Gm. Webb et al., LIE-BACKLUND SYMMETRIES OF DISPERSIONLESS, MAGNETOHYDRODYNAMIC MODEL-EQUATIONS NEAR THE TRIPLE UMBILIC POINT, Journal of physics. A, mathematical and general, 29(16), 1996, pp. 5209-5240
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.