Laboratory «Nonlinear and nonlocal equations and their applications»

Name of the project:

Nonlinear and nonlocal equations and their applications

Goals of project: 

1. To study wave turbulence described by the cubic Schrodinger equation with dissipation and random force on the torus of a large period;
2. To study heat transfer in crystals;
3. To study wave kinetic equations;
4. To study wave turbulence  described by Boltzmann-type equations;
5. To study the kinetics of high-temperature plasma in a thermonuclear reactor and to determine the conditions for plasma retention;
6. To perform a numerical modeling of plasma flow in a mirror trap accounting for an external magnetic field;
7. To study biological and biomedical problems, including models of a viral infection in mathematical immunology and epidemiology using methods of the qualitative theory of reaction-diffusion equations and their mathematical modeling;
8. To study the solvability and smoothness of generalized solutions of nonlocal boundary problems.

Project objective: 

• To study the behavior of solutions of the cubic Schrodinger equation with dissipation and random force on the torus of a large period in the wave turbulence limit. That is, the amplitude of a solution tends to zero and its spatial period tends to infinity, in particular,

(1) to study the behavior of the energy spectra of solutions formed by second moments of their Fourier coefficients at this limit;                                

(2) to study the stabilization of probability characteristics of solutions to the statistical equilibrium   with the growth of the time as well as the stabilization of their energy spectra to the universal limit;

• To study the possibility of the propagation of the results of  points (1) and (2) to heat transfer equations in crystal lattices; a systematic research of the mathematical structure of the main kinetic models of wave turbulence, especially in comparison with equations of the kinetic gas theory;

• To analyze the long-term limits of Boltzmann-type equations, consistent with wave turbulence o the Fourier coefficients when the solutions of wave turbulence equations are interpreted as quasiparticles;     

• To study the qualitative properties of classical and generalized solutions of mixed problems for the Vlasov-Poisson system of equations with an external magnetic field related to the problem of high-temperature plasma retention in a thermonuclear reactor;

• To develop tests for the verification of algorithms of solving the Vlasov-Posson system of equations on the basis of new analytical and qualitative solutions; to conduct computational experiments to investigate the processes in a material exposed to radiation and near-wall plasma in the context of pulse heating as well as the processes of plasma flow in an axisymmetric magnetic field directed along the axis of the trap in the presence of periodic modulation of magnetic field strength; to compare the observed results of numerical computations with new experimental data obtained G. I. Budker Institute of Nuclear Physics of the Siberian Branch of the Russian Academy of Sciences;

• To research the equations and  systems of equation of reaction-diffusion from the viewpoint of the existence and stability of partial types of solutions, such as traveling waves, standing and moving impulses, on the basiss of methods of nonlinear and linear analysis. The application of mathematical results and numerical modeling methods to the research of biological and biomedical problems, including a model of a viral infection in mathematical immunology and epidemiology.