A (2+1)-dimensional generalized Hirota–Satsuma–Ito equations: Lie symmetry analysis, invariant solutions and dynamics of soliton solutions

This paper investigates the exact invariant solutions and the dynamics of soliton solutions to the (2+1)-dimensional generalized Hirota–Satsuma–Ito (g-HSI) equations. By applying the Lie symmetry technique, infinitesimal vectors, the commutation relations, and various similarity reductions are derived from the g-HSI equations. Using the two stages of Lie symmetry reductions, the equation is transformed into various nonlinear ordinary differential equations (NLODEs). After that, by solving the various resulting ODEs, we obtain abundant explicit exact solutions in terms of the involved functional parameters. These closed-form invariant solutions are successfully presented in the form of distinct complex wave-structures of solutions like combo-form solitons, dark-bright solitons, W-shaped solitons, the interaction between multiple solitons, parabolic wave solitons, multi-wave structures, and curved-shaped parabolic solitons. Furthermore, using computerized symbolic computation and numerical simulation, the physical behaviors of some obtained solutions are displayed in three-dimensional graphics. The resulting solutions are found to be useful for understanding the dynamics of the exact closed-form solutions of this model and show the authenticity as well as the effectiveness of the proposed method. Therefore, the gained solutions and their dynamical wave structures are quite significant for understanding the propagation of the excitation waves in shallow water wave models. Furthermore, using the resulting symmetries, the conservation laws of g-HSI equations have been obtained by applying Ibragimov's theorem.