Nonlocal symmetry analysis, exact solutions, and evolution of nonlinear waves for one-dimensional shallow water magnetohydrodynamics equations

In this article, we study the symmetry analysis for a system of hyperbolic quasilinear partial differential equations (PDEs) that describe one-dimensional shallow water magnetohydrodynamics (SMHD) and examine its evolutionary behavior of nonlinear waves. The governing system of PDEs admits a five-dimensional solvable Lie algebra. We construct the optimal system of one-dimensional subalgebras with the help of its adjoint actions. We obtain various closed-form exact solutions by using each of these subalgebras. Furthermore, we establish several nonlocal symmetries of the given system by both the conservation law-based method and the symmetry-based method. Nonlocal symmetries are helpful in finding new exact solutions beyond those of local symmetry reductions. As an application, we study one of the obtained exact solutions of the governing system to analyze the evolutionary behavior of weak discontinuity and its interaction with characteristic shock.