A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation

• Published in: Communications on pure and applied mathematics Vol. 72; no. 8; pp. 1722 - 1805
• Main Authors: Bilman, Deniz, Miller, Peter D.
• Format: Journal Article
• Language: English
• Published: New York John Wiley and Sons, Limited 01.08.2019
• Subjects: Asymptotic methods; Asymptotic properties; Boundary conditions; Dynamic stability; Inverse scattering; Schrodinger equation; Singularity (mathematics); Structural stability; Transformations (mathematics)
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.21819

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. © 2019 Wiley Periodicals, Inc.