{"paper":{"title":"An averaging theorem for nonlinear Schr\\\"odinger equations with small nonlinearities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Guan Huang","submitted_at":"2013-12-03T10:27:44Z","abstract_excerpt":"Consider nonlinear Schr\\\"odinger equations with small nonlinearities\n  \\[\\frac{d}{dt}u+i(-\\triangle u+V(x)u)=\\epsilon \\mathcal{P}(\\triangle u,u,x),\\quad x\\in \\mathbb{T}^d.\\eqno{(*)}\\]\n  Let $\\{\\zeta_1(x),\\zeta_2(x),\\dots\\}$ be the $L_2$-basis formed by eigenfunctions of the operator $-\\triangle +V(x)$. For any complex function $u(x)$, write it as \\mbox{$u(x)=\\sum_{k\\geqslant1}v_k\\zeta_k(x)$} and set $I_k(u)=\\frac{1}{2}|v_k|^2$. Then for any solution $u(t,x)$ of the linear equation $(*)_{\\epsilon=0}$ we have $I(u(t,\\cdot))=const$. In this work it is proved that if $(*)$ is well posed on time-in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.0759","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"}