{"paper":{"title":"Time-averaging for weakly nonlinear CGL equations with arbitrary potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alberto Maiocchi, Guan Huang, Sergei Kuksin","submitted_at":"2014-11-08T17:48:38Z","abstract_excerpt":"Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the form: $$ u_t+i(-\\Delta u+V(x)u)=\\epsilon\\mu\\Delta u+\\epsilon \\mathcal{P}( u),\\quad x\\in {R^d}\\,, \\quad(*)\n  $$ under the periodic boundary conditions, where $\\mu\\geqslant0$ and $\\mathcal{P}$ is a smooth function. Let $\\{\\zeta_1(x),\\zeta_2(x),\\dots\\}$ be the $L_2$-basis formed by eigenfunctions of the operator $-\\Delta +V(x)$. For a complex function $u(x)$, write it as $u(x)=\\sum_{k\\geqslant1}v_k\\zeta_k(x)$ 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.2143","kind":"arxiv","version":4},"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"}