Abundant exact and explicit solitary wave and periodic wave solutions to the Sharma–Tasso–Olver equation
Introduction
In this paper, we consider the following double nonlinear dispersive, integrable equationwhere α is a real parameter and is the unknown function depending on the temporal variable t and the spatial variable x. This equation contain both linear dispersive term and the double nonlinear terms and . 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 equationswhere are two constants. When , we can obtain the following first integral as given
- Step 1.
Given nonlinear partial differential equation, for instance, in two variables, as follows: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 transformationwhere are two constants to be determined later and is an arbitrary constant, and thus Eq. (1) becomesIntegrating the above equations once with regard to ξ, one getswhere 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)
- et al.
Soliton fission and fusion: Burgers equation and Sharma–Tasso–Olver equation
Chaos Solitons Fract.
(2004) New solitons and kinks solutions to the Sharma–Tasso–Olver equation
Appl. Math. Comput.
(2007)The extended hyperbolic function method and exact solutions of the long–short wave resonance equations
Chaos Solitons Fract.
(2008)
Cited by (24)
Auto-Bäcklund transformation for some nonlinear partial differential equation
2016, OptikCitation Excerpt :Jawad et al. [17] obtained exact solutions for STO equation by using modified simple equation method. Shang et al. [18] presented exact explicit travelling wave solutions of STO equation by using the extended hyperbolic function method. Burgers equation is used for mathematical modeling of fluid mechanics, nonlinear acoustic gas dynamics, traffic flow, the theory of shock waves, turbulence problems.
Special polynomials associated with the Burgers hierarchy
2012, Applied Mathematics and ComputationThe Bäcklund transformations and abundant explicit exact solutions for the AKNS-SWW equation
2011, Communications in Nonlinear Science and Numerical SimulationCitation Excerpt :Fan Engui and Zhang Hongqing extended the homogeneous balance method and proposed an approach to obtaining Bäcklund transformation for the nonlinear evolution equations [17]. In a recent paper [18], Shang obtained the Bäcklund transformation and a Lax pair and some new explicit exact solutions of Hirota–Satsuma SWW Eq. (4) by means of the Bäcklund transformations and the extension of the hyperbolic function method presented in [19]. The purpose of this paper is to investigate the AKNS–SWW Eq. (3) by combining the extended homogeneous balance method with the extended hyperbolic function method.
Bäcklund transformations and abundant exact explicit solutions of the Sharma-Tasso-Olver equation
2011, Applied Mathematics and ComputationCitation Excerpt :By the Bäcklund transformation obtained in Section 2 and the Airy function, the exact explicit solution of the initial value problem for the STO equation is obtained. The results presented herein greatly extend known results in previous works [8–17]. In this section, we will utilize the Bäcklund transformations obtained in Section 2 and the computer program Maple 12 to obtain abundant exact explicit solutions to the STO Eq. (1).
Reduction of the Sharma-Tasso-Olver equation and series solutions
2011, Communications in Nonlinear Science and Numerical SimulationNew abundant exact solutions for the (2 + 1)-dimensional generalized Nizhnik-Novikov-Veselov system
2010, Communications in Nonlinear Science and Numerical Simulation