Gm. Webb et al., SYMMETRIES OF THE TRIPLE DEGENERATE DNLS EQUATIONS FOR WEAKLY NONLINEAR DISPERSIVE MHD WAVES, Journal of Plasma Physics, 54, 1995, pp. 201-244
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.