{"paper":{"title":"Some $L^\\infty$ solutions of the hyperbolic nonlinear Schr\\\"odinger equation and their stability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"M\\'ario Figueira, Sim\\~ao Correia","submitted_at":"2015-10-29T15:44:30Z","abstract_excerpt":"Consider the hyperbolic nonlinear Schr\\\"odinger equation (HNLS) over $\\mathbb{R}^d$ $$ iu_t + u_{xx} - \\Delta_{\\textbf{y}} u + \\lambda |u|^\\sigma u=0. $$ We deduce the conservation laws associated with (HNLS) and observe the lack of information given by the conserved quantities. We build several classes of particular solutions, including \\textit{spatial plane waves} and \\textit{spatial standing waves}, which never lie in $H^1$. Motivated by this, we build suitable functional spaces that include both $H^1$ solutions and these particular classes, and prove local well-posedness on these spaces. M"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.08745","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"}