Exact soliton solutions for the fifth-order Sawada–Kotera equation

doi:10.1016/j.amc.2008.08.028

Applied Mathematics and Computation

Volume 206, Issue 1, 1 December 2008, Pages 272-275

https://doi.org/10.1016/j.amc.2008.08.028 Get rights and content

Abstract

Exact soliton solutions for the fifth-order Sawada–Kotera equation are obtained by using the Hirota bilinear method. These solutions include one-soliton solutions, periodic two-soliton solutions and singular periodic soliton solutions. The results show that there exist periodic two-soliton solutions and singular periodic soliton solutions for the fifth-order Sawada–Kotera equation.

Introduction

Integrable and partially integrable nonlinear partial differential equations (NPDE) have attracted much attention of mathematicians as well as physicists for the last 40 years. The investigation of exact travelling wave solutions to nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. Solitons are the most important solutions among travelling wave solutions. The existence of multi-soliton, especially two-soliton solutions, is crucial for information technology: it makes it possible undisturbed simultaneous propagation of many pulses in both directions. The Hirota bilinear method [1], [2], [3] and its multilinear refinements provide simple tools of construction for such solutions, if they exist. The Hirota bilinear method relies on a transformation for considered equation. It is not easy for us to find such a transformation for some equations and sometimes it requires the introduction of new dependent and sometimes even independent variables. Most equations (even non-integrable ones) having Hirota bilinear form possess automatically one-soliton or multi-soliton solutions, such as the KdV equation and the Kadomtsev–Petviashvili (KP) equation [4], [5], [6], [7], [8], [9], [10]. In general, exact solutions including various type of solitary wave solutions and periodic solutions are functions of single variable $ξ = kx - wt$ or $ξ = px + qy - wt$ .

Recently, Dai et al. introduced a new technique, extended homoclinic test technique which is used seeking periodic solitary wave solutions of integrable equation and obtained a new periodic solitary wave solutions for the classical KdV equation [7]. In this paper, based On this idea, we introduce a new ansätz function $f (ξ, η)$ which is a function of two variable $ξ = Qx + kt$ and $η = px - wt$ and consider (1 + 1)-dimensional fifth-order Sawada–Kotera equation by the Hirota bilinear method. We obtain some new exact solutions including one-soliton solution, periodic two-soliton solutions and singular periodic soliton solutions, respectively.

The rest of this paper is organized as follows: in Section 2, procedure for solving the fifth-order Sawada–Kotera equation is given. Finally, some conclusions are given.

Section snippets

Procedure for solving the fifth-order Sawada–Kotera equation

We consider the fifth-order Sawada–Kotera equation $u_{t} + 45 u^{2} u_{x} + 15 (u_{x} u_{xx} + u_{xxx} u) + u_{xxxxx} = 0 .$ Eq. (1) is transformed into the bilinear form $D_{x} (D_{t} + D_{x}^{5}) f \cdot f = 0,$ through the transformation $u = 2 \ln (f)_{xx},$ where bilinear operator $D_{x}^{m} D_{t}^{n}$ is defined as $D_{x}^{m} D_{t}^{n} a \cdot b = {(\frac{\partial}{\partial x} - \frac{\partial}{\partial x^{'}})}^{m} {(\frac{\partial}{\partial t} - \frac{\partial}{\partial t^{'}})}^{n} a (x, t) b (x^{'}, t^{'}) |_{x = x^{'}, t = t^{'}} .$ Eq. (2) can be rewritten as $2 f_{xt} f - 2 f_{x} f_{t} + 2 f_{xxxxxx} f - 12 f_{xxxxx} f_{x} + 30 f_{xxxx} f_{xx} - 20 f_{xxx}^{2} = 0 .$ We let the function f be expressed in the form $f (x, t) = b_{1} e^{(Qx + kt)} + b_{2} \cos (px - wt) + b_{3} e^{- (Qx + kt)}$ Substituting Eq. (6) into Eq. (3), we get $u (x, t) = b_{2} ($

Conclusion

The fifth-order Sawada–Kotera equation is investigated by the Hirota bilinear method. One-soliton solution, periodic two-soliton solutions and singular periodic soliton solutions of this equation are obtained. These results show that there exist one-soliton solution, periodic two-soliton solutions and singular periodic soliton solutions for the fifth-order Sawada–Kotera equation.

Acknowledgement

Authors would like to thank referee for valuable suggestion and help.

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^☆: The project supported by Chinese Natural Science Foundation (Nos. 10361007, 10661002), Yunnan Natural Science Foundation (No. 2004A0001M) and Yunnan Educational Science Foundation (No. 06Y041A).

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Exact soliton solutions for the fifth-order Sawada–Kotera equation☆

Abstract

Introduction

Section snippets

Procedure for solving the fifth-order Sawada–Kotera equation

Conclusion

Acknowledgement

Appl. Math. Comput.

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Appl. Math. Comput.

Exact solution of the KdV equation for multiple collisions of solitons

Phys. Rev. Lett.