{"paper":{"title":"Instability of the solitary wave solutions for the generalized derivative nonlinear Schr\\\"odinger equation in the endpoint case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bing Li, Cui Ning","submitted_at":"2018-04-08T18:48:54Z","abstract_excerpt":"We consider the stability theory of solitary wave solutions for the generalized derivative nonlinear Schr\\\"odinger equation\n  $$\n  i\\partial_{t}u+\\partial_{x}^{2}u+i|u|^{2\\sigma}\\partial_x u=0,\n  $$\n  where $1<\\sigma<2$.\n  The equation has a two-parameter family of solitary wave solutions of the form\n  $$ u_{\\omega,c}(t,x)=e^{i\\omega t+i\\frac c2(x-ct)-\\frac{i}{2\\sigma+2}\\int_{-\\infty}^{x-ct}\\varphi^{2\\sigma}_{\\omega,c}(y)dy}\\varphi_{\\omega,c}(x-ct).\n  $$\n  The stability theory in the frequency region of $|c|<2\\sqrt{\\omega}$ was studied previously. In this paper, we prove the instability of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.02738","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"}