Soliton molecules, novel hybrid interaction solutions and periodic wave solutions to the (3 + 1)-dimensional nonlinear evolution equation for shallow-water waves

Abstract

The central target of this research is looking into some novel solutions of the (3 + 1)-dimensional nonlinear evolution equation (NEE) for the shallow water waves. By manipulating the Hirota bilinear approach (HBA), the N-soliton solutions(N-SSs) are extracted. Then applying the resonance condition on the N-SSs, we successfully derive the soliton molecules (SMS) on the (x,y), (x,z) and (y,z)-planes. In addition, abundant novel hybrid interaction solutions are also studied by assigning the reasonable parameters, including the interaction between 1-soltion and the breather solution, interaction between 1-soltion and the soliton molecule(SM) with two solitons, interaction between two-breather solutions, interaction between breather solution and the SM of two solitons, interaction between two SMS of two solitons, interaction bwtween 1-soliton and the SM of three solitons. In the end, the periodic wave solutions (PWSs) are explored by taking advantage of the test function approach and symbolic calculation. The dynamic attributes of the obtained solutions are described graphically to unveil their physical behaviors. The discoveries in this work can enable us better understand the nonlinear dynamics of the considered equation.