Development of theoretical mathematics

Student of the Faculty of Mechanics and Mathematics, M.V. Lomonosov Moscow State University.

Postgraduate student at the Department of Differential Equations, Faculty of Mechanics and Mathematics, M.V. Lomonosov Moscow State University.

Defended his Postgraduate diploma. PhD thesis: “The Cauchy problem and the mixed problem in a half-strip for equations of the Korteweg-de Vries type”.

Assistant of the Mathematics Department of the Moscow Institute of Radio Engineering, Electronics and Automation (MIREA).

Assistant professor at the Mathematics Department of MIREA.

Associate Professor of the Department of Differential Equations and Functional Analysis of RUDN University.

Defended his Doctoral thesis. Dr.Sci. thesis: “Boundary value problems for the Korteweg-de Vries equation and its generalizations”.

Professor at the Department of Differential Equations and Functional Analysis (Department of Nonlinear Analysis and Optimization after 2005), RUDN University.

Received the academic title of Professor.

Professor of the S. Nikolsky Mathematical Institute.

Winner of the RUDN prize in the field of science and innovations.

### Teaching

- Made training courses, the most significant are the following:
- “Evolutionary type functional spaces” - Moscow: RUDN University, 2011, 2016 - 2nd edition.
- "Selected chapters of the theory of evolutionary equations" - Moscow: RUDN University, 2014.

- Conducts the following lecture courses for bachelor and master’s degree students in RUDN University:
- "Complex analysis" ("Mathematics" direction, bachelor degree),
- "Partial differential equations" ("Mathematics" direction, bachelor degree),
- "Nonlinear evolution equations" ("Mathematics" direction, magistracy)

### Science

- Results on the global solvability and correctness of the Cauchy problem and initial boundary value problems for nonlinear evolution equations of odd order were obtained. These equations include the Korteweg-de Vries, Kawahary, Kadomtsev-Petviashvili, Zakharov-Kuznetsov equations. These equations describe nonlinear wave processes in dispersion media. These results have been established both for wide classes of equations of the type under consideration and for individual equations. For the considered boundary problems, the problems of internal regularity of solutions, the behavior of solutions at large times and controllability are also studied.
- The studies were started in the 1980s together with Professor Stanislav N. Kruzhkov for solving the Cauchy problem for the Korteweg-de Vries equation - the most famous representative of this class of equations. In particular, the so-called local smoothing effect was discovered (at the same time with the famous American mathematician T. Kato), which allowed us to establish the results of the existence of global time solutions with irregular initial data. In addition, the property was found to increase the internal smoothness of solutions depending on the decay rate at infinity of the initial data. The research method essentially used the properties of the corresponding linearized equation, namely, the Airy equation. In particular, the idea of inversion of the linear part of the Korteweg-de Vries equation was applied in order to establish the results on the uniqueness of the considered solutions.
- Further study of the properties of the Airy equation allowed us to construct special solutions of this equation of the type of boundary potentials. The use of these boundary potentials made it possible to obtain results on the global correctness of initial boundary value problems for the Korteweg-de Vries equation with natural or close to natural conditions for the smoothness of boundary data.
- In the following, these methods, developed for the Korteweg-de Vries equation, were applied to a wider class of equations of odd order. Results were obtained on the existence and uniqueness of global solutions. Assumptions on the class of the equations considered allowed us to use such analogs of the properties of the Korteweg-de Vries equation, such as the existence of conservation laws and the effect of local smoothing. Special solutions of the type of boundary potentials of the corresponding linearized equations were also constructed, which were used in the study of initial-boundary value problems.
- In the process of studying the generalizations of the Korteweg-de Vries equation, special attention was paid to those that were model for describing the propagation of nonlinear waves. In particular, for the Kadomtsev-Petviashvili equation in the case of two spatial variables, the first results on the existence of global solutions to the Cauchy problem were obtained.
- Another example of a multidimensional generalization of the Korteweg-de Vries equation is the Zakharov-Kuznetsov equation. For this equation, global correctness classes were constructed, both in the case of the Cauchy problem and in the initial-boundary value problems. In this case, cases of both two-dimensional and three-dimensional equations are considered. The use of boundary potentials made it possible to obtain these results under the natural conditions of smoothness of the boundary data. Results on the internal regularity of solutions were also established.
- Another important equation that generalizes the Korteweg-de Vries equation is the Kawahara equation, which models the propagation of waves in media with a higher order of dispersion. Here, results are also obtained on the global correctness of initial-boundary problems under the natural conditions of smoothness of boundary data and on the internal regularity of solutions.
- In recent years, issues of the behavior of solutions at large times, as well as some control problems, have also been studied for the equations above.

### Scientific interests

- solvability and correctness of boundary problems for nonlinear evolution equations of odd order
- internal regularity of solutions of boundary value problems for nonlinear evolution equations of odd order
- behavior at large times of solutions of boundary value problems for nonlinear evolution equations of odd order
- controllability of boundary value problems for nonlinear evolution equations of odd order

_{s}for s∈[0,2] are established.