Andrey Shafarevich
Publications:
Allilueva A. I., Shafarevich A. I.
Evolution of Lagrangian Manifolds and Asymptotic Solutions to the Linearized Equations of Gas Dynamics
2019, vol. 24, no. 1, pp. 8089
Abstract
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.

Shafarevich A. I.
The Maslov Complex Germ and Semiclassical Spectral Series Corresponding to Singular Invariant Curves of Partially Integrable Hamiltonian Systems
2018, vol. 23, no. 78, pp. 842849
Abstract
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.

Chernyshev V. L., Tolchennikov A. A., Shafarevich A. I.
Behavior of Quasiparticles on Hybrid Spaces. Relations to the Geometry of Geodesics and to the Problems of Analytic Number Theory
2016, vol. 21, no. 5, pp. 531537
Abstract
We review our recent results concerning the propagation of “quasiparticles” 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 timedependent Schrödinger equation with localized initial data.We describe connections between the behavior of quasiparticles with the properties of the corresponding geodesic flows. We also describe connections of our problem with various problems in analytic number theory.

Allilueva A. I., Shafarevich A. I.
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
2015, vol. 20, no. 6, pp. 691700
Abstract
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 2Dsurface. 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 socalled degenerate Alfvén modes; the latter appear if the main term of the magnetic field is tangent to the surface of the jump.
