Lax Pairs and Rational Solutions of Similarity Reductions for Kupershmidt and Sawada – Kotera Hierarchies

IF 0.8 4区数学 Q3 MATHEMATICS, APPLIED

Regular and Chaotic Dynamics Pub Date : 2021-06-03 DOI:10.1134/S1560354721030059

Nikolay A. Kudryashov

{"title":"Lax Pairs and Rational Solutions of Similarity Reductions for Kupershmidt and Sawada – Kotera Hierarchies","authors":"Nikolay A. Kudryashov","doi":"10.1134/S1560354721030059","DOIUrl":null,"url":null,"abstract":"<div><p>Self-similar reductions for equations of the Kupershmidt and Sawada – Kotera hierarchies are\nconsidered. Algorithms for constructing a Lax pair\nfor equations of these hierarchies are presented. Lax pairs for ordinary differential\nequations of the fifth, seventh and eleventh orders\ncorresponding to the Kupershmidt and the Sawada – Kotera hierarchies are given.\nThe Lax pairs allow us to solve these equations by means of the inverse\nmonodromy transform method. The application of the Painlevé test to the seventh order of the similarity reduction for the Kupershmidt hierarchy is\ndemonstrated. It is shown that special solutions of the similarity reductions for the Kupershnmidt and Sawada – Kotera hierarchies are determined via\nthe transcendents of the <span>\\(K_{1}\\)</span> and <span>\\(K_{2}\\)</span> hierarchies. Rational solutions of the similarity reductions of the modified Kupershmidt and Sawada – Kotera\nhierarchies are given. Special polynomials associated with the self-similar reductions of\nthe Kupershmidt and Sawada – Kotera hierarchies are presented.\nRational solutions of some hierarchies are calculated by means of the Miura transformations and taking into account special polynomials.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"26 3","pages":"271 - 292"},"PeriodicalIF":0.8000,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Regular and Chaotic Dynamics","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1134/S1560354721030059","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 2

Abstract

Self-similar reductions for equations of the Kupershmidt and Sawada – Kotera hierarchies are considered. Algorithms for constructing a Lax pair for equations of these hierarchies are presented. Lax pairs for ordinary differential equations of the fifth, seventh and eleventh orders corresponding to the Kupershmidt and the Sawada – Kotera hierarchies are given. The Lax pairs allow us to solve these equations by means of the inverse monodromy transform method. The application of the Painlevé test to the seventh order of the similarity reduction for the Kupershmidt hierarchy is demonstrated. It is shown that special solutions of the similarity reductions for the Kupershnmidt and Sawada – Kotera hierarchies are determined via the transcendents of the \(K_{1}\) and \(K_{2}\) hierarchies. Rational solutions of the similarity reductions of the modified Kupershmidt and Sawada – Kotera hierarchies are given. Special polynomials associated with the self-similar reductions of the Kupershmidt and Sawada – Kotera hierarchies are presented. Rational solutions of some hierarchies are calculated by means of the Miura transformations and taking into account special polynomials.

查看原文本刊更多论文

Kupershmidt和Sawada - Kotera层次相似性约简的Lax对和有理解

研究了Kupershmidt和Sawada - Kotera层次方程组的自相似约简。给出了构造这些层次方程的Lax对的算法。给出了对应于Kupershmidt和Sawada - Kotera层次的五阶、七阶和十一阶常微分方程的松弛对。Lax对允许我们用反单变换方法求解这些方程。对Kupershmidt层次结构的七阶相似性降阶进行了painlev检验。证明了Kupershnmidt和Sawada - Kotera层次的相似性约简的特殊解是通过\(K_{1}\)和\(K_{2}\)层次的超越来确定的。给出了改进的Kupershmidt和Sawada - Koterahierarchies的相似约简的合理解。提出了与Kupershmidt和Sawada - Kotera层次的自相似约简相关的特殊多项式。利用Miura变换并考虑特殊多项式计算了某些层次的有理解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Regular and Chaotic Dynamics 物理-力学

CiteScore

2.50

自引率

7.10%

发文量

审稿时长

>12 weeks

期刊介绍： Regular and Chaotic Dynamics (RCD) is an international journal publishing original research papers in dynamical systems theory and its applications. Rooted in the Moscow school of mathematics and mechanics, the journal successfully combines classical problems, modern mathematical techniques and breakthroughs in the field. Regular and Chaotic Dynamics welcomes papers that establish original results, characterized by rigorous mathematical settings and proofs, and that also address practical problems. In addition to research papers, the journal publishes review articles, historical and polemical essays, and translations of works by influential scientists of past centuries, previously unavailable in English. Along with regular issues, RCD also publishes special issues devoted to particular topics and events in the world of dynamical systems.