Integrable systems with higher genus spectral parameter

具有更高属谱参数的可积系统

• 批准号: 14540172
• 负责人: TAKASAKI Kanehisa
• 金额: $ 2.05万
• 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: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540172/
• 关键词: integrable system; soliton equation; algebraic curve; higher genus; vector bundle; Tyurin parameter; Grassmann manifold; 種類; モジュライ空間; 可換微分作用素対; 共形場理論

There has been progress in several different aspects as follows.1.Soliton equations related to Tyurin parameters: Following Krichever's recent proposal on a construction of integrable systems with Tyurin parameters, we constructed an elliptic analogue of the nonlinear Schroedinger equation. It turned out that this equation, like many other soliton equations, can be mapped to a dynamical system on a submanifold of an infinite dimensional Grassmann manifold. This result result can be to an algebraic curve of arbitrary genus.2. Landau-Lifshitz equation and infinite dimensional Grassmann variety: The Landau-Lifshitz equation is a typical soliton equation related to an elliptic curve. We found that this equation, too, can be interpreted as a dynamical system on a submanifold of an infinite dimensional Grassmann manifold.3.Integrable system formulated by a pair of meromorphic functions on an algebraic curve of arbitrary genus: We constructed an integrable system on the moduli space of a pair of meromorphic functions on an algebraic curve of arbitrary genus. This system may be thought of as a special case of the so called Beauville system.

1.与Tyurin参数有关的孤子方程：根据Krichever最近提出的构造具有Tyurin参数的可积系统的建议，我们构造了一个椭圆型的非线性薛定谔方程。结果表明，这个方程和许多其他孤子方程一样，可以映射到无限维格拉斯曼流形的一个子流形上的动力系统。这一结果可以得到任意类型的代数曲线。Landau-Lifshitz方程和无限维Grassmann变分：Landau-Lifshitz方程是一个典型的与椭圆曲线有关的孤子方程。我们发现这个方程也可以解释为无穷维Grassmann流形上子流形上的动力系统。3.由任意亏格的代数曲线上的一对亚纯函数构成的可积系统：我们在任意亏格的代数曲线上的一对亚纯函数的模空间上构造了一个可积系统。这个系统可以被认为是所谓的博维尔系统的一个特例。

项目成果

高崎金久: "Landau-Lifshitz hierarchy and infinite dimensional Grassmann variety"Letters in Mathematical Physics. (未定).

Kanehisa Takasaki：“Landau-Lifshitz 层次结构和无限维格拉斯曼簇”数学物理学快报（待定）。

Kanehisa Takasaki: "Landau-Lifshitz hierarchy and infinite dimensional Grassmann variety."Letters in Mathematical Physics. (to appear).

Kanehisa Takasaki：“Landau-Lifshitz 层次结构和无限维格拉斯曼簇。”数学物理学快报。

高崎金久: "Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains"Communications Mathematical Physics. 241巻1号. 111-142 (2003)

高崎兼久：“周期修饰链的谱曲线、达布坐标和哈密顿结构”通讯数学物理，第 241 卷，第 111-142 期（2003 年）。

高崎金久: "Integrable systems whose spectral curve is the graph of a function"Journal of Geometry and Physics. 47巻・1号. 1-20 (2003)

Kanehisa Takasaki：“谱曲线是函数图的可积系统”《几何与物理学杂志》第 47 卷，第 1. 1-20 期（2003 年）。

藤井道彦: "An algorithm for solving linear ordinary differential equations of Fuchsian type with three singular points"Interdisciplinary Information Science. 9巻・1号(未定). (2003)

Michihiko Fujii：“求解具有三个奇异点的 Fuchsian 型线性常微分方程的算法”跨学科信息科学，第 9 卷，第 1 期（待定）。