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This paper has been published in Comm. Pure Appl. Math. To download a preprint of this paper, just click here.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1513054. Any opinions, findings and conclusions or recomendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

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A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation

Deniz Bilman

Department of Mathematics, University of Michigan

 Peter D. Miller

Department of Mathematics, University of Michigan

Abstract:

We propose a modification of the standard inverse scattering transform for the focusing nonlinear Schrödinger 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. The modified transform method also allows rogue waves to be generated on top of other structures by elementary Darboux transformations, rather than the generalized Darboux transformations in the literature or other related limit processes.