V5B1 - Advanced Topics in Analysis and PDE - Integrable Systems and Nonlinear Fourier Transforms, Winter term 2022/2023 - Mathematical Institute of the University of Bonn - Mathematical Institute of the University of Bonn

V5B1 - Advanced Topics in Analysis and PDE- Integrable Systems and Nonlinear Fourier transforms

Winter Semester 2022/23

Prof. Dr. Herbert Koch: Instructor

Lectures

Tue 12-14, Thu 14-16 Room 0.008, Endenicher Allee 60. First lecture: Tuesday, October 18 and October 20, via Zoom and streamed to 0.008

Whenever needed: Zoom Meeting-ID: 981 0438 0929, Passcode 48

Office hours

Tue 14-15, Room 2.011, Endenicher Allee 60. Or by appointment.

Content

There is a huge number of equations under the roof of integrable equations: Geodesics on ellipsoids, varies tops, Nonlinear Schrödinger equation, Korteweg-de Vries equation, Kadomtsev-Petviashvili equation, Cubic Szegö equation, derivative nonlinear Schrödinger, sine Gordon and so on, with connection to many areas in mathematics from Algebraic Geometry, Probability and PDEs to Statistical Mechanics.

The notion of integrability is not clearly defined: There is the analytic notion of Liouville integrability for finite dimensional Hamiltonian equations, and a less precise one of algebraic integrability which is motivated by the desire to have formulas. Even without clear definition integrable systems cover nonlinear structures which allow to compute objects exactly. There are not many such structures and it should not be surprising that the same structures occur in different areas of mathematics, many of them relating a study of the spectral properties of operators to nonlinear differential equations.

Many of the basic integrable equations occur as universal asymptotic limits of wave propagation. The lecture will consist of several parts. We will connect analytics and algebraic aspects.

Topics

A) Introduction, Hamiltonian systems, Poisson structure and Liouville integrability

B) The linear Schrödinger operator, its resolvent and the Korteweg-de Vries equations.

C) Structure of algebraic integrability. Toda lattice.

D) The Korteweg-de Vries equation at low regularity: The approach of Killip and Visan

E) An analytic approach to the Nonlinear Schrödinger equation.

Literature

The following four books are recommended but not required:

Babelon, Bernard, Talon: Introduction to classical integrable systems, Cambridge Monographs in Mathematics Physic 2000
Deift: Orthogonal polynomials and random matrices: A Riemann-Hilbert approach Courant Lecture Notes in Math. 1999
Faddeev, Takhtajan: Hamiltonian methods in the theory of solitons, Springer 1987
Forrester: Log gases and random matrices, London Math. Soc. 2010
Ifrim, Tataru: Testing by wave packets and modified scattering in nonlinear disersive pde's arxiv:2204:13285, 2022
Ifrim, Tataru: Global solutions for 1d cubic defocusing dispersive equations 1, arxiv:2205.12212, 2022
Killip, Visan: KdV is wellposed at H^{-1}, Annals of Math (2) 190, 2019, 249-305.
Moser: Integrable Hamiltonian systems and spectral theory, Scuola Normale Superiore Pisa 1981
Segal: Integrable systems and inverse scattering in: Integrable systems, Oxford, 2011

The prerequisites include basic knowledge about functional analysis, partial differential equations and complex analysis.