{"paper":{"title":"Backlund Transformations and Hierarchies of Exact Solutions for the Fourth Painleve Equation and their Application to Discrete Equations","license":"","headline":"","cross_cats":["nlin.SI"],"primary_cat":"solv-int","authors_text":"Andrew P. Bassom (Department of Mathematics, Exeter, Peter A. Clarkson, U.K.), University of Exeter","submitted_at":"1994-12-09T11:48:03Z","abstract_excerpt":"In this paper we describe B\\\"acklund transformations and hierarchies of exact solutions for the fourth Painlev\\'e equation (PIV) $${\\d^2 w\\over\\d z^2}={1\\over2w}\\left(\\d w\\over\\d z\\right)^2 + {{3\\over2}}w^3 + 4zw^2 + 2(z^2-\\alpha)w+{\\beta\\over w},\\eqno(1){\\hbox to 16pt{\\hfill}}$$ with $\\alpha$, $\\beta$ constants. Specifically, a nonlinear superposition principle for PIV, hierarchies of solutions expressible in terms of complementary error or parabolic cylinder functions as well as rational solutions will be derived. Included amongst these hierarchies are solutions of (1) for which $\\alpha=\\pm\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"solv-int/9412002","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/solv-int/9412002/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"}