Nonlinear waves, solitons and integrable systems - From water to optics, Bose-Einstein condensates and beyond

by Gino Biondini (State University of New York at Buffalo)

Tuesday 23 May 2023, 16:00 → 17:00 Europe/Rome

SEMINARIO ONLINE

SEMINARIO DI FISICA MATEMATICA

The development of the theory of nonlinear wave equations, solitons and integrable systems has been one of the great achievements of mathematical physics of the second half of the 20th century, with numerous applications ranging from wavter waves to optics, Bose-Einstein condensation, cosmology, and beyond.  In this talk I will present some of my recent work in this area.  In particular, I will discuss three classes of problems:

1. Nonlinear stage of modulational instability (MI).  MI -- the instability of a constant background to long-wavelength perturbations -- is a ubiquitous nonlinear phenomenon discovered in the 1960’s.  Until recently, however, a characterization of the nonlinear stage of MI induced by localized perturbations of a constant background was missing.  First I will show how one can identify the signature of MI in the inverse scattering transform for the focusing nonlinear Schrodinger (NLS) equation.  Then I will show how one can rigorously characterize the nonlinear stage of MI by computing the long-time asymptotics of the NLS equation for localized perturbations of a constant background.  Finally, I will show that this kind of behavior is not limited to the NLS equation, but it is shared among many different models (including PDEs, nonlocal systems and differential-difference equations).  The corresponding behavior has been observed in nonlinear optics experiments.

2. Whitham modulation theory for (2+1)-dimensional evolution equations and applications.  In 1965, G.B. Whitham formulated a nonlinear modulation theory for evolution equations, which allows one to study the small-dispersion limit by deriving a set of hyperbolic PDEs describing the modulation of the parameters of the traveling-wave solutions of KdV.  While Whitham modulation theory has been applied with success in a variety of settings, most studies were limited to PDEs in one spatial dimension.  I will show how one can formulate a (2+1)-dimensional generalization of Whitham modulation theory for the Kadomtsev-Petviashvili (KP) equation and other physically significant equations.  I will then show how the system can be used to study the stability of the periodic traveling wave solutions of these equations, as well as to characterize a variety of other physically interesting initial value problems.

3. Soliton resonance and web structure in (2+1)-dimensional integrable systems.  One of the striking features of some (2+1)-dimensional soliton systems is that they admit a much wider family of solutions compared to (1+1)-dimensional ones, which describe novel kinds of behavior.  In the last part of the talk I will show (i) how one can characterize analytically a large class of soliton solutions of the KP equation and other physically significant evolution equations such as the Davey-Stewartson system, (ii) how these solutions describe soliton resonance and web structure, and (iii) how different combinations of physical parameters lead to different types of soliton interactions.

Link zoom: https://uniroma1.zoom.us/j/82330282429?pwd=MktKdmlkSUxxdXlraFhKSHhJTDZnUT09