{"paper":{"title":"Transition asymptotics for the real solutions of the sinh-Gordon Painlev\\'e III equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.CA","math.MP"],"primary_cat":"nlin.SI","authors_text":"Kenta Miyahara, Maxim L. Yattselev","submitted_at":"2026-06-26T20:11:35Z","abstract_excerpt":"We consider solutions of the sinh-Gordon Painlev\\'e III equation \\[ u_{xx} + \\frac{1}{x} u_x = \\sinh u \\] that are real on $(0,\\infty)$. They are parametrized by the monodromy parameter $p\\in\\overline{\\mathbb{C}}$, $|p|>1$, and an additional real parameter $s^{\\mathbb{R}}$ when $p=\\infty$. Our previous joint work with A. Its described the asymptotic behavior of these solutions as $x\\to\\infty$. Here, we describe the transition as $x, p\\to \\infty$, $2\\Im(p)=-s^{\\mathbb R}$, between singular solutions ($|p|<\\infty$) and smooth solutions ($p=\\infty$). In short, if we parametrize $|p|^2 = 1 + e^{2\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28579","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.28579/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"}