Anna Allilueva
Institutskii per. 9, Dolgoprudnyi, 141700 Russia
Moscow Institute of Physics and Technology
Publications:
Allilueva A. I., Shafarevich A. I.
Conic Lagrangian Varieties and Localized Asymptotic Solutions of Linearized Equations of Relativistic Gas Dynamics
2019, vol. 24, no. 6, pp. 671681
Abstract
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
spacelike 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.

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.

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.
