{"title":"Riemann Theta Function Solutions to a Hierarchy of Integrable Semi-Discrete Equations","authors":"X.G. Geng, M.X. Jia, X. Zeng, J. Wei","doi":"10.1134/S1061920825600667","DOIUrl":null,"url":null,"abstract":"<p> The theory of tetragonal curves is applied to the study of discrete integrable systems. Based on the discrete Lenard equation, we derive a hierarchy of Blaszak–Marciniak four-field lattice equations associated with the <span>\\(4\\times4\\)</span> matrix spectral problem. Resorting to the characteristic polynomial of the Lax matrix for the hierarchy of Blaszak–Marciniak four-field lattice equation, we introduce a tetragonal curve, a Baker-Akhiezer function, and three meromorphic functions on it. We study algebro-geometric properties of the tetragonal curve and asymptotic behaviors of the Baker-Akhiezer function and meromorphic functions near two infinite points. The straightening out of various flows is exactly given by means of the Abel map and the meromorphic differential. We finally obtain Riemann theta function solutions of the entire Blaszak–Marciniak four-field lattice hierarchy. </p><p> <b> DOI</b> 10.1134/S1061920825600667 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"32 3","pages":"464 - 479"},"PeriodicalIF":1.5000,"publicationDate":"2025-09-29","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/S1061920825600667","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
The theory of tetragonal curves is applied to the study of discrete integrable systems. Based on the discrete Lenard equation, we derive a hierarchy of Blaszak–Marciniak four-field lattice equations associated with the \(4\times4\) matrix spectral problem. Resorting to the characteristic polynomial of the Lax matrix for the hierarchy of Blaszak–Marciniak four-field lattice equation, we introduce a tetragonal curve, a Baker-Akhiezer function, and three meromorphic functions on it. We study algebro-geometric properties of the tetragonal curve and asymptotic behaviors of the Baker-Akhiezer function and meromorphic functions near two infinite points. The straightening out of various flows is exactly given by means of the Abel map and the meromorphic differential. We finally obtain Riemann theta function solutions of the entire Blaszak–Marciniak four-field lattice hierarchy.
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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