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.

在几个不同方面取得了进展。1.soliton方程与三尿蛋白酶参数相关：遵循Krichever关于具有Tyurin参数的可集成系统构建的最新建议，我们构建了非线性Schroedinger方程的椭圆类似物。事实证明，与许多其他孤子方程一样，该方程式可以映射到无限尺寸格拉曼（Grassmann）歧管的子手机上的动态系统。该结果可能是任意属的代数曲线2。 Landau-Lifshitz方程和无限尺寸格拉曼（Grassmann）品种：Landau-Lifshitz方程是与椭圆曲线相关的典型孤子方程。 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.该系统可能被认为是所谓的Beauville系统的特殊情况。

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: "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）。

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）。

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）。