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

Abstract

Abstract 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 ∇(hB)!= 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 f0, related to the constant rotation of the reference frame. For four different cases, namely g = 0, f0 = 0; g ?= 0 , f0 = 0; g = 0, f0 ?= 0; and g ?= 0, f0 ?= 0 the admitted Lie symmetries for the hyperbolic system form different Lie algebras. Specifically the admitted Lie algebras are the G10 = {A3,3 ⊗s A2,1} ⊗s A5a,34; G8 = A2,1 ⊗s A6,22; G7 = A3,5 ⊗s {A2,1 ⊗s A2,1}; and G6 = A3,5 ⊗s A3,3 respectively, where we use the Morozov-Mubarakzyanov-Patera