SYMMETRIES OF THE TRIPLE DEGENERATE DNLS EQUATIONS FOR WEAKLY NONLINEAR DISPERSIVE MHD WAVES

Lie symmetries, conservation laws, and Lagrangian and Hamiltonian form ulations of the triple degenerate, derivative nonlinear Schrodinger (T DNLS) equations for weakly nonlinear dispersive, magnetohydrodynamic ( MHD) waves are derived. The equations describe how Alfven waves propag ating parallel to the background magnetic field B interact with the ma gneto-acoustic modes near the triple umbilic point where the fast, slo w and Alfven mode phase speeds coincide. The Lie point symmetries are used to derive classical similarity solutions of the equations. In par ticular, the similarity solutions corresponding to time translation, s pace translation and rotational invariance symmetries are reduced to q uadrature. The dispersionless TDNLS system is of hydrodynamic type: an d has three families of characteristics analogous to the slow, interme diate and fast modes of MHD. The Riemann invariants corresponding to e ach of these families are obtained in closed analytic form. Examples o f solitary wave and periodic travelling wave solutions are investigate d by plotting the contours of the Hamiltonian H(v, w) in the (v, w) ph ase plane, where the canonical variables v and w correspond to the nor malized transverse magnetic field perturbations. An analysis of the pr olongation Lie algebra is carried out in order to investigate the inte grability of the equations.