Soliton solutions of the (2+1)-dimensional Kaup system for water waves

Soliton solutions in Gram determinant of the (2+1)-dimensional Kaup system have been studied based on the Kadomtsev-Petviashvili hierarchy reduction method. Such system has been used to describe the water waves propagating in an infinite narrow channel of constant mean depth. Propagation of one soliton and interactions between two solitons have been discussed. u and v components are shock profiles, while the R component presents the bell-shaped or the depression profiles. There are three types of the elastic interactions between two solitons of R component: bell-bell, bell-depression and depression-depression. Soliton fission and fusion have also been addressed, and two solitons shares the Y-type resonance. N-soliton solutions in Wronskian determinant and the corresponding Maya diagrams have also been presented. Finally, a conditionally stable finite difference scheme is proposed to numerically study solitons and verify that the soliton solution is stable.