Asymptotic Solutions for Linear and Nonlinear MHD Systems with a Rapid Jump near a Surface. Dynamics of the Surface of the Jump and Evolution of the Magnetic Field

We review our recent results concerning the asymptotic solutions for both linear and nonlinear MHD equations.We describe the asymptotic structure of the solution with a rapid jump near a 2D-surface. For the linear system we demonstrate the effect of the instantaneous growth of the solution. We also study the weak limit of the solution and the corresponding generalized problem. For the nonlinear system we describe the asymptotic division into different modes, the free boundary problem for the movement of the surface and the equation on the moving surface for the profile of the asymptotic solution. We also study the possibility of the instantaneous growth of the magnetic field. It appears that the growth is possible only in the case of the so-called degenerate Alfvén modes; the latter appear if the main term of the magnetic field is tangent to the surface of the jump.