Lie symmetry analysis for obtaining the abundant exact solutions, optimal system and dynamics of solitons for a higher-dimensional Fokas equation. (January 2021) - Kumar, Sachin; Kumar, Dharmendra; Kumar, Amit
Highlights: Lie symmetry analysis is performed on a (4+1)-dimensional Fokas equation. Lie symmetries and one-dimensional optimal systems are derived for a higher-dimensional equation. We obtain abundant exact invariant solutions through the four stages of Lie symmetry reductions. We present a comparison of our results with other works. In addition, we exhibit the dynamical structures of the soliton solutions through 3D illustrations. Abstract: In this article, the Lie group of transformation method via one-dimensional optimal system is proposed to obtain some more exact solutions of the (4+1)-dimensional Fokas equation. Lie infinitesimal generators, possible vector fields, and their commutative and adjoint relations are presented by employing the Lie symmetry method. An optimal system of the one-dimensional subalgebras is also constructed using Lie vectors. Meanwhile, based on the optimal system, Lie symmetry reductions of the Fokas equation is obtained. A repeated process of Lie symmetry reductions, using the single, double, triple, quadruple, and quintuple combinations between the considered vectors, transforms the Fokas equation into nonlinear ordinary differential equations which produce abundant group-invariant solutions. The same problem was studied by Sadat et al. (Chaos Solitons Fractals 140:110134, 2020) using the same Lie symmetry technique via commutative product approach but with the less number of vector fields and therefore could obtain only three exact solutions as compared to the number of analytic solutions in this paper. In order to provide rich localized structures, some solutions are supplemented via numerical simulation, which produces some breather-type solitons, oscillating multi-solitons on the parabolic-shaped surface, fractal dromions, lump-type solitons, and annihilation of different parabolic multi-solitons profiles. The dynamical behaviors of excitation-localized structures are demonstrated graphically via 3D plots for suitable values of the arbitrary free parameters and independent arbitrary functions.