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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1106.1158","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-06-06T19:31:20Z","cross_cats_sorted":["math.AP","math.MP"],"title_canon_sha256":"bdb5c62b52ff2236301b351ab21eff1671d360df0e0e9553d370c41fe1cfb0bf","abstract_canon_sha256":"d990c2086a1c5f971d037ebb49f7a1b31a6e8aecfe84e31bdd9c3c67994a8d28"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:00.628099Z","signature_b64":"6qF/jA0HYpXF//tIrXmMM7+gi7XRAU5/Qtlo+huY6if0wDKK44Wn+mCnMDOUrq6tKtJGUbuh+R/oWEmrrTyLAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b87a789bd8c0eef9bb9003b0c6aee72af2d845b1997d15610dcd27603a06a63e","last_reissued_at":"2026-05-18T03:13:00.627351Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:00.627351Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\nu\\to0$. 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