{"paper":{"title":"Strong confinement limit for the nonlinear Schr\\\"odinger equation constrained on a curve","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Florian M\\'ehats, Nicolas Raymond","submitted_at":"2014-12-02T20:14:38Z","abstract_excerpt":"This paper is devoted to the cubic nonlinear Schr\\\"odinger equation in a two dimensional waveguide with shrinking cross section of order $\\epsilon$. For a Cauchy data living essentially on the first mode of the transverse Laplacian, we provide a tensorial approximation of the solution $\\psi^\\epsilon$ in the limit $\\epsilon\\to 0$, with an estimate of the approximation error, and derive a limiting nonlinear Schr\\\"odinger equation in dimension one. If the Cauchy data $\\psi^{\\epsilon}_0$ has a uniformly bounded energy, then it is a bounded sequence in $H^1$ and we show that the approximation is of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.1049","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}