dorsal/arxiv
View SchemaA Spectral Mapping Theorem and Invariant Manifolds for Nonlinear Schr\"odinger Equations
| Authors | F. Gesztesy, C. K. R. T. Jones, Y. Latushkin, M. Stanislavova |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9907017 |
| URL | https://arxiv.org/abs/solv-int/9907017 |
Abstract
A spectral mapping theorem is proved that resolves a key problem in applying invariant manifold theorems to nonlinear Schr\" odinger type equations. The theorem is applied to the operator that arises as the linearization of the equation around a standing wave solution. We cast the problem in the context of space-dependent nonlinearities that arise in optical waveguide problems. The result is, however, more generally applicable including to equations in higher dimensions and even systems. The consequence is that stable, unstable, and center manifolds exist in the neighborhood of a (stable or unstable) standing wave, such as a waveguide mode, under simple and commonly verifiable spectral conditions.
{
"annotation_id": "34e52962-43df-49f3-810f-a0ca3c3650dc",
"date_created": "2026-03-02T18:02:51.522000Z",
"date_modified": "2026-03-02T18:02:51.522000Z",
"file_hash": "f15f6a1de29c29411c28f0ae79c4923a72bd57450c7b827ff34eb7f4ef676c68",
"private": false,
"record": {
"abstract": "A spectral mapping theorem is proved that resolves a key problem in applying\ninvariant manifold theorems to nonlinear Schr\\\" odinger type equations. The\ntheorem is applied to the operator that arises as the linearization of the\nequation around a standing wave solution. We cast the problem in the context of\nspace-dependent nonlinearities that arise in optical waveguide problems. The\nresult is, however, more generally applicable including to equations in higher\ndimensions and even systems. The consequence is that stable, unstable, and\ncenter manifolds exist in the neighborhood of a (stable or unstable) standing\nwave, such as a waveguide mode, under simple and commonly verifiable spectral\nconditions.",
"arxiv_id": "solv-int/9907017",
"authors": [
"F. Gesztesy",
"C. K. R. T. Jones",
"Y. Latushkin",
"M. Stanislavova"
],
"categories": [
"solv-int",
"nlin.SI"
],
"title": "A Spectral Mapping Theorem and Invariant Manifolds for Nonlinear Schr\\\"odinger Equations",
"url": "https://arxiv.org/abs/solv-int/9907017"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "f4502db6-a637-4ac0-b564-7fa14d9b1d39",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}