Application of the trigonal curve to the Blaszak–Marciniak lattice hierarchy

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• 作者：Xianguo Geng ; Xin Zeng
• 关键词：Blaszak–Marciniak lattice ; algebro ; geometric solution ; trigonal curve
• 刊名：Theoretical and Mathematical Physics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：190
• 期：1
• 页码：18-42
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Theoretical, Mathematical and Computational Physics; Applications of Mathematics;
• 出版者：Pleiades Publishing
• ISSN：1573-9333
• 卷排序：190

文摘

We develop a method for constructing algebro-geometric solutions of the Blaszak–Marciniak (BM) lattice hierarchy based on the theory of trigonal curves. We first derive the BM lattice hierarchy associated with a discrete (3×3)-matrix spectral problem using Lenard recurrence relations. Using the characteristic polynomial of the Lax matrix for the BM lattice hierarchy, we introduce a trigonal curve with two infinite points, which we use to establish the associated Dubrovin-type equations. We then study the asymptotic properties of the algebraic function carrying the data of the divisor and the Baker–Akhiezer function near the two infinite points on the trigonal curve. We finally obtain algebro-geometric solutions of the entire BM lattice hierarchy in terms of the Riemann theta function.