Group classification of the two‐dimensional magnetogasdynamics equations in Lagrangian coordinates

Abstract

The present paper is devoted to the group classification of magnetogasdynamics equations in which dependent variables in Euler coordinates depend on time and two spatial coordinates. It is assumed that the continuum is inviscid and nonthermal polytropic gas with infinite electrical conductivity. The equations are considered in mass Lagrangian coordinates. Use of Lagrangian coordinates allows reducing number of dependent variables. The analysis presented in this article gives complete group classification of the studied equations. This analysis is necessary for constructing invariant solutions and conservation laws on the base of Noether's theorem.

One class of MHD equations: Conservation laws and exact solutions

“…15 We study the case, where all dependent functions in Eulerian coordinates only depend on two space variables 𝑥 1 and 𝑥 2 . Equations (2) reduce to the equations 19 :…”

“…Considering group classification of Equations ( 4), it was derived in Ref. [19] the following model:…”

“…Recently, the group analysis method has been applied to the MHD equations, where all dependent variables in Eulerian coordinates depend on time and two spatial coordinates. 19 The analysis was carried out in mass Lagrangian coordinates, which reduce the MHD equations to the study of two second-order partial differential equations. The use of Lagrangian coordinates make it possible to solve four equations, which led to the form of the reduced equations containing four arbitrary functions: the entropy and a three-dimensional vector associated with the magnetic field.…”

“…The group classification, done in Ref. [19], separated the MHD equations into equivalent classes with respect to these functions. This classification identified several classes of equations that share similar mathematical properties.…”

“…Section 2 introduces the MHD equations in mass Lagrangian coordinates, where all dependent variables in Eulerian coordinates depend on time and two spatial coordinates. 19 These equations serve as the basis for the rest of the study. This section also discusses the admitted Lie group and equivalence transformations, which are used to simplify the solutions of the studied equations.…”