Soliton-shape-preserving and soliton-complex interactions for a (1+1)-dimensional nonlinear dispersive-wave system in shallow water

Abstract

Under investigation in this paper is a (1+1)-dimensional nonlinear dispersive-wave system for the long gravity waves in shallow water. With symbolic computation, we derive the multi-soliton solutions for the system. Four sorts of interactions for the system are discussed: (1) Soliton shape preserving, in which two solitons undergo the fusion behavior while the amplitudes and velocities of the other two remain unchanged during the interaction process;(2) Head-on collisions between the two-soliton complexes; (3) Overtaking collisions between the twosoliton complexes; (4) Two-soliton complexes formed by the inelastic collisions. Such soliton structures might be of certain value in fluid dynamics.

On head-on collision between two solitary waves in shallow water: the use of the extended PLK method

“…To determine the unknown phase shift functions, in their analysis they made the statement that "although certain terms do not cause any secularity at this order but they will cause secularity at the higher order expansion, therefore, those terms must vanish". Several researchers, utilizing the implication of this statement studied the head-on-collision of solitary wave problems in various media [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Unfortunately, our calculations for the higher order expansion show that the terms mentioned in their work do not cause any secularity in the solution.…”

“…Inserting (11) and (13) into equations (8) and (9) and setting the coefficients of like powers of equal to zero the following equations are obtained O ( ) equations:…”