A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation
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 sp...
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| Published in | Communications on pure and applied mathematics Vol. 72; no. 8; pp. 1722 - 1805 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
John Wiley and Sons, Limited
01.08.2019
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| Subjects | |
| Online Access | Get full text |
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| 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. |
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| AbstractList | 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. |
| Author | Miller, Peter D. Bilman, Deniz |
| Author_xml | – sequence: 1 givenname: Deniz surname: Bilman fullname: Bilman, Deniz email: bilman@umich.edu organization: University of Michigan – sequence: 2 givenname: Peter D. surname: Miller fullname: Miller, Peter D. email: millerpd@umich.edu organization: University of Michigan |
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| Snippet | We propose a modification of the standard inverse scattering transform for the focusing nonlinear Schrödinger equation (also other equations by natural... |
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| SubjectTerms | Asymptotic methods Asymptotic properties Boundary conditions Dynamic stability Inverse scattering Schrodinger equation Singularity (mathematics) Structural stability Transformations (mathematics) |
| Title | A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation |
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