Integrable systems with higher genus spectral parameter

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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.