Abundant exact and explicit solitary wave and periodic wave solutions to the Sharma–Tasso–Olver equation

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Abstract

This paper is concerned with a double nonlinear dispersive equation—the Sharma–Tasso–Olver equation. The extended hyperbolic function method is employed to investigate the solitary and periodic travelling waves in this equation. With the aid of Mathematica and Wu-elimination Method, the abundant exact explicit solutions of the nonlinear Sharma–Tasso–Olver equation are derived. The solutions obtained in this paper include (a) solitary wave solutions, (b) the singular travelling wave solutions, and (c) periodic travelling wave solutions of triangle function types. Several entirely new exact solutions to the equation are explicitly obtained, in addition to deriving all known solutions in a systematic way. This work can be regard as an extension to the recent work by Wazwaz.

Introduction

In this paper, we consider the following double nonlinear dispersive, integrable equationut+α(u3)x+32α(u2)xx+αuxxx=0,where α is a real parameter and u(x,t) is the unknown function depending on the temporal variable t and the spatial variable x. This equation contain both linear dispersive term αuxxx and the double nonlinear terms α(u3)x and 32α(u2)xx. Eq. (1) be called Sharma–Tasso–Olver equation in literatures. Many physicists and mathematicians have pay their attentions to the Sharma–Tasso–Olver equation in recently years due to its appearance in scientific applications. In [1], Yan investigated the Sharma–Tasso–Olver equation (1) by using the cole-Hopf transformation method. The simple symmetry reduction procedure is repeatedly used in [2] to obtain exact solutions where soliton fission and fusion were examined. Wang et al. examined the soliton fission and fusion thoroughly by means of the Hirota’s bilinear method and the Bäcklund transformation method in [3]. More recently, the Sharma–Tasso–Olver equation (1) was treated analytically by the tanh method, the extended tanh method, and other ansatze involving hyperbolic and exponential functions in [4]. The multiple solitons and kinks solutions are obtained.

The aim of this paper is to derive more exact solitary wave solutions and periodic wave solutions of the Sharma–Tasso–Olver equation. We will employ the extended hyperbolic function method presented in [5] to solve Sharma–Tasso–Olver equation. Some entirely new exact solitary wave solutions and periodic wave solutions of the Sharma–Tasso–Olver equation are obtained. The method and result can be regarded as an extension of the recent works by Wazwaz.

The paper is organized as follows: in Section 2, we briefly describe what is the extended hyperbolic function method and how to use it to derive the travelling solutions of nonlinear PDEs. In Section 3, we apply the extended hyperbolic function method to the Sharma–Tasso–Olver equation and establish many rational form solitary wave, rational form triangular periodic wave solutions. In last Section, we briefly make a summary to the results that we have obtained.

Section snippets

The extended hyperbolic function method

Now we would like to outline the main steps of our method:

Consider the coupled Riccati equationsf(ξ)=-f(ξ)g(ξ),g(ξ)=Rε-rRεf(ξ)-g2(ξ),where ε=±1,R,r are two constants. When R0, we can obtain the following first integral as giveng2(ξ)=Rε-2rRεf(ξ)+Cf2(ξ),

  • Step 1.

    Given nonlinear partial differential equation, for instance, in two variables, as follows:P(u,ux,ut,uxx,uxt,)=0,where P is in general a nonlinear function of its arguments, the subscripts denote the partial derivatives. We seek for the

Exact solutions of the Sharma–Tasso–Olver equation

We now apply the extended hyperbolic function method presented above on the STO equation. According to the above method, to seek travelling wave solutions of (1), we make the transformationu(x,t)=u(ξ),ξ=kx+ωt+ξ0,where k,ω are two constants to be determined later and ξ0 is an arbitrary constant, and thus Eq. (1) becomesωu(ξ)+αk(u3)(ξ)+3α2k2(u2)(ξ)+αk3u(ξ)=0.Integrating the above equations once with regard to ξ, one getsωu(ξ)+αku3(ξ)+3α2k2(u2)(ξ)+αk3u(ξ)=E,where E is an integration constant.

Summary and conclusions

In summary, by use of the extended hyperbolic function method, some more general forms of exact explicit travelling wave solutions for the STO equation are obtained. We not only obtain all known exact solitary wave solutions, periodic wave solutions, and singular travelling wave solutions but also find some new exact solitary wave solutions, singular travelling wave solutions, and periodic travelling wave solutions of triangle function. The method presented here can also be regarded as an

Acknowledgements

The authors would like to express their thanks to Professor Changzheng Qu for his encouragements and helpful suggestions. This work is supported by the National Natural Science foundation of China (10771041) and the Scientific Project Program of Guangzhou Education Bureau (No. 62035).

References (6)

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