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Inverse Scattering Transform (IST) (Summer Semester 2023)

The preliminary meeting takes place at 13pm, Friday, February 10th, 2023, in SR 1.067 (Mathematical Building).

Presentation: 90min. (questions included). Blackboard and/or slides.

Registration: Interested students are welcome to send an email directly to Xian.liao(at)kit.edu or Robert.wegner(at)kit.edu for registration. Registered students will be added to MS Team "IANA_Liao_SeminarIST". The deadline for registration is 01.04.

Preparation: Contact Robert.wegner(at)kit.edu at least one month before the presentation & Upload Handout/Slides in MS Team in advance.

Description: The cubic Nonlinear Schrödinger (NLS) equation appears as a prototypical model in various physical contexts such as water waves, nonlinear optics and Bose-Einstein condensation. The long-time behaviour of the solutions of (NLS) is of high physical relevance. The analysis in the one-dimensional case is, however, very mathematical challenging, due to the weak dispersion effects. The inverse scattering transform (IST) method provides a new point of view, and formulates the solutions of (NLS) as the asymptotics of the solutions to a related Riemann-Hilbert (RH) problem. This is remarkable, as it reduces the nonlinear problem to a linear problem. In this seminar, we are going to solve the RH-problem, and transfer analytical results concerning the long-time behaviour of the solutions to the (NLS) setting. Contrary to the delicate (harmonic) analysis tools developed for (NLS), the IST-method requires very elementary analysis techniques, and can be grasped easily by Bachelor and Master students.

Prerequisites: Analysis I-4 (in particular complex analysis), Functional Analysis.

Preliminary Schedule (To be completed if there would be more registrations)
The three introductory talks in May consist of the resolution of an ODE system (direct scattering transform), the Cauchy integral formula in Hölder spaces and the resolution of a scalar Riemann-Hilbert problem.
The first talk in June explains the inverse scattering method intuitively by solving the linear Schrödinger equation. The rest three talks in June will apply the IST strategy to Nonlinear Schrödinger equation step by step.

Date Topic Main Reference Additional References
03.05. Direct scattering transform for NLS (1) Lecture 1
10.05. The Cauchy integral (1) Lecture 2 (p. 1-45) (4) Sec. 1.1
17.05. Riemann-Hilbert problems and a scalar model (1) Lecture 2 (p.46 - end), (2) Sec. 2.1 (3) Sec. 3.2
07.06. Asymptotics for the linear Schrödinger equation (2) Introduction, (3) Sec. 2
14.06. A matrix Riemann-Hilbert problem (2) Sec. 2.2 (3) Sec. 3.2 - 3.5
21.06. Reduction of NLS to a dbar problem (2) Sec. 2.3 (3) Sec. 3.2 - 3.5
28.06. Analysis of the dbar problem and asymptotics for NLS (2) Sec. 2.4 - 2.5 (3) Sec. 3.5 - 3.6

(1) P. Miller. Script on Riemann-Hilbert methods in integrable systems. Link
(2) Momar Dieng, K. D. T-R Mclaughlin. Long-time asymptotics for the NLS equation via dbar methods. Link
(3) P. Miller, P. A. Perry, J.-C. Saut, C. Sulem. Nonlinear Dispersive Partial Differential Equations and Inverse Scattering (p. 253 - 295). Link
(4) X. Zhou. Riemann-Hilbert problems and integrable systems - a preliminary version. Link

Seminar: Wednesday 9:45-11:15 SR 3.068 Begin: 3.5.2023, End: 28.6.2023
Lecturer M. Sc. Robert Wegner
Office hours: on Appointment
Room 3.028 Kollegiengebäude Mathematik (20.30)
Email: robert.wegner@kit.edu
Lecturer M. Sc. Rebekka Zimmermann
Office hours: by appointment
Room 3.030 Kollegiengebäude Mathematik (20.30)
Email: rebekka.zimmermann@kit.edu