Speaker: Deniz Bilman, University of Michigan
Date: November 8, 2018
Title: A robust inverse scattering transform and extreme superposition of rogue waves
Abstract: We propose a modification of the standard inverse scattering transform for the focusing nonlinear Schr\"odinger equation (also other equations by natural generalization) formulated with nonzero boundary conditions at infinity. The purpose is to deal with arbitrary-order poles and potentially severe spectral singularities in a simple and unified way. As an application, we use the modified transform to place the Peregrine solution and related higher-order ``rogue wave'' solutions in an inverse-scattering context for the first time. This allows one to directly study properties of these solutions such as their dynamical or structural stability, or their asymptotic behavior in the limit of high order. We establish the existence of a limiting profile of the rogue wave in the high-order limit and that this profile is a new particular solution of the focusing nonlinear Schr\"odinger equation in the rescaled variables --- the rogue wave of infinite order --- which also satisfies ordinary differential equations with respect to space and time. The spatial differential equations are identified with certain members of the Painlev\'e-III hierarchy.