Superposition Formulas and Evolution Behaviors of Multi-Solutions to the (3+1)-Dimensional Generalized Shallow Water Wave-like Equation

Abstract

The superposition formulas of multi-solutions to the (3+1)-dimensional generalized shallow water wave-like Equation (GSWWLE) are proposed. There are arbitrary test functions in the superposition formulas of the mixed solutions and the interaction solutions, and we generalized to the sum of any N terms. By freely selecting the test functions and the positive integer N, we have obtained abundant solutions for the GSWWLE. First, we introduced new mixed solutions between two arbitrary functions and the multi-kink solitons, and the abundant mixed solutions were obtained through symbolic computation. Next, we constructed the multi-localized wave solutions which are the superposition of N-even power functions. Finally, the novel interaction solutions between the multi-localized wave solutions and the multi-arbitrary function solutions for the GSWWLE were obtained. The evolution behaviors of the obtained solutions are shown through 3D, contour and density plots. The received results have immensely enriched the exact solutions of the GSWWLE in the available literature.

Keywords

• mixed solution
• multi-localized wave solution
• interaction solution
• generalized bilinear equation
• (3+1)-dimensional generalized shallow water wave-like equation