{"paper":{"title":"Surface Gap Soliton Ground States for the Nonlinear Schr\\\"{o}dinger Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["nlin.PS"],"primary_cat":"math.AP","authors_text":"Michael Plum, Tom\\'a\\v{s} Dohnal, Wolfgang Reichel","submitted_at":"2010-11-12T11:39:34Z","abstract_excerpt":"We consider the nonlinear Schr\\\"{o}dinger equation $(-\\Delta +V(x))u = \\Gamma(x) |u|^{p-1}u$, $x\\in \\R^n$ with $V(x) = V_1(x) \\chi_{\\{x_1>0\\}}(x)+V_2(x) \\chi_{\\{x_1<0\\}}(x)$ and $\\Gamma(x) = \\Gamma_1(x) \\chi_{\\{x_1>0\\}}(x)+\\Gamma_2(x) \\chi_{\\{x_1<0\\}}(x)$ and with $V_1, V_2, \\Gamma_1, \\Gamma_2$ periodic in each coordinate direction. This problem describes the interface of two periodic media, e.g. photonic crystals. We study the existence of ground state $H^1$ solutions (surface gap soliton ground states) for $0<\\min \\sigma(-\\Delta +V)$. Using a concentration compactness argument, we provide an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.2886","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"}