{"paper":{"title":"Reduction of weakly nonlinear parabolic partial differential equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hayato Chiba","submitted_at":"2013-02-04T01:16:11Z","abstract_excerpt":"It is known that the Swift-Hohenberg equation $\\partial u/\\partial t = -(\\partial_x^2 + 1)^2u + \\varepsilon (u-u^3)$ can be reduced to the Ginzburg-Landau equation (amplitude equation) $\\partial A/\\partial t = 4\\partial_x^2 A + \\varepsilon (A-3A|A|^2)$ by means of the singular perturbation method. This means that if $\\varepsilon >0$ is sufficiently small, a solution of the latter equation provides an approximate solution of the former one. In this paper, a reduction of a certain class of a system of nonlinear parabolic equations $\\partial u/\\partial t = \\mathcal{P}u + \\varepsilon f(u)$ is prop"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.0562","kind":"arxiv","version":2},"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"}