Lax representations for the magnetohydrodynamics equations

Abstract

We find two Lax representations for the reduced magnetohydrodynamics equations ({\sc rmhd}) and derive a B{\"a}cklund transformation between the tangent and the cotangent coverings of these equations. Then, we study the action of the B{\"a}cklund transformation on the second-order cosymmetries and the action of its inverse on the Lie symmetries of {\sc rmhd}. The action of the inverse transformation produces another Lax representation for {\sc rmhd}. The reduction of {\sc rmhd} by the symmetry of shifts along the $z$-axis coincides with the equations of two-dimensional ideal magnetohydrodynamics ({\sc imhd}). Applied to the Lax representations and the B{\"a}cklund transformation of {\sc rmhd}, the reduction provides analogous constructions for {\sc imhd}. The action of the inverse B{\"a}cklund transformation on the Lie symmetries of {\sc imhd} is expressed in terms of a new four-parameter Lax representation of these equations.