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 \(\mathbf{\mathbf{\nabla }}\left( h\mathbf{B} \right) \neq 0\) 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 \(g=0,~f_{0}=0\); \(g\neq 0\,,~f_{0}=0\); \(g=0\), \(f_{0}\neq 0\); and \(g\neq 0\), \(f_{0}\neq 0\) the admitted Lie symmetries for the hyperbolic system form different Lie algebras. Specifically the admitted Lie algebras are the \(L^{10}=\left\{ A_{3,3}\rtimes A_{2,1}\right\} \otimes _{s}A_{5,34}^{a}\); \(% L^{8}=A_{2,1}\rtimes A_{6,22}\); \(L^{7}=A_{3,5}\rtimes\left\{ A_{2,1}\rtimes A_{2,1}\right\} \); and \(L^{6}=A_{3,5}\rtimes A_{3,3}~\)respectively, where we use the Morozov-Mubarakzyanov-Patera classification scheme. For the general case where \(f_{0}g\neq 0\), 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.