Seminar

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One of the prototypical integrable nonlinear evolution equations is the nonlinear Schrodinger (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 a Dirac operator. This is because the associated Dirac 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 Dirac 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 Dirac 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) some spectral theory of non-self-adjoint Dirac operators on the circle, (ii) the existence of an explicit two-parameter family of elliptic finite-gap potentials of said operator, and time permitting (iii) soliton gases in the semiclassical limit of the focusing NLS equation on the circle.

Spectral Theory of Non-Self-Adjoint Dirac Operators on the Circle 597 KB application/pdf

Spectral Theory of Non-Self-Adjoint Dirac Operators on the Circle