Lecture Course : Riemann-Hilbert problems and their applications to integrable partial differential equations

Start: Monday, April 5, 2027, 14:00 - 15:30 h
Further dates:  Every Wednesday, April 7, 14, 21, 2027
                               and every Monday, April 12, 19, 26, 2027 14:00 - 15:30 h 
End: Monday, April 26, 2027

Abstract: 
In contrast with linear partial differential equations (PDE), there is no “general theory” of nonlinear partial differential equations. But for some classes of nonlinear PDE, it turns out that one can not only construct a vast amount of exact solutions but also solve, in a quite efficient way, general problems of certain type, like the Cauchy  (initial value) problems. This holds for a class of nonlinear PDE called “integrable equations”. They are also known as  “soliton equations”, because they support solutions (solitons) which are localized,  stable nonlinear travelling waves that retain their shape and speed after interactions. 

In this series of lectures, we introduce a basic property which characterize a PDE as being integrable – the Lax pair representation, and discuss how to use an analogous property of linear PDE in order to obtain a representation of the solution of the Cauchy problem (for a linear PDE that can be viewed as a linearization of an integrable PDE) in terms of the solution of an associated Riemann-Hilbert problem. For this, we discuss some basics of Riemann-Hilbert problems, which are boundary-value problems for (piece-wise) analytic functions in the complex plane.

Further, we discuss how to obtain a representation of the solution of the Cauchy problem for a nonlinear PDE in terms of the solution of an appropriate Riemann-Hilbert problem. As a prototype model, we use the nonlinear Schrödinger equation, which is a basic model of nonlinear wave propagation in various situations. In particular, we discuss the soliton solutions in the framework of the RH problem approach. 

Perhaps the most important advantage of the Riemann–Hilbert approach to integrable partial differential equations is that it allows us to develop a highly effective method for studying the long-term behavior of solutions to Cauchy problems for such equations using the so-called “nonlinear steepest descent method.” We will discuss how this works for problems “on various backgrounds,” starting with the simplest case of a zero background (provided that the solution decays at the spatial infinities) and moving on to nonzero backgrounds, including “step-like backgrounds.”

Finally, we will consider the Uniform Transform Method (or the Fokas method) for studying  initial-boundary value problems for integrable PDEs.

Contents of the course:
TBA

Requirements
TBA

Lecture Course Announcement (pdf) - will follow

Link to the course directory - will follow