PLANE WAVE STABILITY OF THE SPLIT-STEP FOURIER METHOD FOR THE NONLINEAR SCHRÖDINGER EQUATION
Plane wave solutions to the cubic nonlinear Schrödinger equation on a torus have recently been shown to behave orbitally stable. Under generic perturbations of the initial data that are small in a high-order Sobolev norm, plane waves are stable over long times that extend to arbitrary negative power...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2014-04-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509414000048/type/journal_article |
| _version_ | 1811156221982408704 |
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| author | ERWAN FAOU LUDWIG GAUCKLER CHRISTIAN LUBICH |
| author_facet | ERWAN FAOU LUDWIG GAUCKLER CHRISTIAN LUBICH |
| author_sort | ERWAN FAOU |
| collection | DOAJ |
| description | Plane wave solutions to the cubic nonlinear Schrödinger equation on a torus have recently been shown to behave orbitally stable. Under generic perturbations of the initial data that are small in a high-order Sobolev norm, plane waves are stable over long times that extend to arbitrary negative powers of the smallness parameter. The present paper studies the question as to whether numerical discretizations by the split-step Fourier method inherit such a generic long-time stability property. This can indeed be shown under a condition of linear stability and a nonresonance condition. They can both be verified in the case of a spatially constant plane wave if the time step-size is restricted by a Courant–Friedrichs–Lewy condition (CFL condition). The proof first uses a Hamiltonian reduction and transformation and then modulated Fourier expansions in time. It provides detailed insight into the structure of the numerical solution. |
| first_indexed | 2024-04-10T04:47:58Z |
| format | Article |
| id | doaj.art-2255a4f412194357828ab4a1d9fbf0e7 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:58Z |
| publishDate | 2014-04-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-2255a4f412194357828ab4a1d9fbf0e72023-03-09T12:34:33ZengCambridge University PressForum of Mathematics, Sigma2050-50942014-04-01210.1017/fms.2014.4PLANE WAVE STABILITY OF THE SPLIT-STEP FOURIER METHOD FOR THE NONLINEAR SCHRÖDINGER EQUATIONERWAN FAOU0LUDWIG GAUCKLER1CHRISTIAN LUBICH2INRIA and ENS Cachan Bretagne, Avenue Robert Schumann, F-35170 Bruz, France Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, F-75230 Paris Cedex 05, FranceInstitut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, GermanyMathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, GermanyPlane wave solutions to the cubic nonlinear Schrödinger equation on a torus have recently been shown to behave orbitally stable. Under generic perturbations of the initial data that are small in a high-order Sobolev norm, plane waves are stable over long times that extend to arbitrary negative powers of the smallness parameter. The present paper studies the question as to whether numerical discretizations by the split-step Fourier method inherit such a generic long-time stability property. This can indeed be shown under a condition of linear stability and a nonresonance condition. They can both be verified in the case of a spatially constant plane wave if the time step-size is restricted by a Courant–Friedrichs–Lewy condition (CFL condition). The proof first uses a Hamiltonian reduction and transformation and then modulated Fourier expansions in time. It provides detailed insight into the structure of the numerical solution.https://www.cambridge.org/core/product/identifier/S2050509414000048/type/journal_articleprimary 65P1065P40; secondary 65M70 |
| spellingShingle | ERWAN FAOU LUDWIG GAUCKLER CHRISTIAN LUBICH PLANE WAVE STABILITY OF THE SPLIT-STEP FOURIER METHOD FOR THE NONLINEAR SCHRÖDINGER EQUATION Forum of Mathematics, Sigma primary 65P10 65P40; secondary 65M70 |
| title | PLANE WAVE STABILITY OF THE SPLIT-STEP FOURIER METHOD FOR THE NONLINEAR SCHRÖDINGER EQUATION |
| title_full | PLANE WAVE STABILITY OF THE SPLIT-STEP FOURIER METHOD FOR THE NONLINEAR SCHRÖDINGER EQUATION |
| title_fullStr | PLANE WAVE STABILITY OF THE SPLIT-STEP FOURIER METHOD FOR THE NONLINEAR SCHRÖDINGER EQUATION |
| title_full_unstemmed | PLANE WAVE STABILITY OF THE SPLIT-STEP FOURIER METHOD FOR THE NONLINEAR SCHRÖDINGER EQUATION |
| title_short | PLANE WAVE STABILITY OF THE SPLIT-STEP FOURIER METHOD FOR THE NONLINEAR SCHRÖDINGER EQUATION |
| title_sort | plane wave stability of the split step fourier method for the nonlinear schrodinger equation |
| topic | primary 65P10 65P40; secondary 65M70 |
| url | https://www.cambridge.org/core/product/identifier/S2050509414000048/type/journal_article |
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