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，离散可积系统在差分格式和算法方面得到了广泛的关注。关于连续时间可积系统与正交多项式之间的关系，已有许多讨论。对全离散（离散时空）系统与正交多项式之间的关系的研究很少，如Spiridonov和Zhedanov提出的离散可积系统（户田，Lotka-Volterra）与经典正交多项式之间的关系。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，并发展与正交性有关的数值算法。首先，我们选取离散的饥饿Lotka-Volterra方程和耦合的KP方程。我们通过Hirota的τ函数来探讨它们，并发展了它们之间的关系。

项目成果

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

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

C.R. Gilson, J.J.C. Nimmo and S. Tsujimoto: "Pfaffianisation of the discrete KP Equation"J. Phys. A: Math. Gen.. 34. 10569-10575 (2001)

C.R.吉尔森，J.J.C.

S. Tsujimoto: "On a discrete analogue of the two-dimensional Toda lattice hierarchy"Publ. RIMS Kyoto University. 38. 113-133 (2002)

S. Tsujimoto：“关于二维 Toda 晶格层次结构的离散模拟”Publ。

C.R.Gilson, J.J.C.Nimmo, S.Tsujimoto: "Pfaffianisation of the discrete KP Equation"J. Phys. A : Math. Gen.. 34. 10569-10575 (2001)

C.R.Gilson、J.J.C.Nimmo、S.Tsujimoto：“离散 KP 方程的普法夫化”J。

S.Tsujimoto: "On a discrete analogue of the two-dimensional Toda lattice hierarchy"Publ. RIMS Kyoto University. 38. 113-133 (2002)

S.Tsujimoto：“关于二维 Toda 晶格层次结构的离散模拟”Publ。