Eur. Phys. J. Plus (2016) 131: 98
https://doi.org/10.1140/epjp/i2016-16098-2

Regular Article

On Lie symmetries, exact solutions and integrability to the KdV-Sawada-Kotera-Ramani equation

¹ Department of Mathematics, China University of Mining and Technology, 221116, Xuzhou, People’s Republic of China
² Center of Nonlinear Equations, China University of Mining and Technology, 221116, Xuzhou, People’s Republic of China
³ Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CB3 0WA, Cambridge, UK

^* e-mail: sftian@cumt.edu.cn
^** e-mail: ttzhang@cumt.edu.cn
^*** e-mail: zhangxingyong163@163.com

Received: 24 November 2015
Revised: 16 January 2016
Accepted: 28 February 2016
Published online: 15 April 2016

Abstract

In this paper, the KdV-Sawada-Kotera-Ramani equation is investigated, which is used to describe the resonances of solitons in one-dimensional space. By using the Lie symmetry analysis method, the vector field and optimal system of the equation are derived, respectively. The optimal system is further used to study the symmetry reductions and exact solutions. Furthermore, the exact analytic solutions of the equation can be obtained by considering the power series theory. Finally, the complete integrability of the equation is systematically presented by using binary Bell’s polynomials, which includes the bilinear representation, bilinear Bäcklund transformation, Lax pair and infinite conservation laws. Based on its bilinear representation, the N-soliton solutions of the equation are also constructed with exact analytic expression.