Integrable systems with higher genus spectral parameter

在以下几个不同方面取得了进展。与Tyurin参数相关的孤子方程：根据krichhever最近关于构造具有Tyurin参数的可积系统的建议，我们构造了非线性薛定谔方程的椭圆模拟。结果表明，这个方程，像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子流形上的动力系统。该结果可适用于任意属的代数曲线。Landau-Lifshitz方程和无限维Grassmann变分：Landau-Lifshitz方程是与椭圆曲线相关的典型孤子方程。我们发现，这个方程也可以解释为一个无限维格拉斯曼流形的子流形上的动力系统。由任意格代数曲线上的一对亚纯函数表示的可积系统：在任意格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。这个系统可以看作是所谓的波维尔系统的一个特例。

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.