{"paper":{"title":"Modified scattering for the cubic Schr\\\"odinger equation on product spaces and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Benoit Pausader, Nicola Visciglia, Nikolay Tzvetkov, Zaher Hani","submitted_at":"2013-11-10T14:14:46Z","abstract_excerpt":"We consider the cubic nonlinear Schr\\\"odinger equation posed on the spatial domain $\\mathbb{R}\\times \\mathbb{T}^d$. We prove modified scattering and construct modified wave operators for small initial and final data respectively ($1\\leq d\\leq 4)$. The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when $d\\geq 2$. As a consequence, we obtain global solutions to the defocusing and focusing problems on $\\mathbb{R}\\times \\mathbb{T}^d$ (for any $d\\geq 2$) with infinitely growing high Sob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.2275","kind":"arxiv","version":3},"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"}