{"title":"Inner Transition Layer in Solutions of the Discrete Painlevé II Equation","authors":"V.Yu. Novokshenov","doi":"10.1134/S1061920824030130","DOIUrl":null,"url":null,"abstract":"<p> We study real-valued asymptotic solutions of the discrete Painlevé equation of second type (dPII) </p><p> In the case of <span>\\(n/\\nu = O(1)\\)</span>, and as <span>\\(n\\to\\infty\\)</span>, the asymptotics is nonuniform. Near the point <span>\\(n= 2\\nu\\)</span>, an <i> inner transition layer</i> occurs, which matches regular asymptotics to the left and to the right of this point. The matching procedure involves classical Painlevé II transcendents. The asymptotics are applied to discrete gap probabilities and random matrix theory. </p><p> <b> DOI</b> 10.1134/S1061920824030130 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"31 3","pages":"517 - 525"},"PeriodicalIF":1.5000,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Journal of Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1134/S1061920824030130","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
We study real-valued asymptotic solutions of the discrete Painlevé equation of second type (dPII)
In the case of \(n/\nu = O(1)\), and as \(n\to\infty\), the asymptotics is nonuniform. Near the point \(n= 2\nu\), an inner transition layer occurs, which matches regular asymptotics to the left and to the right of this point. The matching procedure involves classical Painlevé II transcendents. The asymptotics are applied to discrete gap probabilities and random matrix theory.
期刊介绍：
Russian Journal of Mathematical Physics is a peer-reviewed periodical that deals with the full range of topics subsumed by that discipline, which lies at the foundation of much of contemporary science. Thus, in addition to mathematical physics per se, the journal coverage includes, but is not limited to, functional analysis, linear and nonlinear partial differential equations, algebras, quantization, quantum field theory, modern differential and algebraic geometry and topology, representations of Lie groups, calculus of variations, asymptotic methods, random process theory, dynamical systems, and control theory.
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