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

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.

Conservative invariant finite‐difference schemes for the modified shallow water equations in Lagrangian coordinates

“…Further we consider the dimensionless equations, and symbol ˜is omitted for brevity. We also assume g 1 = 2 by means of equivalence transformations [8].…”

“…The local conservation laws of ( 9) have been obtained in [8]. They are listed below depending on the bottom topography according to the results of the group classifications with respect to the function b .…”

“…For the linearization purposes, we need to know the set of variables on which approximations B may depend. According to the group classification [8], there the following possibilities are of interest:…”

“…The consideration of the shallow water equations in the presence of a magnetic field (SMHD) is a relatively new area of magnetohydrodynamics. One of the first models to describe SMHD was presented in [1] (the Gilman model), and since then, many papers by various authors have been devoted to studying this model, including its stability [2], numerical simulation [3][4][5][6][7], and conservation laws [8,9] (see, for example, a more detailed review in [8]).…”

“…In this case, most of the equations of the SMHD system are integrated, and only one equation remains unintegrated. For this equation in [8], a group classification was carried out according to the function describing the topography of the bottom. Conservation laws and invariant solutions in Lagrangian and Eulerian coordinates were obtained.…”