Calculus of moving surfaces and singular wave-fronts in ideal magnetohydrodynamics

Jacques Hadamard [1] introduced a method of studying propagating wave-fronts in linear media, treating them as mathematical singular surfaces of discontinuities of various field functions. Contrary to the shock-like wave-fronts in nonlinear media, the analysis of the Hadamard's (weak) fronts does not require any additional physical conditions across the fronts but only some purely geometric and kinematic conditions, the so called, compatibility conditions. The approach of Hadamard appears to be very flexible permitting analysis of the fronts in nonhomogeneous and nonlinear media also. Hadamard method has a very essential shortcoming: the structure of the compatibility conditions remains to be very cumbersome and their complexity rapidly grows with the growing order of the discontinuous derivatives. Unfortunately, the higher terms in the Hadamard method demand using those higher derivatives. Levi-Civita [2] and Thomas [3] made essential contribution to cope with this sort of difficulties. This work was continued in the monograph [4], devoted to the so called Calculus of Moving Surfaces (CMS). Earlier, the results, recently presented in [4], permitted to explore and express in a very transparent form the so-called transport (Ricatti) equation for nonlinear ideal hydrodynamics. In this paper, we present further analysis for the Alfven wave in ideal magnetohydrodynamics.