Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

TL;DR

This paper analyzes the symmetries and conservation laws of one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates, revealing new conservation laws and solutions for different bottom topographies.

Contribution

It provides a comprehensive symmetry classification and derives new conservation laws for SMHD equations considering various bottom topographies in Lagrangian coordinates.

Findings

Symmetry classification identifies all relevant bottom topographies.

New conservation laws are derived for different topographies.

Invariant solutions are constructed for the equations.

Abstract

Symmetries of the one-dimensional shallow water magnetohydrodynamics equations (SMHD) in Gilman's approximation are studied. The SMHD equations are considered in case of a plane and uneven bottom topography in Lagrangian and Eulerian coordinates. Symmetry classification separates out all bottom topographies which yields substantially different admitted symmetries. The SMHD equations in Lagrangian coordinates were reduced to a single second order PDE. The Lagrangian formalism and Noether's theorem are used to construct conservation laws of the SMHD equations. Some new conservation laws for various bottom topographies are obtained. The results are also represented in Eulerian coordinates. Invariant and partially invariant solutions are constructed.