Hetero-Bäcklund transformation, bilinear forms and multi-solitons for a (2＋1)-dimensional generalized modified dispersive water-wave system for the shallow water

Abstract

This shallow-water-directed paper plans to consider a (2＋1)-dimensional generalized modified dispersive water-wave (2DGMDWW) system, which describes the nonlinear and dispersive long gravity waves travelling along two horizontal directions in the shallow water of uniform depth. With symbolic computation, (1) a hetero-Bäcklund transformation is constructed, coupling the solutions as for the 2DGMDWW system with the solutions as for a known (2＋1)-dimensional Boiti-Leon-Pempinelli system describing the water waves in an infinitely narrow channel of constant depth, with that hetero-Bäcklund transformation dependent on the shallow-water coefficients in the 2DGMDWW system, with the former solutions indicating certain shallow-water-wave patterns for the height of the water surface and the horizontal velocity of the water wave, while with the latter solutions related to the horizontal velocity and elevation of the water wave; (2) two sets of the bilinear forms are obtained, each set of which is shown to depend on the shallow-water coefficients in the 2DGMDWW system and to be linked to certain shallow-water-wave patterns for the height of the water surface and the horizontal velocity of the water wave; and (3) two sets of the $N$-soliton solutions are also worked out, each set of which is seen to rely on the shallow-water coefficients in the 2DGMDWW system and to represent the existence of $N$-solitonic shallow-water-wave patterns with respect to the height of the water surface and the horizontal velocity of the water wave, with $N$ as a positive integer.