Symmetries of the triple degenerate DNLS equations for weakly nonlinear dispersive MHD waves

Abstract

Lie symmetries, conservation laws, and Lagrangian and Hamiltonian formulations of the triple degenerate, derivative nonlinear Schrödinger (TDNLS) equations for weakly nonlinear dispersive, magnetohydrodynamic (MHD) waves are derived. The equations describe how Alfvén waves propagating parallel to the background magnetic field B interact with the magneto-acoustic modes near the triple umbilic point where the fast, slow and Alfvén mode phase speeds coincide. The Lie point symmetries are used to derive classical similarity solutions of the equations. In particular, the similarity solutions corresponding to time translation, space translation and rotational invariance symmetries are reduced to quadrature. The dispersionless TDNLS system is of hydrodynamic type, and has three families of characteristics analogous to the slow, intermediate and fast modes of MilD. The Riemann invariants corresponding to each of these families are obtained in closed analytic form. Examples of solitary wave and periodic travelling wave solutions are investigated by plotting the contours of the Hamiltonian H(v, w) in the (v, w) phase plane, where the canonical variables v and w correspond to the normalized transverse magnetic field perturbations. An analysis of the prolongation Lie algebra is carried out in order to investigate the integrability of the equations.