{"paper":{"title":"Orbital stability of solitary waves for generalized derivative nonlinear Schr\\\"odinger equations in the endpoint case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Qing Guo","submitted_at":"2017-05-12T07:56:34Z","abstract_excerpt":"We consider the following generalized derivative nonlinear Schr\\\"odinger equation \\begin{equation*} i\\partial_tu+\\partial^2_xu+i|u|^{2\\sigma}\\partial_xu=0,\\ (t,x)\\in\\mathbb R\\times\\mathbb R \\end{equation*} when $\\sigma\\in(0,1)$. The equation has a two-parameter family of solitary waves $$u_{\\omega,c}(t,x)=\\Phi_{\\omega,c}(x)e^{i\\omega t+\\frac{ic}2x-\\frac i{2\\sigma+2}\\int_0^x\\Phi_{\\omega,c}(y)^{2\\sigma}dy},$$ with $(\\omega,c)$ satisfying $\\omega>c^2/4$, or $\\omega=c^2/4$ and $c>0$. The stability theory in the frequency region $\\omega>c^2/4$ was studied previously. In this paper, we prove the sta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.04458","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"}