Laboratory «Nonlinear and nonlocal equations and their applications»

Scientific results:

For conservative nonlinear equations arising in modern physics (i.e. for the nonlinear equations that describe processes without energy dissipation) and for their small perturbations by means of random force and friction, a method of quasi-solutions has been developed. This method allows constructive obtaining approximate solutions of the indicated equations, which we call "quasi-solutions", and studying the behavior of quasi-solutions under various limiting regimes, considered by physicists. In particular, for the limiting regime, when firstly the spatial period of the quasi-solution tends to infinity and then the magnitude of the perturbation tends to zero, it has been proven that the distribution of the energy of a quasi-solution between different oscillatory frequencies (called the "energy spectrum of the quasi-solution") is approximately described by the solution of the wave kinetic equation. Using the results of modern number theory, developed and refined by us for the purposes of our study, it is shown that the change of the order of limit transitions (firstly the perturbation tends to zero, then the period tends to infinity) leads to the fact that the energy spectrum of the quasi-solution becomes described by a different kinetic equation, previously unknown to physicists.

There is no doubt that quasi-solutions are close to an exact solution. Thus, their energy spectra are close to the energy spectrum of the exact solution, so that the latter is also well approximated by a solution of the corresponding kinetic equation. In order to rigorously prove the closeness of the solution and quasi-solutions, we have developed a new version of the Newton-Kantorovich-Nash-Moser (NKNM) method. The NKNM method allows one to prove the closeness of approximate and exact solutions of various equations, and we are currently working on applying our version of the NKNM method to the equations in the first paragraph.

New methods have been developed for studying the behavior of complex physical systems containing randomness, over large periods of time. In a number of important cases, these methods make it possible to prove that the behavior of the main statistical characteristics of such systems is universal, i.e. is independent of the initial configuration of the system. If the system under consideration is close to linear or integrable, then it is possible to obtain an effective description of these characteristics. Since the developed methods are abstract, they are applicable to systems of diverse nature arising in various sections of natural sciences. 

In order to further study of the properties of wave kinetic equations, which, by virtue of the results mentioned in Section 1) describe the energy spectra of quasi-solutions, a class of generalized kinetic equations is introduced and studied. This class includes both wave kinetic equations and the famous Boltzmann equation. Discretizations of generalized kinetic equations are considered and the behavior of solutions of the obtained discrete equations under unlimited growth of time is examined. In a number of cases it is proved that with an increase of the discretization dimension, the solutions of discrete equations converge to a solution of the original generalized kinetic equation.

Cooperation:

• Mathematical Institute named after. V.A. Steklov Russian Academy of Sciences
• Institute of Nuclear Physics named after G.I. Budker SB RAS
• Institute of Mathematics named after. S.L. Sobolev of the Siberian Branch of the Russian Academy of Sciences
• National Research University "Higher School of Economics"
• St. Petersburg branch of the Mathematical Institute named after. V. A. Steklov of the Russian Academy of Sciences