{"paper":{"title":"On integrals of the tronqu\\'{e}e solutions and the associated Hamiltonians for the Painlev\\'{e} II equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.MP"],"primary_cat":"math-ph","authors_text":"Dan Dai, Lun Zhang, Shuai-Xia Xu","submitted_at":"2019-08-05T09:30:47Z","abstract_excerpt":"We consider a family of tronqu\\'{e}e solutions of the Painelv\\'{e} II equation \\begin{equation*} q''(s)=2q(s)^3+sq(s)-(2\\alpha+\\frac12), \\qquad \\alpha > -\\frac12, \\end{equation*} which is characterized by the Stokes multipliers $$s_1=-e^{-2\\alpha \\pi i },\\quad s_2=\\omega, \\quad s_3=-e^{2 \\alpha \\pi i} $$ with $\\omega$ being a free parameter. These solutions include the well-known generalized Hastings-McLeod solution as a special case if $\\omega=0$. We derive asymptotics of integrals of the tronqu\\'{e}e solutions and the associated Hamiltonians over the real axis for $\\alpha > -1/2$ and $\\omega"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01532","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.01532/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}