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 SchrÃ¶dinger, 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.

Notes

Notes 18.10.2022

Notes 20.10.2022

Exam

There will be an oral exam at the end of the semester. Further details will be forthcoming.