A new method for finding traveling wave solutions of the generalized KdV equation with variable coefficient

doi:10.1016/j.amc.2007.09.065

Applied Mathematics and Computation

Volume 199, Issue 1, 15 May 2008, Pages 268-274

https://doi.org/10.1016/j.amc.2007.09.065 Get rights and content

Abstract

In this paper, a transformation between two ansätz equations is presented and applied to discuss the generalized KdV equation with variable coefficient. A lot of exact traveling wave solutions are obtained in terms of elliptic functions. The method can also be used to study other nonlinear partial differential equations.

Introduction

Now we consider the generalized KdV equation with variable coefficient [1], [6] as follows: $u_{t} + 2 β (t) u + [α (t) + β (t) x] u_{x} - 3 η γ (t) {uu}_{x} + γ (t) u_{xxx} = 0 .$ During the past several years, quite a few papers studied the traveling wave solutions of Eq. (1) [1], [2], [3], [4], [5], [6]. Many nonlinear equations of mathematical physics that describe some important physical and dynamic processes are the special cases of Eq. (1). For example, when $γ (t) = - K_{0} (t), η = - 2, β (t) = h (t), α (t) = - 4 K_{1} (t)$ , Eq. (1) can be degenerated to the following variable coefficient and nonisospectral KdV equation [2], [3]: $u_{t} = K_{0} (t) (u_{xxx} + 6 {uu}_{x}) + 4 K_{1} (t) u_{x} - h (t) (2 u + {xu}_{x}),$ and when $γ (t) = 1, η = - 2, α (t) = C_{0}$ , Eq. (1) can also be degenerated to the following variable coefficient KdV equation: $u_{xxx} + 6 {uu}_{x} + [(C_{0} + β (t) x) u]_{x} + β (t) u + u_{t} = 0 .$

In [6], we obtain some new soliton-like solutions of Eq. (1) by the improved Jacobi elliptic function expansion method. In this paper, we will use a transformation of two ansätz equations to find traveling wave solutions of Eq. (1).

Section snippets

Solutions of a ansätz equation

In this section, we will discuss the following ansätz equation: $(ψ^{'} (ξ))^{2} = ψ^{3} (ξ) + E ψ^{2} (ξ) + F ψ (ξ) + G$ with the Jacobi elliptic function. Let $f (ψ) = ψ^{3} (ξ) + E ψ^{2} (ξ) + F ψ (ξ) + G$ , then we consider the following four cases:

(i)
If $f (ψ)$ has a double real zero at $ψ = A_{1}$ and a simple real zero at $ψ = A_{2}$ , then $f (ψ) = (ψ - A_{1})^{2} (ψ - A_{2})$ .
(ii)
If $f (ψ)$ has three real zeros $B_{1}, B_{2}$ and $B_{3}$ , satisfying $B_{1} < B_{2} < B_{3}$ , then $f (ψ) = (ψ - B_{1}) (ψ - B_{2}) (ψ - B_{3})$ .
(iii)
If $f (ψ)$ has a real simple zero A and two complex zeros with negative realpants, then $f (ψ) = (ψ - A) [(ψ + B)^{2} + C^{2}]$ ,

Solutions of the generalized KdV equation with variable coefficient

Considering homogeneous balance between ${uu}_{x}$ and $u_{xxx}$ in (1), we assume that the solution of Eq. (1) is of the form $u = g_{0} + g_{1} φ (ξ) + g_{2} φ^{2} (ξ),$ $ξ = px + q,$ ${(\frac{d φ}{d ξ})}^{2} = a φ^{4} + b φ^{2} + c,$ where $a$ , $b$ and c are constants, and $g_{0} = g_{0} (t), g_{1} = g_{1} (t), g_{2} = g_{2} (t), p = p (t)$ and $q = q (t)$ are functions of t to be determined. Substituting (9), (10), (11) into (1) and collecting coefficients of $φ^{k} φ^{(l)}, k = 0, 1, 2, 3, l = 0, 1$ , then setting each coefficients to zero, we have the following set of over-determined equations: $2 β g_{0} + g_{0}^{'} = 0,$ $2 β g_{1} + g_{1}^{'} = 0,$ $2 β g_{2} + g_{2}^{'} = 0,$ $g_{1} [c$

Conclusions

A transformation between two ansätz equations is presented and applied to the study of the generalized KdV equation with variable coefficient and many new exact traveling wave solutions are obtained. Our method can be used to solve other nonlinear partial differential equations.

Acknowledgement

This work has been supported by the National Natural Science Foundation of China (19972026).

References (7)

T. Brugarino et al.
Phys. Lett. A
(1980)
X. Zhao et al.
Phys. Lett. A
(2005)
Z.S. Feng
Chaos, Solitons Fractals
(2005)

There are more references available in the full text version of this article.

Cited by (5)

Integrable properties for a generalized non-isospectral and variable-coefficient Korteweg-de Vries model
2010, Modern Physics Letters B
Darboux transformation for the nonisospectral and variable-coefficient KdV equation
2017, arXiv
The extended auxiliary equation method for the KdV equation with variable coefficients
2011, Chinese Physics B
Explicit solutions of (2+1)-dimensional canonical generalized KP, KdV, and (2+1)-dimensional burgers equations with variable coefficients
2009, Communications in Theoretical Physics
N-soliton solution in Wronskian form for a generalized variable-coefficient Korteweg-de Vries equation
2008, Chinese Physics Letters

View full text

Applied Mathematics and Computation

A new method for finding traveling wave solutions of the generalized KdV equation with variable coefficient

Abstract

Introduction

Section snippets

Solutions of a ansätz equation

Solutions of the generalized KdV equation with variable coefficient

Conclusions

Acknowledgement

Phys. Lett. A

Phys. Lett. A

Chaos, Solitons Fractals

Integrable properties for a generalized non-isospectral and variable-coefficient Korteweg-de Vries model

Darboux transformation for the nonisospectral and variable-coefficient KdV equation

The extended auxiliary equation method for the KdV equation with variable coefficients

Explicit solutions of (2+1)-dimensional canonical generalized KP, KdV, and (2+1)-dimensional burgers equations with variable coefficients

N-soliton solution in Wronskian form for a generalized variable-coefficient Korteweg-de Vries equation