Symbolic computation on the long gravity water waves: scaling transformations, bilinear forms, [formula omitted]-soliton solutions and auto-Bäcklund transformation for a variable-coefficient variant Boussinesq system

Abstract

Water waves are one of the most common phenomena in nature. Hereby, on a variable-coefficient variant Boussinesq system for the nonlinear and dispersive long gravity waves travelling in two horizontal directions in the shallow water with varying depth, with respect to the horizontal velocity of the water and height deviating from the equilibrium position of the water, our symbolic computation leads to the scaling transformations, bilinear forms, N-soliton solutions and auto-Bäcklund transformation with the sample solitons, where N is a positive integer. Our results are dependent on the water-wave variable coefficients and under the relevant variable-coefficient constraints.