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Announcement for Seminar Inverse Scattering Transform (IST), Summer Semester 2023

Instructor: JProf. Dr. Xian Liao

Time & Place: 09:45-11:15, Wednesday, SR 3.068

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

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, Functional Analysis.

Please see here for more information.

Dieng and McLaughlin, Long-time asymptotics for the NLS equation via dbar methods, arXiv preprint, 2008.