For the 1D shallow-water MHD system with a relaxed magnetic-divergence constraint, the admitted Lie symmetry algebras are classified into four cases, and an optimal system of subalgebras is used to build similarity solutions for the rotating, gravitating case.
Kurganov, Finite-volume schemes for shallow-water equations, Acta Numerica 27, 289 (2018)
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Lie Symmetries for the Shallow Water Magnetohydrodynamics Equations in a Rotating Reference Frame
For the 1D shallow-water MHD system with a relaxed magnetic-divergence constraint, the admitted Lie symmetry algebras are classified into four cases, and an optimal system of subalgebras is used to build similarity solutions for the rotating, gravitating case.