Lump solutions to an integrable (3 + 1)-dimensional Boussinesq equation and its dimensionally reduced equations in shallow water

Abstract

This study uses the Hirota bilinear method and Maple, a symbolic computation program, to derive lump solutions for a new integrable (3 + 1)-dimensional Boussinesq equation and its dimensionally reduced equations. Furthermore, lump solutions with free parameters have been constructed using the dimensionally reduced new form of the (3 + 1)-dimensional Boussinesq equation. The derived lump solutions show it has two trough positions and one crest position. The amplitudes and shapes of lump waves don't vary during propagation but they change their positions. By making three-dimensional, two-dimensional, and density plots for specific values of the relevant free parameters, the propagations of the obtained lump wave solutions are displayed. They also demonstrate how the trough and crest positions of a lump wave change over time with constant velocity. The phase shifts, propagation directions, and energy distributions can be seen from the graphical outputs that the free parameters of the model play a significant role in changing the shapes and amplitudes of the waves. The resulting solutions and their physical characteristics may help to understand how the waves propagate in shallow water in oceanography.