{"paper":{"title":"Weakly nonlinear stochastic CGL equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Sergei B. Kuksin","submitted_at":"2011-06-06T19:31:20Z","abstract_excerpt":"We consider the linear Schr\\\"odinger equation under periodic boundary condition, driven by a random force and damped by a quasilinear damping: $$ \\frac{d}{dt}u+i\\big(-\\Delta+V(x)\\big) u=\\nu \\Big(\\Delta u-\\gr |u|^{2p}u-i\\gi |u|^{2q}u \\Big) +\\sqrt\\nu\\, \\eta(t,x).\\qquad (*) $$ The force $\\eta$ is white in time and smooth in $x$. We are concerned with the limiting, as $\\nu\\to0$, behaviour of its solutions on long time-intervals $0\\le t\\le\\nu^{-1}T$, and with behaviour of these solutions under the double limit $t\\to\\infty$ and $\\nu\\to0$. We show that these two limiting behaviours may be described i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.1158","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"}