Andrey Shafarevich

We study asymptotic solution of the Cauchy problem for the linearized system of relativistic gas dynamics. We assume that initial condiditiopns are strongly localized near a space-like surface in the Minkowsky space. We prove that the solution can be decomposed into three modes, corresponding to different routsb of the equations of characteristics. One of these roots is twice degenerate and the there are no focal points in the corresponding miode. The other two roots are simple; in order to describe the corresponding modes, we use the modificication of the Maslov’s canonical operator which was obtained recently.

We study asymptotic solution of the Cauchy problem for linearized equations of gas dynamics with rapidly oscillating initial data. We construct the formal serie, satisfying this problem. This serie is naturally divided into three parts, corresponding to the hydrodynamic mode and two acoustic modes. The summands of the serie are expressed in terms of the Maslov canonic operator on moving Lagrangian manifolds. Evolution of the manifolds is governed by the corresponding classical Hamiltonian systems.

We study semiclassical eigenvalues of the Schroedinger operator, corresponding to singular invariant curve of the corresponding classical system. The latter system is assumed to be partially integrable. We describe geometric object corresponding to the eigenvalues (comlex vector bundle over a graph) and compute semiclassical eigenvalues in terms of the corresponding holonomy group.

We review our recent results concerning the propagation of “quasi-particles” in hybrid spaces — topological spaces obtained from graphs via replacing their vertices by Riemannian manifolds. Although the problem is purely classical, it is initiated by the quantum one, namely, by the Cauchy problem for the time-dependent Schrödinger equation with localized initial data.We describe connections between the behavior of quasi-particles with the properties of the corresponding geodesic flows. We also describe connections of our problem with various problems in analytic number theory.

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.