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Application of the discrete integrable systems on the semi-infinite lattice to the system of the bi-orthogonal polynomials

半无限格上离散可积系统在双正交多项式系统中的应用

基本信息

批准号：
15540119
负责人：
TSUJIMOTO Satoshi
金额：
$ 1.92万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

The purpose of this research is to construct the theory of the spectral transformations of the bi-orthogonal polynomials and the continued fractions from the point of the view of the discrete integrable systems on the semi-infinite lattice. We mainly study the relationship between the bi-orthogonal polynomials which have the three-term relations and the Toda-type discrete integrable systems. We give main results:・The R1 type and the R2 type discrete integrable systems associated with the generalized eigen value problemsWe study the bilinear equations of the R1 type and the R2 type discrete integrable systems. Then we clarify the relationship with the Toda type discrete systems including the nonautonomous discrete Toda equation and the relativistic discrete Toda equation, and derive their Backlund transformations.・The discrete integrable system associated with the rational interpolation functionsWe derive a discrete integrable system associated with Frobenius-Stickelberger's pioneering research on the rational interpolation functions and Thiele's theory on the Pade approximations. Moreover we give the bilinear equations of this integrable system and show that these bilinear equations are derived from the discrete KP equation and the two-dimensional discrete Toda equation. A generalised Lotka-Volterra equation and a generalized epsilon algorithm are presented.・The nonautonomous discrete Toda equatioFrom the research on the bilinear equation of the R1 chain and the R2 chain, we have new results on the nonautonomous discrete Toda equation. The Darboux transformations of the nonautonomous discrete Toda equation are derived from the Darboux transformation of the discrete KP equation and the two-dimensional discrete Toda equation.

本研究的目的是从半无限点阵上离散可积系统的角度构建双正交多项式和连分数的谱变换理论。我们主要研究具有三项关系的双正交多项式与户田型离散可积系统之间的关系。我们给出的主要结果：·与广义特征值问题相关的R1型和R2型离散可积系统我们研究了R1型和R2型离散可积系统的双线性方程。然后我们阐明了与Toda型离散系统（包括非自治离散Toda方程和相对论离散Toda方程）的关系，并导出了它们的Backlund变换。・与有理插值函数相关的离散可积系统我们导出了与Frobenius-Stickelberger对有理插值函数的开创性研究和Thiele的Pade理论相关的离散可积系统近似值。此外，我们还给出了该可积系统的双线性方程，并表明这些双线性方程是由离散KP方程和二维离散Toda方程导出的。提出了广义Lotka-Volterra方程和广义epsilon算法。・非自治离散Toda方程通过对R1链和R2链双线性方程的研究，我们对非自治离散Toda方程有了新的结果。非自治离散Toda方程的达布变换是由离散KP方程和二维离散Toda方程的达布变换推导出来的。