19 Mar 2003

Einar Mjoelhus "Soliton theory of quasi parallel MHD waves - I".

MHD (Magnetohydrodynamics) is a model for the slowest and most large-scale phenomena in plasmas. It is popular for the large scale phenomena in space plasmas like the interplanetary and interstellar space, the sun, as well as the magnetosphere, but also for the containment and stability in confined plasmas studied for thermonuclear fusion. (In those cases, "large scale" is much smaller than in the former cases, because the densities are so much higher.) The wave modes for MHD are well known. There are three "modes". The system of equations is symmetric-hyperbolic. A feature is that for propagation along the ambient magnetic field, two of the three characteristic velocities coincide. 

According to "classical" theory of weakly nonlinear waves, there is a "reduction" near each characteristic velocity, which, when weak dispersive terms are added, leads to the Korteweg -de Vries' (KdV) equation. There is an analoguous reduction for the situation around the double velocity at nearly parallel propagation. Reduction in this case leads to two coupled equations, which when written in a complex form has been termed the "Derivative Nonlinear Schroedinger equation (DNLS). As it has turned out, this equation is a "soliton equation", like the KdV equation.

Finally, when extending from the strictly one-dimensional to a weakly three-dimensional situation, the KdV equation extends to the so-called KP (Kadomtsev - Petviashvili) equation. We have derived an analoguous extension to the DNLS equation. The latter has been little studied, and may represent an interesting testbed for exercises into finding and applying symmetries of nonlinear partial differential equations."