# Travelling Wave Solutions for the KdV-Burgers-Kuramoto and  Nonlinear Schrödinger Equations Which  Describe Pseudospherical Surfaces

Authors: S. M. Sayed; O. O. Elhamahmy; G. M. Gharib
Journal: Journal of Applied Mathematics
 (2008)

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## Abstract

We use the geometric notion of a differential system describing surfaces of a constant negative curvature and describe a family of pseudospherical
surfaces for the KdV-Burgers-Kuramoto and nonlinear Schrödinger equations with constant Gaussian curvature−1. Travelling wave solutions for the above equations are obtained by using a sech-tanh method and   Wu's elimination method.


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## Body

## 1. Introduction

The theory of
integrable systems has been an active area of mathematics for the  past thirty years. Different aspects of the subject have
fundamental relations with mechanics and dynamics, applied mathematics,
algebraic structures, theoretical physics, analysis including spectral theory
and geometry [1–3]. Most differential geometers have some knowledge and
experience with finite dimensional integrable systems as they appear in
sympectic geometry (mechanics) or ordinary differential equations (ODEs),
although the reformulation of part of this theory as algebraic geometry is not
commonly known [4].There are two
quite separate methods of extension of these ideas to partial differential
equations (PDEs): one based on algebraic constructions and one based on
spectral theory and analysis. These are less familiar still to geometers.Many geometric equations are known to have
integrable aspects, especially if one takes into account that most experts do
not have a good definition of “integrable” as applied to PDEs,
particularly elliptic examples. In addition to those we mention in our
historical discussion, the equations for harmonic maps (sigma models) from
surfaces into groups [5–7], harmonic tori in symmetric spaces [8],
constant mean curvature surfaces in space forms [9], isometric immersions of
space forms in other space forms [10, 11], and the theory of affine spheres [12]
and affine minimal surfaces [13] are all examples of “elliptic”
integrable systems.The ideas
surrounding string theory resulted in a series of deep and not completely
understood connections between representation theory of certain algebras and
many of the more classical theories of integrable systems in mathematics
[14–16]. Most recently, supersymmetric quantum field theories produce in a
natural way moduli spaces of vacua or ground states which have new geometry
generated by the supersymmetry. Since the supersymmetry generalizes the
classical symmetries which produce integrals for the Euler-Lagrange equations
via Noether's theorem, the connection with integrability is perhaps not
surprising [17–19]. However, this does not explain entirely the use of
integrable systems in hyper-Kähler geometry [20], Seiberg-Witten theory [21],
special Kähler geometry, and quantum cohomology [22].The 19th century geometers were mainly
interested in the local theory of surfaces inR3,
which we might regard as the prehistory of these modern constructions. The
sine-Gordon equation arose first through the theory of surfaces of constant
Gauss curvature −1,
and the reduced 3-wave equation can be found in Darboux's work
on triply orthogonal systems of R3 [23].It is well-known that a differential
equation (DE) for a real-valued functionu(x,t),
or a differential system for a 2-vector-valued function u(x,t),
is said to describe pseudospherical surfaces (pss) if it is the necessary and
sufficient condition for the existence of smooth real functions fij, 1≤i≤3, 1≤j≤2, depending only on u and a finite number of derivatives, such that
the one-forms(1.1)ωi=fi1dx+fi2dt,1≤i≤3,
satisfy the
structure equations of a surface of constant Gaussian curvature −1(1.2)dω1=ω3∧ω2,dω2=ω1∧ω3,dω3=ω1∧ω2.A DE for a real valued function u(x,t) is kinematically integrable if it is the
integrability condition of a one-parameter family of linear problems
[24–30](1.3)νx=P(η)ν,νt=Q(η)νin which P(η) and Q(η) are SL(2,R)-valued functions of x, t and (u and its derivatives) up to a finite order.
Thus, an equation is kinematically integrable if it is equivalent to the zero
curvature condition(1.4)∂P(η)∂t−∂Q(η)∂x+[P(η),Q(η)]=0,where trP(η)=trQ(η)=0, for each η (spectral parameter or eigenvalue). In
addition, a DE will be said to be strictly kinematically integrable if it is
kinematically integrable and diagonal entries of the matrix P(η) introduced above  are η and −η.The main aim of this paper is to explain the
relationships between local differential geometry of surfaces and integrability
of evolutionary nonlinear evolution equations (NLEEs). New travelling wave
solutions for the KdV-Burgers-Kuramoto (KBK) and non-linear Schrödinger (NLS)
equations which describe pss are obtained.The paper is organized as follows. The correspondence between KBK, NLS
equations and their families of pss is established in Section2. In Section 3,
a new exact soliton solution is obtained for the KBK equation by using a
sech-tanh method and Wu's elimination method. In Section 4, we construct a new
travelling wave solution for the NLS equation by using the same above way.
Finally, we give some conclusions in Section 5.


## 2. The KBK and NLS Equations that Describe pss

The inverse scattering method (ISM) was introduced
first for the Korteweg-de Vries equation (KdVE) [18]. Later it was extended by
Zakharov and Shabat [31] to a 2×2 scattering problem for the NLS equation and
that was subsequently generalized by Ablowitz, Kaup, Newell, and Segur (AKNS)
[32] to include a variety of NLEEs. Khater et al. [28] generalized the results
of Konno and Wadati [33] by considering ν as a three-component vector and Ω as a traceless 3×3 matrix one-form. The above definition of a DE
is equivalent to saying that the DE for u is the integrability condition for the problem(2.1)dν=Ων,ν=(ν1ν2ν3),where ν is a vector and the 3×3 matrix Ω (Ωij, i,j=1,2,3) is traceless(2.2)trΩ=0,and consists of a one-paramter (η), family of one-forms in the independent
variables (x,t), the dependent variable u, and its derivatives. Khater et al. [28]
introduced the inverse scattering problem (ISP):(2.3)ν1x=f31ν2−f11ν3,ν2x=−f31ν1−ην3,ν3x=−f11ν1−ην2,ν1t=f32ν2−f12ν3,ν2t=−f32ν1−f22ν3,ν3t=−f12ν1−f22ν2.The associated integrability
conditions for (1.3) or (2.1), which are obtained by cross-differentiation, then
take the matrix form(2.4)dΩ−Ω∧Ω=0,or the component
form(2.5)f12,x−f11,t=f31f22−ηf32,f22,x=f11f32−f12f31,f32,x−f31,t=f11f22−ηf12.We will restrict ourselves to the case wheref21=η.
More precisely, we say that  a DE for u(x,t) describes a pss if it is a necessary and
sufficient condition for the existence of functions fij, 1≤i≤3, 1≤j≤2, depending on u(x,t) and its derivatives, f21=η,
such that the one-forms in (1.1) satisfy the structure equations (1.2) of a pss. It
follows from this definition that for each nontrivial solution u of the DE, one gets a metric defined on M2,
whose Gaussian curvature is −1.It has been known, for a long time, that the
sine-Gordon (SG) equation describes a pss. In this paper, we extend the same
analysis to include the KBK and NLS equations.Examples: letM2 be a differentiable surface, parametrized by
coordinates x, t. (a) The KBK Equation
Consider(2.6)ω1=(−12u+g(x,t))dx+(12ux+14u2+f(x,t))dt,ω2=ηdx+(12ηu−ηg(x,t))dt,ω3=−ηdx+(−12ηu+ηg(x,t))dt, in which the functions g(x,t) and f(x,t) satisfy the equations(2.7)gx+g2+f=0,fx−gt=12(αuxx+βuxxx+γuxxxx).Then M2 is a pss if and only if u satisfies the KBK equation(2.8)ut+uux+αuxx+βuxxx+γuxxxx=0,where α, β, and γ are constants.(b) The NLS
Equation
Consider(2.9)ω1=2wdx+(−4ηw+2vx)dt,ω2=2ηdx+(2(v2+w2)−4η2)dt,ω3=−2vdx+(4ηv+2wx)dt. Then M2 is a pss if and only if u satisfies the NLS equation(2.10)iut+uxx+2|u|2u=0,whereu=v+iw.


## 3. Travelling Wave Solutions for the KBK Equation

Now we will
find travelling wave solutionsu(x,t) for the KBK equation (2.8). The solutions of
KBK equation possess their actual physical application; this is the reason why
so many methods, such as Wiss-Tabor-Carnevale transformation method [34],
tanh-function method [35], truncated expansion method [36], and so on, have
been applied to obtain the solution of KBK equation. In this section, we obtain
a new travelling wave solution class for KBK equations by using a sech-tanh
method [37, 38] and Wu's elimination method [39]. The main idea of the algorithm
is as follows.
Suppose there is a PDE of the form(3.1)f(u,ux,ut,uxx,uxt,utt,…)=0,where f is a polynomial. By assuming travelling wave
solutions of the form(3.2)u(x,t)=ϕ(ρ),ρ=λ(x−kt+c),where k, λ are constant parameters to be determined, and c is an arbitrary constant, from the two
equations (3.1) and (3.2) we obtain an ODE
(3.3)f(ϕ′,ϕ′′,ϕ′′′,…)=0,where ϕ′=dϕ/dρ.
According to the sech-tanh method [37–41], we suppose that (3.3) has
the following formal travelling wave solution:(3.4)ϕ(ρ)=∑i=1nsechi−1ρ(Bisechρ+Aitanh⁡ρ)+A0,where A0,…,An and B1,…,Bn are constants to be determined. Then we proceed as follows. (i)
Equating the highest-order nonlinear term
and highest-order linear partial derivative in (3.3) yields the value of n.(ii)
Substituting (3.4) into (3.3), we obtain a polynomial equation involving tanhρsechiρ, sechiρ for i=0,1,2,…,n (with n being positive integer).(iii)
Setting the
constant term and coefficients ofsechρ,tanhρ,sechρtanhρ,sech2ρ,…,
in the equation obtained in (ii) to zero, we obtain a system of algebraic
equations about the unknown numbers k, λ, A0, Ai, Bi for  i=1,2,…,n.(iv)
Using the Mathematica and Wu's
elimination methods, the algebraic equations in (iii) can be solved. These yield the solitary wave solutions for
the system (3.3). We remark that the above method yields solutions that include
terms sechρ or tanhρ,
as well as their combinations. There are different forms of those obtained by
other methods, such as the homogenous balance method [42, 43]. We assume formal
solutions of the form(3.5)u(x,t)=ϕ(ρ),ρ=λ(x−kt+c),where k, λ are constant parameters to be determined
later, and c is an arbitrary constant. Substituting from
(3.5) and (2.8), we obtain an ODE(3.6)−kϕ′+ϕϕ′+λαϕ′′+βλ2ϕ′′′+γλ3ϕ′′′′=0.(i) We suppose that (3.6) has the
following formal solution:
(3.7)ϕ(ρ)=A0+A1sechρ+B1tanh⁡ρ+A2sech2ρ+B2sechρtanh⁡ρ+A3sech3ρ+B3sech2ρtanh⁡ρ.(ii) From (3.6) and (3.7), we get(3.8)−kϕ′+ϕϕ′+λαϕ′′+βλ2ϕ′′′+γλ3ϕ′′′′=(−A1B1−A0B2+kB2+λαA1−βλ2B2+γλ3A1)sechρ+(−A0A1−B1B2+kA1+αλB2−βλ2A1+γλ3B2)sechρtanh⁡ρ+(A0B1−2A2B1−2A1B2−2A0B3−kB1+2kB3+4αλA2+4βλ2B1−8βλ2B3+16γλ3A2)sech2ρ+(−A12−2A0A2+B12−B22−2B1B3+2kA2−2αλB1+4αλB3−8βλ2A2−8γλ3B1+16γλ3B3)sech2ρtanh⁡ρ+(2A1B1−3A3B1+2A0B2−3A2B2−3A1B3−2kB2−2αλA1+9αλA3+20βλ2B2−20γλ3A1+81γλ3A3)sech3ρ+(−3A1A2−3A0A3+3B1B2−3B2B3+3kA3−6αλB2+6βλ2A1−27βλ2A3−60γλ3B2)sech3ρtanh⁡ρ+(3A2B1+3A1B2−4A3B2+3A0B3−4A2B3−3kB3−6αλA2−6βλ2B1+60βλ2B3−120γλ3A2)sech4ρ+(−2A22−4A1A3+2B22+4B1B3−2B32−12αλB3+24βλ2A2+24γλ3B1−240γλ3B3)sech4ρtanh⁡ρ+(4A3B1+4A2B2+4A1B3−5A3B3−12αλA3−24βλ2B2+24γλ3A1−408γλ3A3)sech5ρ+(−5A2A3+5B2B3+60βλ2A3+120γλ3B2)sech5ρtanh⁡ρ+(5A3B2+5A2B3−60βλ2B3+120γλ3A2)sech6ρ+(−3A32+3B32+360γλ3B3)sech6ρtanh⁡ρ+(6A3B3+360γλ3A3)sech7ρ=0.(iii) Setting the coefficients
ofsechjρtanhiρ for i=0,1 and j=1,2,…,7 to zero, we have the following set of
overdetermined equations in the unknowns A0, A1, A2, A3, B1, B2, B3, λ, and    k.(iv) We now solve the above set of equations (3.8) by
using Mathematica and Wu's elimination method, and obtain the following solutions:
(3.9)λ2=(5±19)α64γ,β2=16αγ,k=5βλ2+αβ4γ,A0=A1=B1=B2=A3=0,A2=15βλ2,B3=−120γλ3.Substituting (3.9) into (3.7), we
obtain(3.10)u(x,t)=15λ2sech2ρ(β−8γλtanh⁡ρ),whereρ=λ(x−kt+c).


## 4. Travelling Wave Solutions for the NLS Equation

Now we will
find travelling wave solutionsu(x,t) for the NLS equation (2.10). Equation (2.10) can
be written in the real form
u=v+iw as follows:
(4.1)vt+wxx+2(v2+w2)w=0,−wt+vxx+2(v2+w2)v=0.We assume formal solutions of
the form(4.2)v(x,t)=ϕ(ρ),w(x,t)=θ(ρ),ρ=λ(x−kt+c),where k, λ are constant parameters to be determined
later, and c is an arbitrary constant. Substituting from
(4.2) into (4.1), we obtain two ODEs:(4.3)−kλϕ′+λ2θ′′+2(ϕ2+θ2)θ=0,kλθ′+λ2ϕ′′+2(ϕ2+θ2)ϕ=0.(i) Equating the highest-order
nonlinear term and highest-order linear partial derivative in (4.3) yields n=1.
Then (4.3) has the following formal solutions:(4.4)ϕ(ρ)=A0+A1sechρ+B1tanh⁡ρ,θ(ρ)=a0+a1sechρ+b1tanh⁡ρ.(ii) With the
aid of Mathematica, substituting (4.4) into (4.3), then we obtain a polynomial
equation involving tanhiρsechjρ for i=0,1,  j=0,1,2,3.(4.5)−kλϕ′+λ2θ′′+2(ϕ2+θ2)θ=(2a03+2a0A02+4b1A0B1+6a0b12+2a0B12)+(λ2a1+6a02a1+2a1A02+4a0A0A1+6a1b12+4b1A1B1+2a1B12)sechρ+(2b13+6a02b1+4a0A0B1+2b1A02+2b1B12)tanh⁡ρ+(kλA1+12a0a1b1+4b1A0A1+4a1A0B1+4a0A1B1)sechρtanh⁡ρ+(6a0a12+4a1A0A1+2a0A12−6a0b12−kλB1−4b1A0B1−2a0B12)sech2ρ+(−2λ2b1+6a12b1−2b13+2b1A12+4a1A1B1−2b1B12)sech2ρtanh⁡ρ+(−2λ2a1+2a13+2a1A12−6a1b12−4b1A1B1−2a1B12)sech3ρ=0,kλθ′+λ2ϕ′′+2(ϕ2+θ2)ϕ=(2a02A0+6A0B12+2A0B12+4a0b1B1+2A03)+(λ2A1+6A02A1+2A1b12+4a0a1A0+6A1B12+4a1b1B1+2A1a02)sechρ+(2B13+6A02B1+4a0A0b1+2B1a02+2b12B1)tanh⁡ρ+(−kλa1+12A0A1B1+4a1A0b1+4a0a1B1+4a0A1b1)sechρtanh⁡ρ+(6A0A12+4a0a1A1+2A0a12−6A0B12+kλb1−4b1a0B1−2A0b12)sech2ρ+(−2λ2B1+6A12B1−2B13+2B1a12+4a1A1b1−2B1b12)sech2ρtanh⁡ρ+(−2λ2A1+2A13+2a12A1−6A1B12−4b1a1B1−2A1b12)sech3ρ=0.(iii) Setting
the constant term and coefficients oftanhiρsechjρ for i=0,1,  j=0,1,2,3,
in the equation obtained in (ii) to zero, we obtain a system of algebraic
equations about the unknown numbers A0, A1, B1, a0, a1, b1,
and k.(iv) Now we
solve the above set of (4.5) by using Mathematica and Wu's elimination
method, and we obtain the following
solution:(4.6)A0=A1==a1=b1=0,a0=±λB1=±iλ,k=∓⁢iλ.Substituting (4.6) into (4.4), we
obtain(4.7)v(x,t)=ϕ(ρ)=±iλtanh⁡ρ,w(x,t)=θ(ρ)=±λ,then the solution of NLS
equation (2.10) takes the form(4.8)u(x,t)=±iλ(1+tanh⁡ρ),ρ=λ(x±iλt+c).


## 5. Conclusions

We find the relationship between KBK, NLS equations
and their families of pss. With the help of Mathematica, many travelling
solutions for the KBK and NLS equations of the pseudospherical class
are obtained by using a sech-tanh method and Wu's
elimination method. We obtained some new solitary wave solutions and periodic
solutions.



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