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Nonlinear stage of modulational instability, solitons and coherent structures

Modulational instability (or MI, a.k.a. the Benjamin-Feir instability in deep water waves), is the instability of a constant background to long-wavelength perturbations, and is a ubiquitous nonlinear phenomenon which arises in many different physical contexts.Ìý Even though MI was discovered in the 1960's, a characterization of the nonlinear stage of MI -- i.e., the behavior of solutions once the perturbations have become comparable with the background -- was missing until recently.Ìý This talk will describe recent progress on this subject.Ìý The most common tool to study MI is the nonlinear Schroedinger (NLS) equation.Ìý In the first part of the talk, I will briefly describe the inverse scattering transform (IST) for the focusing NLS equation with non-zero background, and I will identify the signature of MI within the IST.Ìý Contrary to a recent conjecture, the main mechanism for the instability is not the formation of solitons, but rather the growth of the Jost eigenfunctions along a certain portion of the continuous spectrum, which provide the precise nonlinear analogue of the unstable Fourier modes.Ìý Then I will show how one can characterize the nonlinear stage of MI by using the IST to compute the long-time asymptotics of solutions of the NLS equation with localized perturbations of the constant background.Ìý At large times, the space-time plane divides into three regions: a far left field and far right field and a central region.Ìý In the left far field and right far field the solution equals the background to leading order up to a phase.Ìý In the central region the solution is described by a coherent oscillation structure comprised of a slow modulation of the periodic traveling wave solutions of the focusing NLS equation.Ìý Importantly, I will also show that this kind of behavior is not limited to the NLS equation, but is instead shared by many different nonlinear models that exhibit MI (including several PDEs, nonlocal systems and differential-difference equations).Ìý Finally, I will briefly discuss the solution behavior for more general scenarios, including interactions between solitons and these coherent oscillation structures.