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Analytical Methods of Function Theory and Applications


Work number - P 3 ALLOWED TO PARTICIPATE

Presented B. Verkin Institute for Low Temperature Physics and Engineering of National Academy of Sciences of Ukraine.

Authors:
1. L. Golinskii, DSc (B. Verkin Institute for Low Temperature Physics and Engineering of NAS of Ukraine);
2. V. Gutlyanskii, DSc (Institute of Applied Mathematics and Mechanics of NAS of Ukraine);
3. I. Egorova, DSc (B. Verkin Institute for Low Temperature Physics and Engineering of NAS of Ukraine);
4. V.P. Kotlyarov, DSc (B. Verkin Institute for Low Temperature Physics and Engineering of NAS of Ukraine);
5. A.S. Romanyuk, DSc (Institute of Mathematics of NAS of Ukraine);
6. I.I. Scrypnik, DSc (Institute of Applied Mathematics and Mechanics of NAS of Ukraine);
7. D. Shepelsky, DSc (B. Verkin Institute for Low Temperature Physics and Engineering of NAS of Ukraine).

The work aims at developing methods and approaches of function theory, which can be efficiently applied for solving problems in various areas of mathematics, particularly, in mathematical physics and theory of partial differential equations. Following this aim, we study actual problems of the theory of integrable nonlinear differential and difference equations, conformal, quasi-conformal and quasi-regular mappings, nonlinear elliptic and parabolic equations, complex analysis of non-self-adjoint operators, spectral theory of matrix difference equations, linear and nonlinear approximations of classes of periodic functions of many variables.  In turn, solving these problems has led us to the development of efficient methods for studying mathematical models of nature, particularly, of hydromechanics (theory of waves on shallow and deep water), optics, combustion theory, diffusion and absorption in chemical reactions in anisotropic and inhomogeneous media. 

In the framework of the project, we have initiated, developed and generalized a number of the function theory methods, including the following: the finite-gap integration of nonlinear equations; methods of the inverse scattering problem, in the form of the Marchenko equations and the Riemann-Hilbert factorization problem, for solving the Cauchy problems with step-like initial data for integrable nonlinear differential and difference equations as well as for solving initial-boundary value problems for such equations; methods of complex analysis and the geometrical theory of conformal and quasiconformal mappings with applications to partial differential equations, methods for studying interior and boundary properties of solutions of nonlinear parabolic and elliptic equations; applications of the potential theory and analytic functions to the perturbation theory of linear operators; methods for studying characteristics of the best  trigonometrical and bi-linear approximations of the Sobolev and Nikolskii-Besov functional classes; the theory of orthogonal polynomials and Jacobi matrices, unitary CMV matrices and matrix models.

Number of publications: 139, including 6 monographs and 133 research papers.

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