Lie Symmetries for the Shallow Water Magnetohydrodynamics Equations in a Rotating Reference Frame

We perform a detailed Lie symmetry analysis for the hyperbolic system of partial differential equations that describe the one-dimensional Shallow Water magnetohydrodynamics equations within a rotating reference frame. We consider a relaxing condition $\mathbf{\nabla }\left(h\mathbf{B}\right)\mathrm{\ne }0$ for the one-dimensional problem, which has been used to overcome unphysical behaviors. The hyperbolic system of partial differential equations depends on two parameters: the constant gravitational potential $g$ and the Coriolis term $f_0$, related to the constant rotation of the reference frame. For four different cases, namely $g=0,f_0=0$; $g\mathrm{\ne }0,f_0=0$; $g=0$, $f_0\mathrm{\ne }0$; and $g\mathrm{\ne }0$, $f_0\mathrm{\ne }0$ the admitted Lie symmetries for the hyperbolic system form different Lie algebras. Specifically the admitted Lie algebras are the $L^{10}=\{A_{3,3}⋊A_{2,1}\}{\otimes }_s{A}_{5,34}^a$; $$; $L^7=A_{3,5}⋊\{A_{2,1}⋊A_{2,1}\}$; and $L^6=A_{3,5}⋊A_{3,3}$respectively, where we use the Morozov-Mubarakzyanov-Patera classification scheme. For the general case where $f_{0}g\mathrm{\ne }0$, we derive all the invariants for the Adjoint action of the Lie algebra $L^6$ and its subalgebras, and we calculate all the elements of the one-dimensional optimal system. These elements are then considered to define similarity transformations and construct analytic solutions for the hyperbolic system.