Multiple periodic-soliton solutions of the $$(3+1)$$ ( 3 + 1 ) -dimensional generalised shallow water equation

Abstract

Based on the extended variable-coefficient homogeneous balance method and two new ansatz functions, we construct auto-Backlund transformation and multiple periodic-soliton solutions of $$(3\,{+}\,1)$$ -dimensional generalised shallow water equations. Completely new periodic-soliton solutions including periodic cross-kink wave, periodic two-solitary wave and breather type of two-solitary wave are obtained. In addition, cross-kink three-soliton and cross-kink four-soliton solutions are derived. Furthermore, propagation characteristics and interactions of the obtained solutions are discussed and illustrated in figures.