{"paper":{"title":"Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schr\\\"odinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Daniele Valeri, Giuseppe Genovese, Renato Luc\\`a","submitted_at":"2015-02-20T18:43:59Z","abstract_excerpt":"We study the one dimensional periodic derivative nonlinear Schr\\\"odinger (DNLS) equation. This is known to be a completely integrable system, in the sense that there is an infinite sequence of formal integrals of motion $\\int h_k$, $k\\in \\mathbb{Z}_{+}$. In each $\\int h_{2k}$ the term with the highest regularity involves the Sobolev norm $\\dot H^{k}(\\mathbb{T})$ of the solution of the DNLS equation. We show that a functional measure on $L^2(\\mathbb{T})$, absolutely continuous w.r.t. the Gaussian measure with covariance $(\\mathbb{I}+(-\\Delta)^{k})^{-1}$, is associated to each integral of motion"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.05967","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"}