LIE-BACKLUND SYMMETRIES OF DISPERSIONLESS, MAGNETOHYDRODYNAMIC MODEL-EQUATIONS NEAR THE TRIPLE UMBILIC POINT

Lie-Backlund symmetries and conservation laws are derived for weakly n onlinear magnetohydrodynamic (MHD) equations describing the interactio n of the Alfven and magnetoacoustic modes propagating parallel to the ambient magnetic field, in the parameter regime near the triple umbili c point, where the gas sound speed a(g) matches the Alfven speed V-A. The dispersive form of the equations can be expressed in Hamiltonian f orm and admit four Lie point symmetries and conservation laws associat ed with space-translation invariance (momentum conservation), time tra nslation 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 charact eristics analogous to the slow, intermediate and fast modes of MHD and the Riemann invariants for each of these modes are given in closed fo rm. 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 dis persionless equations and a hodograph transformation for a restricted version of the equations.