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Alexander Tovbis: Focusing Nonlinear Schroedenger Equation (fNLS): from small dispersion limit to soliton/breather gases

Time: Tue 2019-10-01 15.15 - 16.15

Location: F11, KTH

Participating: Alexander Tovbis


1D fNLS is a nonlinear PDE that can be integrated through the inverse scattering transform method. The small dispersion (semiclassical) limit of the fNLS is a convenient tool to study large scale space-time behavior of the fNLS evolution of a given initial data (potential). In the process of such evolution, a potential typically develops an increasingly complicated patterns that can be locally approximated by some special quasi-periodic solutions of the ​fNLS known as finite-gap (nonlinear multi phase wave) solutions. These solutions can be defined and expressed in terms of the corresponding hyperelliptic Riemann surfaces. Rigorous derivations of such semiclassical approximations was a big success of the nonlinear steepest descent method of Deift ​and Zhou, applied to corresponding matrix Riemann Hilbert Problems.

However, as the number of phases (genus) typically grows with the time, the approximation of ​a particular solution becomes meaningless after some point. Then it starts to make more sense to look for some ``in large" characteristics of the solutions, such as , for example, the probability density function, the kinetic and potential energy, etc. At this point it also makes sense to talk about gas dynamics of some fNLS related gases, where elementary fNLS solutions such as solitons, breathers, etc. play the role of gas particles. From this point of view, the theory of soliton, breather and more general gases can be described in terms of the special high genus limits of hyperelliptic Riemann surfaces, which parametrize the finite gap solutions. I will talk about my recent work in that direction.