Collision dynamics between breather and lump-type localized waves in the (3+1)-dimensional shallow water wave equation

Abstract

Abstract In this paper, the collision dynamics of breather and lump-type localized waves in the (3+1)-dimensional shallow water wave equation are investigated in detail. Firstly, the auto-Bäcklund transformation and the linear representation of the (3+1)-dimensional shallow water wave equation are derived in virtue of the truncated Painlevé expansion method, which provide convenience in solving the (3+1)-dimensional shallow water wave equation. Secondly, based on the linear representation and the principle of linear superposition, the rational solutions in the exponential and polynomial forms are constructed. Tuning the free parameters of the rational solutions, localized waves of various patterns are obtained such as breather, lump-type localized waves and their hybrid structure. The anomalous inelastic interaction phenomenons of breather and lump-type localized waves are exhibited. Thirdly, combining the large-time behaviors of solution with the velocity relationship of localized waves, the dynamical properties and the classification of localized wave solutions are discussed in detail. Finally, we discuss the bound state of breather and lump-type localized waves under the velocity resonance condition, three different types of lump-breather molecules are displayed. The obtained results further enrich the structures and dynamical behaviors of localized waves. It is expected that the interaction phenomena taking place in the (3+1)-dimensional shallow water wave equation will be helpful in predicting or controlling some related shallow water wave phenomena.