Dynamics near the origin of the long range scattering for the one-dimensional Schrödinger equation
We consider the cubic Schrödinger equation on the line, for which the scattering theory requires modifications due to long range effects. We revisit the construction of the modified wave operator, and recall the construction of its inverse, in order to describe the asymptotic behavior of these opera...
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| Language: | English |
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.676/ |
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| author | Carles, Rémi |
| author_facet | Carles, Rémi |
| author_sort | Carles, Rémi |
| collection | DOAJ |
| description | We consider the cubic Schrödinger equation on the line, for which the scattering theory requires modifications due to long range effects. We revisit the construction of the modified wave operator, and recall the construction of its inverse, in order to describe the asymptotic behavior of these operators near the origin. At leading order, these operators, whose definition includes a nonlinear modification in the phase compared to the linear dynamics, correspond to the identity. We compute explicitly the first corrector in the asymptotic expansion, and justify this expansion by error estimates. |
| format | Article |
| id | doaj-art-7d1cb995e93f410699078b23893f58e8 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-7d1cb995e93f410699078b23893f58e82025-02-07T11:26:37ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G121717174210.5802/crmath.67610.5802/crmath.676Dynamics near the origin of the long range scattering for the one-dimensional Schrödinger equationCarles, Rémi0Univ Rennes, CNRS, IRMAR – UMR 6625, F-35000 Rennes, FranceWe consider the cubic Schrödinger equation on the line, for which the scattering theory requires modifications due to long range effects. We revisit the construction of the modified wave operator, and recall the construction of its inverse, in order to describe the asymptotic behavior of these operators near the origin. At leading order, these operators, whose definition includes a nonlinear modification in the phase compared to the linear dynamics, correspond to the identity. We compute explicitly the first corrector in the asymptotic expansion, and justify this expansion by error estimates.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.676/Nonlinear Schrödinger equationlong range scatteringasymptotic expansionerror estimate |
| spellingShingle | Carles, Rémi Dynamics near the origin of the long range scattering for the one-dimensional Schrödinger equation Comptes Rendus. Mathématique Nonlinear Schrödinger equation long range scattering asymptotic expansion error estimate |
| title | Dynamics near the origin of the long range scattering for the one-dimensional Schrödinger equation |
| title_full | Dynamics near the origin of the long range scattering for the one-dimensional Schrödinger equation |
| title_fullStr | Dynamics near the origin of the long range scattering for the one-dimensional Schrödinger equation |
| title_full_unstemmed | Dynamics near the origin of the long range scattering for the one-dimensional Schrödinger equation |
| title_short | Dynamics near the origin of the long range scattering for the one-dimensional Schrödinger equation |
| title_sort | dynamics near the origin of the long range scattering for the one dimensional schrodinger equation |
| topic | Nonlinear Schrödinger equation long range scattering asymptotic expansion error estimate |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.676/ |
| work_keys_str_mv | AT carlesremi dynamicsneartheoriginofthelongrangescatteringfortheonedimensionalschrodingerequation |