2022 International Conference on Applied Mathematics, Informatics, and Computing Sciences (AMICS 2022)

Abstract: The report discusses a new method for classifying integrable nonlinear differential-difference equations with three independent variables. Integrable equations have a wide range of applications in mathematical physics, so the problem of describing integrable equations is in demand. In our recent papers, we have developed an efficient approach to the problem of classifying integrable nonlinear equations with three independent variables, at least one of which is discrete. The idea of the method is based on the observation that by appropriately terminating a given integrable equation, one can reduce it to a system of hyperbolic type equations in two independent variables, which is Darboux integrable. A system of equations with two independent variables of hyperbolic type is called Darboux integrable if it admits a complete set of independent integrals in each of the characteristic directions. In turn, systems that allow a complete set of integrals have finite-dimensional characteristic Lie-Rinehart algebras. We use this fact as a classification criterion.

Biography: Education
1972-1977 Bashkir State University, Department of Mathematics. Diploma, speciality “Mathematics and teaching of Mathematics”. Master’s degree.
1977-1980 Bashkir State University (post-graduate course) Supervisor Prof. Alexey Shabat.
1982 Ph.D. in Differential Equations and Mathematical Physics.
1996 The second Doctor dissertation.
Research Interests
Methods for solving nonlinear differential and discrete equations of mathe-matical physics. Symmetries and their applications in integrability theory. Integrability criteria for non-linear partial differential equations and their discrete analogs, classification algorithms for integrable models.
Research Experience
To this moment I have more than 100 articles published in Russian and foreign world-famed scientific journals. In my articles the following problems are studied:
The stable recurrent algorithm for solvng the Riemann-Hilbert problem of analytical conjugation of functions is suggested. Based on this algorithm a numerical realization of the direct and inverse scattering transform method for NLS type equations is developed (in collaboration with A.G. Shagalov).
The phenomenon of boundary condition for soliton equations preserving the integrability property by using symmetries is investigated in collaboration with M. Gurses, V.E. Adler, S.I. Svinolupov and others.
The notion of the characteristic operators introduced for the case of PDE by A.B. Shabat is adopted to the discrete models and applied to the problem of the classification of semidiscrete models integrable in sense of Darboux (in collaboration with N. Zheltukhina, A. Pekcan).
The notion of generalized invariant manifold for nonlinear PDE and their discrete analogs was introduced. It was observed that such manifolds provide a very effective tool for constructing Lax pairs and recursion operators, as well as for searching particular solutions to nonlinear integrable models (in
collaboration with M.N. Kuznetsova and A.R. Khakimova).
The effective method for integrable classification of nonlinear lattices with three independent variables is worked out based on the using hierarchies of the integrable in sense of Darboux finite field reductions (in collaboration with M.N. Kuznetsova, A.R. Khakimova and A.U. Sakieva).
List of publications can be found in http://www.mathnet.ru/eng/person17529.