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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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3248_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(4\times4\)</EquationSource> </InlineEquation> 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>

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Riemann Theta Function Solutions to a Hierarchy of Integrable Semi-Discrete Equations

  • 期刊论文

  • 发表日期：

    2025年9月29日

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  • X.G. Geng,
  • M.X. Jia,
  • X. Zeng,
  • J. Wei

摘要

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.

DOI 10.1134/S1061920825600667