{"title":"ELLIPTIC ASYMPTOTIC REPRESENTATION OF THE FIFTH PAINLEVÉ TRANSCENDENTS","authors":"S. Shimomura","doi":"10.2206/kyushujm.76.43","DOIUrl":null,"url":null,"abstract":"For the fifth Painleve transcendents an asymptotic representation by the Jacobi $\\mathrm{sn}$-function is presented in cheese-like strips along generic directions near the point at infinity. Its elliptic main part depends on a single integration constant, which is the phase shift and is parametrised by monodromy data for the associated isomonodromy deformation. In addition, under a certain supposition, the error term is also expressed by an explicit asymptotic formula, whose leading term is written in terms of integrals of the $\\mathrm{sn}$-function and the $\\vartheta$-function, and contains the other integration constant. Instead of the justification scheme for asymptotic solutions of Riemann-Hilbert problems by the Brouwer fixed point theorem, we begin with a boundedness property of a Lagrangian function, which enables us to determine the modulus of the $\\mathrm{sn}$-function satisfying the Boutroux equations and to construct deductively the elliptic representation.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2020-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Kyushu Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2206/kyushujm.76.43","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4
Abstract
For the fifth Painleve transcendents an asymptotic representation by the Jacobi $\mathrm{sn}$-function is presented in cheese-like strips along generic directions near the point at infinity. Its elliptic main part depends on a single integration constant, which is the phase shift and is parametrised by monodromy data for the associated isomonodromy deformation. In addition, under a certain supposition, the error term is also expressed by an explicit asymptotic formula, whose leading term is written in terms of integrals of the $\mathrm{sn}$-function and the $\vartheta$-function, and contains the other integration constant. Instead of the justification scheme for asymptotic solutions of Riemann-Hilbert problems by the Brouwer fixed point theorem, we begin with a boundedness property of a Lagrangian function, which enables us to determine the modulus of the $\mathrm{sn}$-function satisfying the Boutroux equations and to construct deductively the elliptic representation.
期刊介绍:
The Kyushu Journal of Mathematics is an academic journal in mathematics, published by the Faculty of Mathematics at Kyushu University since 1941. It publishes selected research papers in pure and applied mathematics. One volume, published each year, consists of two issues, approximately 20 articles and 400 pages in total.
More than 500 copies of the journal are distributed through exchange contracts between mathematical journals, and available at many universities, institutes and libraries around the world. The on-line version of the journal is published at "Jstage" (an aggregator for e-journals), where all the articles published by the journal since 1995 are accessible freely through the Internet.