamma)^{2}+p](\frac{1}{a}+.\frac{1}{b})=-\frac{8\lambda\gamma}{\theta^{2}+1}$(3.19),

where is an integration constant which can be determined by the initial conditions forand .
equation. Using
Lastly, we shall comment on an analogous work based on aa second-order perturbation theory, Grimshaw et al [16] obtained the equations for theamplitude and position of a soliton that correspond to Eqs. (3.15), where they introduced
a simple sech-type forcing. However, they restricted their analysis to the two extreme
cases when the external force has a lengthscale much greater than the soliton (broad
forcing) and when it is much less (narrow forcing). They found that the second-order
perturbation equations turn out to be integrable only for the broad forcing where thesoliton behaves like a delta function. On the other hand, the present second-order equations (3.15) become to be integrable without any assumption on the lengthscale of theforcing as just proved. It is not clear, however, whether the structural difference between
the two dynamical systems mentioned above should be attributed to the perturbation
schemes employed or to any other reason such as the structure of the underlying forced
soliton equations.
$p$

$a$

$\theta$$\mathrm{f}\mathrm{K}\mathrm{d}\mathrm{V}$D. Properties of solutions
1. Stationary solutions and their linear stability
Before discussing the properties of solutions which evolve from various initial conditions, we look for stationary solutions of Eqs. (3.15) and consider their stability characteristics. In the following analysis, we put $b=1$ without loss of generality. If we47

and , we immediately find from the equations
denote the stationary solutions by$da_{s}/dt=d\xi_{s}/dt=0$ that these are determined by the relations$a_{s}$$\xi_{s}$$\Gamma-a_{s}=\frac{4\lambda\gamma}{(a_{S}+1)2},$

Figure 2 is a plot of the stationary amplitude$\gamma=0.05$ and.

$a_{s}$$\xi_{s}=0$.

$(3.20a, b)$

as a function of the parameter$\Gamma$for$\lambda=\pm 1$

FIG. 2. Plot of the stationary amplitude

as a function of .
$\Gamma$$a_{\mathrm{s}}$In the case of $\lambda=-1$ , we can see that for any value of within the range $\Gamma>-4\gamma$ , there
exists one stationary amplitude. On the other hand, in the case of $\lambda=+1$ , there exists. To discuss the linear stability of the solution
one stationary amplitude only if(3.20), we linearize Eqs. (3.15) around the solution stationary solution as$\Gamma$$\Gamma>4\gamma$$a=a_{s}+\hat{a},$$\xi=\xi_{s}+\hat{\xi}$.

Then we obtain a system of linearized equations for \^a and$\frac{d\hat{a}}{dt}=-\frac{8\lambda\gamma a_{s}^{3}\hat{\xi}}{(a_{s}+1)3},$

Elimination of the variable
differential equation for \^a$\hat{\xi}$$(3.21a, b)$$\hat{\xi}$$\frac{d\hat{\xi}}{dt}=-[1-\frac{8\lambda\gamma}{(a_{s}+1)^{3}}]\hat{a}$.

$(3.22a, b)$from the above system of equations yields a second-order$\frac{d^{2}\hat{a}}{dt^{2}}=\frac{8\lambda\gamma a_{s}^{3}}{(a_{s}+1)3}[1-\frac{8\lambda\gamma}{(a_{\mathrm{s}}+1)3}]\hat{a}$.

(3.23)

Since $a_{s}>0$ and $0<\gamma<<1$ , the coefficient of \^a on the right-hand side of (3.23) becomes
negative (positive) for $\lambda=-1(\lambda=+1)$ . Substituting the solution of the formand examining the sign of the eigenvalue , we find that the solution corresponding to$\lambda=-1(\lambda=+1)$ is stable (unstable) against infinitesimal perturbations. In the case of$\lambda=-1$ , the stable points represent centers since the corresponding eigenvalues are pure$\hat{a}\propto\exp(\sigma t)$$\sigma$48

imaginary while for $\lambda=+1$ , the unstable points are seen to be unstable nodes with oneof the two eigenvalues being real and positive.
As an important remark, we point out that Eq. (3.2) exhibits an exact stationary
solution for a special value of . Indeed, careful investigation of the stationary form ofEq. (3.2) shows that an exact solution which satisfies the boundary conditionas$|x|arrow\infty$ exists when
. Explicitly, it reads in the form$\Gamma$$uarrow \mathrm{O}$$\Gamma=1+\lambda\gamma$$u_{s}= \frac{1}{x^{2}+1}$.

(3.24)

It turns out that this solution corresponds to the stationary solutions
and $\xi_{s}=0$
Eq.
(3.15) as seen from (3.20).
of
2. Analysis of solutions in phase plane
Let us now investigate the properties of the general solution. As already demonstrated, the system of equations (3.16) exhibits a first integral given by (3.19). Using thisfact, the explicit time dependence of the variables and can be obtained in principle
by integrating the equations of the form $da/dt=f(a)$ and $d\xi/dt=g(\xi)$ whereandare known functions. These integrals are, however, found to be difficult to perform
analytically. Here we shall analyze the solutions in phase plane
for the following
four different combinations of the parameters and :. $\lambda=-1,$
This case represents a bottom topography which is concave downward. Figure 3
shows a typical phase portrait of the solution (3.19) for various values of the constantwhere the parameters and have been chosen as 1.0 and 0.05, respectively. The arrowin each trajectory indicates the direction of the motion of a soliton. Note in this casethat there exists only one linearly stable stationary amplitude for a given as seen from
Fig. 2.
$a_{\mathrm{s}}=1$

$a$

$\xi$

$f$

$g$

$(a, \theta)$$\lambda$$\Gamma$$\Gamma>-4\gamma$$\mathrm{a}$$p$$\Gamma$$\gamma$$\Gamma$FIG. 3. Phase portrait of the solution (3.19) with $\lambda=-1,$ $\Gamma=1.0$ and $\gamma=0.05$ for various
values of .
$p$

If the initial amplitude of the soliton is either much larger or much less than unity,
the soliton interacts with the bottom topography whereby it suffers a small change of theamplitude near the origin $\theta=0$ . After passing through topography, the soliton gradually

49

recovers its initial amplitude. The orbit of the solution is then described by the equation$(a-a_{\infty})(a+a_{\infty}-2)( \frac{1}{a}+1)=\frac{0.4}{\theta^{2}+1}$(3.25),. Whileis the amplitude of the soliton whenin the present example, where
there is no net change in the amplitude after interaction, the soliton exhibits a phase shift
which may be given by the relation$\thetaarrow\pm\infty$$a_{\infty}$$\triangle\xi=\int_{-\infty}^{\infty}[\frac{d\xi}{dt}-(\Gamma-a_{0})]dt$(3.26),

in the brackets
is the initial amplitude of the soliton. The second term
where
in
the
absenceoftheinteraction
and
hence
(3.26) yields a
velocitythesolitonrepresents
net phase shift due to the interaction. Substituting (3.15b) into (3.26), we can write it in
the form$\Gamma-a_{0}$$a_{0}$.

(3.27)$\triangle\xi=\int_{-\infty}^{\infty}[a-a_{0}+\frac{4\lambda\gamma}{a^{2}}\frac{\xi^{2}-(\frac{1}{a}+1)^{2}}{[\xi^{2}+(\frac{1}{a}+1)^{2}]2}]dt$

If we use (3.15a) and (3.19), we can transform (3.27) into an integral with respect to .
The resulting expression is, however, too complicated to calculate analytically. Instead of
performing numerical integration, we shall evaluate (3.27) by a successive approximation.
We readily find from (3.15) that in the approximation up to order$a$$\gamma$$a=a_{0}+ \frac{4\lambda\gamma}{\Gamma-a_{0}}\frac{\frac{1}{a_{0}}+1}{[(\Gamma-a\mathrm{o})t+\xi 0]^{2}+(\frac{1}{a_{0}}+1)^{2}}+O(\gamma)2$$\xi=(\Gamma-a_{0})t+\xi_{0}+O(\gamma)$,,$(3.28a)$$(3.28b)$where is the initial position of the soliton. Introducing (3.28) into (3.27) and performing
correct up to order as
the integral with respect to , we obtain the expression of$\xi_{0}$$t$$\triangle\xi$$\gamma$$\triangle\xi=-\frac{4\pi\lambda\gamma}{(\Gamma-a_{0})^{2}}\mathrm{s}\mathrm{g}\mathrm{n}(\Gamma-a\mathrm{o})$.

(3.29)

Note that this quantity is proportional to the cross-sectional area of the bottom topogfrom (3.3). The phase shift of the soliton predicted byraphy because(3.29) is quite interesting since algebraic solitons usually exhibit no phase shift after theircollisions [21-23].
Figure 3 also indicates another fascinating aspect of the solutions. To be more
specific, if the soliton enters from the right with an amplitude a bit larger than unity, itinteracts with topography and then propagates to the left direction while decreasing its
amplitude. After a lapse of time, the soliton changes its direction and goes back to theright. The motion is repeated in the same way. One observes that the orbit of the soliton
becomes periodic around a stationary state $a_{s}=$ 1.048 and $\xi_{s}=0$ which turns out tobe a center in the present example. This phenomenon implies that the soliton has been$\int_{-\infty}^{\infty}B(x)dX=\pi\lambda$

50

captured by topography while oscillating around the stationary state indicated above.
We find from (3.19) that the capture occurs inside the orbit$(a-1)^{2}( \frac{1}{a}+1)=\frac{0.4}{\theta^{2}+1}$(3.30).

since it always possesses
Note that Eq. (3.30) represents two curves coalescing at
two positive amplitude solutions for a given .
. $\lambda=-1,$ $\Gamma<-4\gamma$
Different from the case a, there exist no stationary amplitudes in the parameter range$\Gamma<-4\gamma$ .
The figure corresponding to Fig. 3 is drawn in Fig. 4 where the parameters
are given by $\Gamma=-1.0$ and $\gamma=0.05$ .
$\theta=\pm\infty$$\theta$$\mathrm{b}$FIG. 4. As in FIG. 3 except 1 $=-1$ .U.

This case is not so interesting because solitons propagate to the left direction irrespective of their initial amplitudes and positions. In fact, one can see that the velocityof the soliton given by (3.15b) always becomes negative provided that $\gamma<<1$ . The only
effect due to the interaction with topography is a phase shift given by (3.29).
$\mathrm{c}$. $\lambda=+1,$$\Gamma>4\gamma$

This corresponds to a bottom topography which is convex upward. In the parameter, there is only one linearly unstable stationary amplitude. A typical phaserange
portrait is depicted in Fig. 5 for various values of where we have put $\Gamma=1.0$ and$\Gamma>4\gamma$$p$$\gamma=0.05$.

As seen from the figure, solitons with larger or smaller amplitudes compared to unity
pass through topography while suffering small phase shifts. On the other hand, solitons
with amplitudes a bit larger than unity decrease gradually their amplitudes in the courseof the propagation to the left direction and eventually they are repelled by topography
so that they propagate back to the right direction with amplitudes smaller than unity.
The two types of the propagation characteristics of the solutions described above are
separated by the orbit$[(a-1)^{2}-0.197]( \frac{1}{a}+1)=-\frac{0.4}{\theta^{2}+1}$,(3.31)

51

in the present example. Equation (3.31) represents two curves intersecting at a stationary
state given by $a_{s}=$ 0.947 and $\theta_{s}=0$ which is seen to be an unstable node from thepreceding stability analysis. Thus, inside the above orbit, the repulsion of solitons occurs
due to topographic effect.
$\mathrm{d}$. $\lambda=+1,$$\Gamma<4\gamma$, there exist no stationary amplitudes. The figure
In the parameter range
corresponding to Fig. 5 is drawn in Fig. 6 for the values of $\Gamma=-1.0$ and $\gamma=0.05$ .
The properties of the orbits of solitons are similar to those of the case . Whensolitons enter from the right, they always pass through topography. The net effect is aphase shift given by (3.29).
$\Gamma<4\gamma$$\mathrm{b}$FIG. 5. As in FIG. 3 except $\lambda=+1.0$ . FIG. 6. As in FIG. 3 except $\lambda=+1.0$ and $\Gamma=-1.0$ .
E. Effect of small dissipation
In the analysis developed so far, the effect of dissipation has been neglected. In real
physical phenomena, however, there exist many situations where the dissipation would
play an important role. For instance, in the description of long internal waves in thestratified lower atmosphere, turbulent dissipation would become to be significant [24].
In order to elucidate the effect of dissipation on the dynamics of algebraic solitons inaddition to topographic effect, we consider the following model equation$u_{t}+\Gamma u_{x}-4uu_{x}-Huxx=\gamma B_{x}+\mu u_{xx}$,(3.32)

where is a small positive parameter which measures the magnitude of dissipation. When
, Eq. (3.32) reduces to the so-called BO-Burgers equation introduced in $[24,25]$ .
While Eq. (3.32) can be used to investigate various problems associated with it, weshall here focus our attention on the study of the dynamics of an algebraic soliton under
the action of small dissipation. For the purpose, it is appropriate to employ the soliton
perturbation theory developed previously.
If we introduce again the profile of the bottom topography given by (3.3), the perturbation equations for the amplitude and position of the soliton are now written as
follows:$\mu$$\gamma=0$$\frac{da}{dt}=-\frac{8\lambda\gamma(\frac{1}{a}+\frac{1}{b})\xi}{[\xi^{2}+(\frac{1}{a}+\frac{1}{b})^{2}]^{2}}-\mu a^{3}$,$(3.33a)$52$\frac{d\xi}{dt}=\Gamma-a+\frac{4\lambda\gamma}{a^{2}}\frac{\xi^{2}-(\frac{1}{a}+\frac{1}{b})^{2}}{[\xi^{2}+(\frac{1}{a}+\frac{1}{b})^{2}]^{2}}$.

$(3.33b)$

When compared to the corresponding equations (3.15) in the absence of dissipation, thedissipative effect appears only in the amplitude equation (3.33a). Owing to the dissipative, however, the complete integrability of the system is lost perfectly. Instead, thetermdynamics of the soliton are found to exhibit a variety of phenomena which have neverbeen observed in the absence of dissipation. Here we shall first seek stationary solutionsof Eqs. (3.33) and subsequently examine their stability against small perturbations.
1. Stationary solution
It now follows from the equations $da_{s}/dt=d\xi_{s}/dt=0$ that the amplitude andposition of the soliton corresponding to the stationary state are obtained by solving thenonlinear algebraic equations$\mu a^{3}$$(3.34a)$$\frac{(\Gamma-a_{s})^{2}+(\mu a_{s})2}{\Gamma-a_{s}+\mathrm{s}\mathrm{g}\mathrm{n}(\Gamma-a_{S})\sqrt{(\Gamma-a_{s})2+(\mu a_{s})^{2}}}=\frac{2\lambda\gamma}{(a_{s}+1)^{2}})$$\xi_{s}=\frac{1}{\mu a_{s}^{2}}(a_{s}+1)[\Gamma-a_{S^{-}}\mathrm{s}\mathrm{g}\mathrm{n}(\Gamma-a_{s})\sqrt{(\Gamma-a_{s})^{2}+(\mu a_{s})2}]$,$(3.34b)$$r$

where we have put $b=1$ . When tends to zero, these expressions reduce to Eqs. (3.20).
It is evident that the characteristics of stationary states given by (3.34) depend on
and . We shall particularly pay our attention to elucidating thethe parameters
effect of while fixing and to typical values. A plot of the stationary amplitude
as a function of is shown in Fig.
for the values of the parameters $\Gamma=1.0$ and$\gamma=0.05$ where Newton’s method has been used to solve Eq. (3.34a).
$\mu$$\Gamma,$$\mu$$\gamma$$\Gamma$$\mu$$a_{s}$$\gamma$$7(\mathrm{a})$$\mu$FIG.7. Plot of the stationary amplitude$a_{s}$and the stationary position$\xi_{s}$as a function of$\mu$

for the values of the parameters $\Gamma=1.0$ and $\gamma=0.05$ .

Also, the corresponding plot of the stationary positionis given in Fig.
for thesame parameter values. The branch OAB in the figures represents the stationary states for$\lambda=-1$ whereas the branchcorresponds to those for $\lambda=+1$ . In the case of $\lambda=-1$ ,that for the range of the parameter $0<\mu<0.025$ , there is only
one can see from Fig.
one stationary amplitude for a given . If excee&the value 0.025, a new amplitude$\xi_{s}$$\mathrm{O}\mathrm{A}’\mathrm{B}’$$7(\mathrm{a})$

$\mu$

$\mu$

$7(\mathrm{b})$53appears and hence we find a total of two stationary amplitudes. This happens until
reaches a value $\mu=0.0314$ which corresponds to the point A in the figure. Furthermore, if
exceeds the point , no stationary amplitudes are found to exist. This last phenomenon
is easily understood from the physical point of view. Indeed, when the magnitude ofexceeds certain critical value ( $\mu=$ 0.0314 in the present example), the dissipative
effect becomes dominant and hence it suppresses both nonlinear and topographic effects.
Consequently, the amplitude of the soliton decreases indefinitely without approaching any
stationary value. The similar explanation is given to the case $\lambda=+1$ where the value ofis 0.0338.
corresponding to the point
2. Linear stability
In order to study the stability of the stationary solutions obtained above, we have
performed the linear stability analysis. Applying the same procedure as that used forthe system of equations (3.15), we have arrived at the following conclusion.
For $\lambda=-1$ , the solutions on the branch AB are stable, representing stable focuses
because in this case the corresponding two eigenvalues are found to be complex conjugate
with a negative real part. On the other hand, the solutions on the branch OA become
unstable with one of the two eigenvalues being real and positive and hence these may be
become unstabletermed unstable nodes. For $\lambda=+1$ , the solutions on the branch
are seen to be stable focuses.
nodes whereas those on the branch$\mu$$\mathrm{A}$

$\mu$

$\mu$

$\mathrm{A}’$$\mu$$\mathrm{A}’\mathrm{B}’$$\mathrm{O}\mathrm{A}’$IV. SUMMARY AND OUTLOOKIn this paper, we have studied the interaction of an algebraic soliton with localized
equation which is derived from a simple
bottom topography on the basis of the
two-layer fluid system by means of an asymptotic approach. Under appropriate scaling
hypothesis, the topographic effect is shown to become a small pertur.b ation to the BO
apply a direct soliton perturbation theory to the systemequation. This enables us
under consideration.
In spite of the introduction of a simple profile of the bottom topography, the present
analysis exhibits a variety of phenomena. In particular, the prediction of the capture
and repulsion of a soliton by topography is interesting because of the possibility of theirappearance in lower atmospheric systems, for instance where a mountain may be modeledby a localized topography. We have also found that the above phenomena are closely
related to the stability characteristic of the system while employing the linear stability
analysis.
Although the dynamical system describing the change of the amplitude and positionof the soliton is shown to be completely integrable, an addition of small dissipation alters drastically the properties of solutions. In fact, for certain range of the dissipation
parameter, we have found the two types of stationary states for each bottom topography
in which one is stable and another is unstable against small perturbation.
In conclusion, we shall point out that in comparison with a large number of works
concerning the ffldV equation, there is a dearth of study devoted to the BO equation.
Indeed, many subjects are left for future studies. Among them, of practical importance is$\mathrm{f}\mathrm{B}\mathrm{O}$$\mathrm{t}\mathrm{o}\backslash$54

to examine whether disturbances generated in the vicinity of topography can propagate
far upstream under the condition that the basic flow is near resonance, i.e. when the
Froude number is nearly unity. This problem may be resolved by integrating Eq. (3.2)
numerically with zero initial condition $u(x, 0)=0$ . In the analysis developed in thispaper, we have addressed ourselves to soliton solutions. As is well-known, however,the BO equation exhibits an exact periodic solution expressed in terms of trigonometricfunctions $[26,27]$ . The modulation of periodic wave train due to external forcing is aninteresting problem which may be analyzed by the similar procedure to that used for thecorresponding problem within the framework of the ffldV equation $[6,28]$ .

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