Eur. Phys. J. Plus (2018) 133: 144
https://doi.org/10.1140/epjp/i2018-11990-3

Regular Article

Nonlocal symmetry and explicit solutions from the CRE method of the Boussinesq equation

¹ Department of Mathematics, Harbin Institute of Technology, 150001, Harbin, China
² Department of Mathematics, University of British Columbia, V6T 1Z2, Vancouver, Canada

^* e-mail: zhaozlhit@163.com

Received: 28 July 2017
Accepted: 8 March 2018
Published online: 12 April 2018

Abstract

In this paper, we analyze the integrability of the Boussinesq equation by using the truncated Painlevé expansion and the CRE method. Based on the truncated Painlevé expansion, the nonlocal symmetry and Bäcklund transformation of this equation are obtained. A prolonged system is introduced to localize the nonlocal symmetry to the local Lie point symmetry. It is proved that the Boussinesq equation is CRE solvable. The two-solitary-wave fusion solutions, single soliton solutions and soliton-cnoidal wave solutions are presented by means of the Bäcklund transformations.