Multiply Kink and Anti-Kink Solutions for a Coupled Camassa–Holm Type Equation**Supported by the National Natural Science Foundation of China under Grant No. 11261037, the Natural Science Foundation of Inner Mongolia Autonomous Region under Grant No. 2014MS0111, the Caoyuan Yingcai Program of Inner Mongolia Autonomous Region under Grant No. CYYC2011050, the Program for Young Talents of Science and Technology in Universities

Abstract

This article amis at revealing dynamical behavior of a coupled Camassa–Holm type equation, which was proposed by Geng and Wang based on a 4 × 4 matrix spectral problem with two potentials. Its kink and anti-kink solutions are presented explicitly. In particular, some exact multi-kink and anti-kink wave solutions are discussed and under some conditions, the kink and anti-kinks look like hat-shape solitons. The dynamic characters of the obtained solutions are investigated by figures. The method used in this paper can be widely applied to looking for the multi-kinks for Camassa–Holm type equations possessing cubic nonlinearity.