Bilinear form, auto-Bäcklund transformations, Pfaffian, soliton, and breather solutions for a (3 + 1)-dimensional extended shallow water wave equation

Abstract

Study of the water waves remains central to fluid physics, ocean dynamics, and engineering. In this paper, a (3 + 1)-dimensional extended shallow water wave equation is investigated via symbolic computation. Bilinear form and two kinds of the bilinear auto-Bäcklund transformations with some solutions are given via the Hirota method. The Nth-order Pfaffian solutions are worked out by means of the Pfaffian technique, where N is a positive integer. N-soliton solutions are exported through the Nth-order Pfaffian solutions. By virtue of the asymptotic analysis, elastic and inelastic interactions between the two solitons on some periodic backgrounds are discussed. Interaction among the three solitons is illustrated graphically. The higher-order breather solutions are investigated via the complex parameter relation. Elastic and inelastic interactions between the two breathers on the periodic backgrounds are depicted graphically. Hybrid solutions consisting of the solitons and breathers are obtained. Interaction between the one soliton and one breather on a periodic background is presented.