{"paper":{"title":"Long-time dynamics of resonant weakly nonlinear CGL equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Guan Huang","submitted_at":"2014-07-04T09:07:51Z","abstract_excerpt":"Consider a weakly nonlinear CGL equation on the torus~$\\mathbb{T}^d$: \\[u_t+i\\Delta u=\\epsilon [\\mu(-1)^{m-1}\\Delta^{m} u+b|u|^{2p}u+ ic|u|^{2q}u].\\eqno{(*)}\\] Here $u=u(t,x)$, $x\\in\\mathbb{T}^d$, $0<\\epsilon<<1$, $\\mu\\geqslant0$, $b,c\\in\\mathbb{R}$ and $m,p,q\\in\\mathbb{N}$. Define \\mbox{$I(u)=(I_{\\dk},\\dk\\in\\mathbb{Z}^d)$}, where $I_{\\dk}=v_{\\dk}\\bar{v}_{\\dk}/2$ and $v_{\\dk}$, $\\dk\\in\\mathbb{Z}^d$, are the Fourier coefficients of the function~$u$ we give. Assume that the equation $(*)$ is well posed on time intervals of order $\\epsilon^{-1}$ and its solutions have there a-priori bounds, indep"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.1156","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"}