Three-wave lump solutions and their dynamic behaviors for the (3+1)-dimensional constant-coefficient and variable-coeffcient differential equations

Abstract

In this paper, we propose two theorems to illustrate the types of equations that can be solved using the quadratic function method to derive the lump solutions localized in the whole plane, which are called three-wave lump solutions, and provide two constant-coefficient equations to illustrate. We further extend the quadratic function method to the variable-coefficient differential equations and obtain the three-wave lump solutions for two (3+1)-dimensional variable-coefficient equations. Moreover, the amplitudes of these lump waves and the distances between the two valleys of each lump are also obtained. Meanwhile, the motion trails, displacements and the velocities of these lump waves are analyzed in detail by virtue of numerical simulation. The study can be used to describe the motion of nonlinear waves in shallow water under the influence of time, and the results can enrich the types of solutions for the KdV-type equations. In addition, the 3d plots and corresponding density plots of the lump waves are displayed to show their spatial structures.