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A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation
Author(s) -
Bilman Deniz,
Miller Peter D.
Publication year - 2019
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21819
Subject(s) - inverse scattering transform , mathematics , gravitational singularity , limit (mathematics) , inverse scattering problem , context (archaeology) , mathematical analysis , generalization , nonlinear system , inverse , boundary (topology) , boundary value problem , scattering , nonlinear schrödinger equation , inverse problem , schrödinger equation , physics , quantum mechanics , geometry , paleontology , biology
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. © 2019 Wiley Periodicals, Inc.