{"paper":{"title":"Random data Cauchy problem for the nonlinear Schr\\\"{o}dinger equation with derivative nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hiroyuki Hirayama, Mamoru Okamoto","submitted_at":"2015-08-10T08:15:03Z","abstract_excerpt":"We consider the Cauchy problem for the nonlinear Schr\\\"{o}dinger equation with derivative nonlinearity $(i\\partial _t + \\Delta ) u= \\pm \\partial (\\overline{u}^m)$ on $\\R ^d$, $d \\ge 1$, with random initial data, where $\\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\\frac{\\partial}{\\partial x_1} , \\, \\dots , \\, \\frac{\\partial}{\\partial x_d}$ or $|\\nabla |= \\mathcal{F}^{-1}[|\\xi | \\mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\\R ^d)$ with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02161","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"}