Lie - Bäcklund symmetries of dispersionless, magneto- hydrodynamic model equations near the triple umbilic point

Abstract

Lie - B?cklund symmetries and conservation laws are derived for weakly nonlinear magnetohydrodynamic (MHD) equations describing the interaction of the Alfv?n and magnetoacoustic modes propagating parallel to the ambient magnetic field, in the parameter regime near the triple umbilic point, where the gas sound speed matches the Alfv?n speed . The dispersive form of the equations can be expressed in Hamiltonian form and admit four Lie point symmetries and conservation laws associated with space-translation invariance (momentum conservation), time translation invariance (energy conservation), rotational invariance about the magnetic field B (helicity conservation), plus a further symmetry that is associated with accelerating wave similarity solutions of the equations. The main aim of the paper is a study of the symmetries and conservation laws of the dispersionless equations. The dispersionless equations are of hydrodynamic type and have three families of characteristics analogous to the slow, intermediate and fast modes of MHD and the Riemann invariants for each of these modes are given in closed form. The dispersionless equations are shown to be semi-Hamiltonian, and to possess two infinite families of symmetries and conservation laws. The analysis emphasizes the role of the Riemann invariants of the dispersionless equations and a hodograph transformation for a restricted version of the equations.