Studies on the orthogonal polynomials, the continued fractions and the discrete integrable systems

正交多项式、连分式和离散可积系统的研究

• 批准号: 12640121
• 负责人: TSUJIMOTO Satoshi
• 金额: $ 1.98万
• 依托单位: Kyoto University (2001)Osaka University (2000)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640121/
• 关键词: discrete integrable system; orthogonal polynomial; Toda equation; Lotka-Volterra equation; algorithm; eigen-value problem; identity of determinants; continued fractions; ロトカ・ボルテラ方程式; ソリトン; 離散化; 行列式; 可積分系; 有理多項式; Volterra方程式

In the last decades, discrete integrable system has been getting a lot of attention from the viewpoints of difference scheme and algorithm. Already there have been many discussions about relations between continuous-time integrable system and orthogonal polynomials. There are only a few studies on the relations between full-discretised (discrete time and space) systems and orthogonal polynomials fully clarified, such as the discrete integrable system (Toda, Lotka-Volterra) and classical orthogonal polynomials by Spiridonov and Zhedanov. Hence the purpose of this studies is to clarify the relations between the discrete integrable system and the orthogonal polynomials and to develop the Numerical algorithms related to the orthogonality. At first, we pick up the discrete hungry Lotka-Volterra equation and the coupled KP equation. We probe them by means of Hirota's tau-function and develop the relations.

近几十年来，离散可积系统从差分格式和算法的角度得到了广泛的关注。关于连续时间可积系统和正交多项式之间的关系已经有很多讨论。关于完全离散（离散时间和空间）系统与正交多项式之间关系的研究很少，例如离散可积系统（Toda、Lotka-Volterra）以及Spiridonov和Zhedanov的经典正交多项式。因此，本研究的目的是阐明离散可积系统与正交多项式之间的关系，并开发与正交性相关的数值算法。首先，我们采用离散饥饿 Lotka-Volterra 方程和耦合 KP 方程。我们通过 Hirota 的 tau 函数来探究它们并发展它们的关系。

项目成果

S.Tsujimoto, Y.Nakamura, M.Iwasaki: "The discrete Lotka-Volterra system computes singular values"Inverse Problem. 17. 53-58 (2001)

S.Tsujimoto、Y.Nakamura、M.Iwasaki：“离散 Lotka-Volterra 系统计算奇异值”反问题。

R. Hirota, M. Iwao and S. Tsujimoto: "Soliton equations exhibiting "Pfaffian Solutions""Glasgow Math. J.. 43A. 33-41 (2001)

R. Hirota、M. Iwao 和 S. Tsujimoto：“展示“普法夫解”的孤子方程”格拉斯哥数学。

A. Mukaihira and Y. Nakamura: "Schur flow for orthogonal polynomials on the unit circle and its integrable discretization"J. Comput. Appl. Math.. 139. 75-94 (2000)

A. Mukaihira 和 Y. Nakamura：“单位圆上正交多项式的 Schur 流及其可积离散化”J。

永井敦: "可積分系の応用数理(中村佳正編著)第6章「離散可積分系と数列の加速法」"裳華房. 328(36) (2000)

永井敦：《可积系统的应用数学》（中村义正主编）第6章“离散可积系统和序列的加速方法” Shokabo. 328(36) (2000)

A.Mukaihira, Y.Nakamura: "Schur flow for orthogonal polynomials on the unit circle and its integralble discretization"J. Comput. Appl. Math.. 139. 75-94 (2000)

A.Mukaihira，Y.Nakamura：“单位圆上正交多项式的 Schur 流及其可积分离散化”J。