{"paper":{"title":"Laguerre polynomials and transitional asymptotics of the modified Korteweg-de Vries equation for step-like initial data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CV","math.MP","nlin.SI"],"primary_cat":"math-ph","authors_text":"Alexander Minakov, Marco Bertola","submitted_at":"2017-11-07T09:57:46Z","abstract_excerpt":"We consider the compressive wave for the modified Korteweg--de Vries equation with background constants $c>0$ for $x\\to-\\infty$ and $0$ for $x\\to+\\infty.$ We study the asymptotics of solutions in the transition zone $4c^2t-\\varepsilon t<x<4c^2t-\\beta t^{\\sigma}\\ln t$ for $\\varepsilon>0,$ $\\sigma\\in(0,1),$ $\\beta>0.$ In this region we have a bulk of nonvanishing oscillations, the number of which grows as $\\frac{\\varepsilon t}{\\ln t}.$ Also we show how to obtain Khruslov--Kotlyarov's asymptotics in the domain $4c^2t-\\rho\\ln t<x<4c^2t$ with the help of parametrices constructed out of Laguerre pol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.02362","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"}