{"paper":{"title":"Asymptotic stability of small solitons to 1D NLS with potential","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tetsu Mizumachi","submitted_at":"2006-05-01T05:59:38Z","abstract_excerpt":"We consider asymptotic stability of a small solitary wave to supercritical 1-dimensional nonlinear Schr\\\"{o}dinger equations $$ iu_t+u_{xx}=Vu\\pm |u|^{p-1}u \\quad\\text{for $(x,t)\\in\\mathbb{R}\\times\\mathbb{R}$,}$$ in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai \\cite{GNT} in the 3-dimensional case using the endpoint Strichartz estimate.\n  To prove asymptotic stability of solitary waves, we need to show that a dispersive part $v(t,x)$ of a solution belongs to $L^2_t(0,\\infty;X)$ for some space $X$. In the 1-dimensional case, this property does not follow from the Strich"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0605031","kind":"arxiv","version":2},"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"}