New exact solutions for a generalization of the Korteweg–de Vries equation (KdV6)

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Abstract

The generalized tanh–coth method is used to construct periodic and soliton solutions for a new integrable system, which has been derived from an integrable sixth-order nonlinear wave equation (KdV6). The system is formed by two equations. One of the equations may be considered as a Korteweg–de Vries equation with a source and the second equation is a third-order linear differential equation.

Introduction

As we know, the nonlinear evolution and wave equations have been studied with much interest in the last two decades. The modern theory of solitary waves plays a very important role in the study of these equations providing methods to solve them. Some analytical methods such as the inverse scattering method [1], Hirota bilinear method [2], Bäcklund transformations [3] and Painlevé analysis [4], have been used in a satisfactory way to solve nonlinear partial differential equations. Other methods that have been also used to obtain exact solutions to these models are the tanh method [5], generalized tanh method [6], [7], tanh–coth method [8], generalized tanh–coth method [9], projective Riccati equations method [10], and the generalized projective Riccati equations method [11], [12].

In this paper we consider the Korteweg–de Vries equation with a source satisfying a third-order differential equationvt+vxxx+12vvx-wx=0wxxx+8vwx+4wvx=0,from the point of view of constructing exact solutions for it. The system (1.1) has been derived from the new integrable sixth-order KdV equationx3+8uxx+4uxxut+uxxx+6ux2=0,by Karasu-Kalcanli et al. [13]. With the change of variablev(x,t)=12u(x,-t)w(x,t)=12w(x,t),the system (1.1) reduces tout-6uux-uxxx+wx=0wxxx+4uwx+2uxw=0.

Recently, Kupershmidt [14] has showed that the system (1.4) is integrable in the usual sense, which is a remarkable fact, since w=0 leaves only the unperturbed KdV itself. Using the generalized tanh–coth method [9], we obtain new exact solutions to (1.4) and therefore, new exact solutions to (1.2), (1.1) can be found. The paper is organized as follows: in Section 2 we review briefly the generalized tanh–coth method. In Section 3 we give the mathematical framework to search exact solutions to (1.4). Finally, some conclusions are given.

Section snippets

The generalized tanh–coth method

Given the nonlinear partial differential equation (NLPDE)P(u,ux,ut,uxx,uxt,utt,)=0,we assume that (2.5) has traveling wave solution in the formu(x,t)=v(ξ),ξ=x+λt+ξ0,where ξ is the independent variable of the traveling wave hypothesis, λ is the wave velocity and ξ0 is a constant which indicates displacement phase.

In this case, (2.5) reduces to ordinary differential equation in the function v(ξ)P(v,v,v,)=0,where “′” is the derivative with respect to ξ. Using the idea of tanh–coth method

Exact solutions to new system

Using the transformationu(x,t)=v(ξ)w(x,t)=w(ξ)ξ=x+λt+ξ0,the system (1.4) reduces to(λv(ξ)-3v2(ξ)-v(ξ)+w(ξ))=0w(ξ)+4v(ξ)w(ξ)+2v(ξ)w(ξ)=0.

The second equation in (3.12) can be written asd(v(ξ)w2(ξ))=-12(w(ξ)w(ξ)).

Integrating (3.13) respect to ξ we havev(ξ)w2(ξ)=14(w(ξ))2-12w(ξ)w(ξ)+K,with K the integration constant. As we seek exact traveling wave solutions we assume that w(ξ)0,w(ξ)0 when ξ. Therefore we will take K=0. Then, from (3.14) finally we obtainv(ξ)=(w(ξ))2-2w(ξ)w(ξ)4w2(ξ).

Conclusions

Using the generalized tanh–coth method, we have found new exact solutions for the new integrable system (1.4). The method used here has some advantages. For instance: changing the value of k in (2.9) we obtain several forms of the solutions; the use of computer memory is minimum compared to other methods; the method can be also applied to other types of nonlinear wave equations.

Acknowledgments

The authors thanks the referees for providing valuable comments and suggestions.

References (14)

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    The so-called KdV6 equation [1] has recently generated much interest, and many of its properties have been studied, e.g., [2–5].

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