Seminar Event Detail

Integrable Systems and Random Matrix Theory

Date:  Monday, October 11, 2021
Location:  ZOOM ID: 926 6491 9790 Virtual (4:00 PM to 5:00 PM)

Title:  The focusing nonlinear Schrödinger equation on the circle: spectral theory, elliptic finite-gap potentials, and soliton gases.

Abstract:   One of the prototypical integrable nonlinear evolution equations is the nonlinear Schr\"odinger (NLS) equation, which is a universal model for weakly nonlinear dispersive wave packets, and as such it arises in a variety of physical settings, including deep water, optics, acoustics, plasmas, condensed matter, etc. A key role in many studies of the NLS equation is played by the Zakharov-Shabat (ZS) spectral problem. This is because the associated ZS operator, which is a first-order matrix differential operator, makes up the first half of the Lax pair of the NLS equation. There are two variants of the NLS equation, referred to as focusing and defocusing, respectively. The corresponding ZS operators are also referred to as focusing and defocusing. In optics, the focusing NLS equation arises when the refraction increases with increasing wavelength, i.e., in the case of anomalous dispersion. Solutions of the focusing and defocusing NLS equation have very different physical behavior. In turn, these differences reflect a markedly different mathematical structure. In particular, the ZS operator for the defocusing NLS equation is self-adjoint, while that for the focusing NLS equation is not. In this seminar I will discuss (i) the existence of an explicit two-parameter family of elliptic finite-gap potentials of the focusing ZS operator, and (ii) soliton gases in the semiclassical limit of the focusing NLS equation on the circle.

A recording can be found here.


Speaker:  Jeffrey Oregero
Institution:  MSRI

Event Organizer:   Ahmad Barhoumi


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