{"paper":{"title":"On Traveling Solitary Waves and Absence of Small Data Scattering for Nonlinear Half-Wave Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Enno Lenzmann, Jacopo Bellazzini, Nicola Visciglia, Vladimir Georgiev","submitted_at":"2018-08-24T13:31:32Z","abstract_excerpt":"We consider nonlinear half-wave equations with focusing power-type nonlinearity $$ i \\pt_t u = \\sqrt{-\\Delta} \\, u - |u|^{p-1} u, \\quad \\mbox{with $(t,x) \\in \\R \\times \\R^d$} $$ with exponents $1 < p < \\infty$ for $d=1$ and $1 < p < (d+1)/(d-1)$ for $d \\geq 2$. We study traveling solitary waves of the form $$ u(t,x) = e^{i\\omega t} Q_v(x-vt) $$ with frequency $\\omega \\in \\R$, velocity $v \\in \\R^d$, and some finite-energy profile $Q_v \\in H^{1/2}(\\R^d)$, $Q_v \\not \\equiv 0$. We prove that traveling solitary waves for speeds $|v| \\geq 1$ do not exist. Furthermore, we generalize the non-existence"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.08134","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"}