A series of 10 lectures by Prof. Dmitry Shepelsky of the B. I. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, Mathematical division

Schedule (with slides):

Wednesday, 9.10.2019 (slides): 15:15-17:00
Thursday, 10.10.2019 (slides): 14:15-16:00
Friday, 11.10.2019 (slides): 14:15-16:00

Tuesday, 15.10.2019 (slides): 14:15-16:00
Wednesday, 16.10.2019 (slides): 15:15-17:00

Abstract:

Riemann-Hilbert (RH) problems are boundary-value problems for sectionally analytic functions in the complex plane. It is a remarkable fact that a vast array of problems in mathematics, mathematical physics, and applied mathematics can be posed as Riemann-Hilbert problems. These include radiation, elasticity, hydrodynamic, diffraction problems, orthogonal polynomials and random matrix theory, nonlinear ordinary and partial differential equations. In applications, the data for a RH problem depend on external parameters, which are physical variables (space, time, matrix size, etc), and, in turn, the solution depends on these parameters as well. It is this dependence that we are interested in, when speaking about the RH problem as a method for studying problems from one or another domain.

The representation of a solution to a nonlinear partial differential equation (PDE) in terms of a solution of the associated RH problem can be viewed as a nonlinear analogue of the contour integral representation for a linear PDE. It provides means to efficiently study not only the existence and uniqueness problems for a class of nonlinear PDE, but (i) to derive detailed asymptotics of solutions of initial value problems and initial boundary value problems for such equations and (ii) to accurately evaluate the solutions inside as well as outside asymptotic regimes. Recent literature on applications of the RH problem includes the monographs [1-3].

The main aim of the proposed course is to introduce the Inverse Scattering Transform method, in the form of the RH problem, for studying integrable nonlinear differential equations and to illustrate the fruitfulness of the method by studying the long-time asymptotics of solutions of such equations. As a prototype model, we will use the nonlinear Schrödinger equation, which is a basic models of nonlinear wave propagation (for instance, in the fiber optics).

References:

[1] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, AMS, Providence, Rhode Island, 2000.
[2] A. S. Fokas, A. R. Its, A. A. Kapaev and V. Yu. Novokshenov, Painleve Transcendens: The Riemann-Hilbert Approach, Mathematical surveys and monographs 128, AMS, 2006.
[3] T. Trogdon and S. Olver, Riemann–Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special  Functions, SIAM, Philadelphia, 2016.