The Computational Theory of Riemann-Hilbert Problems - Lecture 4

Explore the computational theory of Riemann-Hilbert problems in this advanced mathematics lecture by Thomas Trogdon. Delve into topics such as computing Cauchy integrals, controlled bases, generalized contours, and singular integral equations. Examine Sobolev spaces, zero-sum spaces, and the regularity of jump matrices. Learn about numerical solutions for Riemann-Hilbert problems and their applications to nonlinear equations like the defocusing nonlinear Schrödinger equation and the KdV equation. Gain insights into steepest descent methods, code implementation, and various deformations. This in-depth lecture is part of a program on integrable systems in mathematics, condensed matter, and statistical physics organized by the International Centre for Theoretical Sciences.

Integrable systems in Mathematics, Condensed Matter and Statistical Physics
The computational theory of Riemann-Hilbert problems Lecture 4
Computing Cauchy integrals
A controlled basis
Generalizing the contours
A definition and a singular integral equation
Sobolev spaces
Zero-sum space
Regularity of the jump matrix
Associated operators
Smoothness
Some notes on numerical solutions
The numerical solution of Riemann- Hilbert problems
The defocusing nonlinear Schrodinger equation
The initial value problem
An important calculation
Steepest descent
[Code Walkthrough]
A deformation
The KdV equation
The KdV equation with decaying data
Nonlinear superposition
With some solitons
Other work
Deformations