Analysis of Lie invariance, analytical solutions, conservation laws, and a variety of wave profiles for the (2+1)-dimensional Riemann wave model arising from ocean tsunamis and seismic sea waves

Nonlinear phenomena are common in nonlinear sciences and physical engineering, such as ocean engineering and marine physics, as well as fluid dynamics. Nonlinear partial differential equations (PDEs) are the most suitable model to describe the nonlinear behavior of wave phenomena. In this work, the Lie symmetry method and the generalized exponential rational function (GERF) technique are applied to investigate the (2+1)-dimensional Riemann wave (RW) model, which has essential physical properties in oceanography and marine engineering. The RW model is fully integrable and has numerous applications in ocean tsunami and tidal wave propagation. Adapting the Lie group method, the infinitesimals, commutator table, adjoint table, and similarity invariants are constructed for the RW model. Thereafter, an optimal system is constructed using the adjoint table. Performing the symmetry reductions, the (2+1)-dimensional RW model is converted into a system of ordinary differential equations (ODEs) using the symmetry variables. Under some parametric constraints, this obtained system of ODEs is solved to determine the closed-form solutions for the RW model. As a result, we obtained some new types of exact solutions for the RW model. Further, using the GERF technique, some more interesting closed-form solutions are obtained. We have also plotted 3-dimensional (3D) graphics to demonstrate the dynamics of the obtained solutions, which show the elastic behavior of solitary waves, and dark-bright multi-soliton profiles of the waves. The conservation laws for the RW model are also deduced using the Noether operators. These conservation laws are helpful to investigate the internal properties, existence, and uniqueness analysis of the solutions to the system of differential equations.