Different dynamics of invariant solutions to a generalized (3+1)-dimensional Camassa-Holm-Kadomtsev-Petviashvili equation arising in shallow water-waves

Abstract

Using the Lie symmetry technique, this paper studies a (3+1)-dimensional generalized Camassa-Holm-Kadomtsev-Petviashvili (GCHKP) equation that can possess various dynamical structures of solitary waves. The GCHKP equation is a special case of the internal shallow-water wave equation, which has recently gained popularity in ocean physics and hydrodynamics engineering. By taking advantage of the Lie symmetry technique, we derive Lie infinitesimals, geometric vector fields, commutator table, adjoint table, and an optimal system for the considered equation. Based on the one-dimensional optimal system, various attractive Lie symmetry reductions are carried out, and the GCHKP equation is reduced to a number of nonlinear ordinary differential equations (NODEs). Consequently, innumerable explicit exact solutions of a (3+1)-D GCHKP equation are obtained employing symmetry reductions via a one-dimensional optimal system of Lie subalgebra. With the assistance of symbolic computation, several 3D and 2D graphics exhibited their dynamic structures, and graphical illustrations were physically interpreted. For the first time, we furnish exact invariant solutions in terms of a variety of functions such as exact rational solutions, trigonometric and hyperbolic function solutions, V-shaped solitons, dark and bright solitons, and solitary waves, singular solitons, breather waves, and parabolic waves. In addition, the obtained results have significantly supplemented the exact invariant solutions of the GCHKP equation in the literature, allowing us to gain a better understanding of the dynamics of complex physical systems. Finally, the proposed method will significantly boost obtaining the exact explicit solutions of the nonlinear evolution equations.