Dynamical Soliton Wave Structures of One-Dimensional Lie Subalgebras via Group-Invariant Solutions of a Higher-Dimensional Soliton Equation with Various Applications in Ocean Physics and Mechatronics Engineering

Abstract

Having realized various significant roles that higher-dimensional nonlinear partial differential equations (NLPDEs) play in engineering, we analytically investigate in this paper, a higher-dimensional soliton equation, with applications particularly in ocean physics and mechatronics (electrical electronics and mechanical) engineering. Infinitesimal generators of Lie point symmetries of the equation are computed using Lie group analysis of differential equations. In addition, we construct commutation as well as Lie adjoint representation tables for the nine-dimensional Lie algebra achieved. Further, a one-dimensional optimal system of Lie subalgebras is also presented for the soliton equation. This consequently enables us to generate abundant group-invariant solutions through the reduction of the understudy equation into various ordinary differential equations (ODEs). On solving the achieved nonlinear differential equations, we secure various solitonic solutions. In consequence, these solutions containing diverse mathematical functions furnish copious shapes of dynamical wave structures, ranging from periodic, kink and kink-shaped nanopteron, soliton (bright and dark) to breather waves with extensive wave collisions depicted. We physically interpreted the resulting soliton solutions by imploring graphical depictions in three dimensions, two dimensions and density plots. Moreover, the gained group-invariant solutions involved several arbitrary functions, thus exhibiting rich physical structures. We also implore the power series technique to solve part of the complicated differential equations and give valid comments on their results. Later, we outline some applications of our results in ocean physics and mechatronics engineering.