Loop groups and equations of KdV type
Loop groups and equations of KdV type
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DOI：
10.1007/bf02698802
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发表时间：
1985-12
期刊：
影响因子：
--
通讯作者：
G. Segal;G. Wilson
中科院分区：
文献类型：
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作者：
G. Segal;G. Wilson
本文的目的是找出M.和Y.Sato最近关于Korteweg-de Vries(KdV)方程和相关的非线性偏微分方程解的一些启示。我们从Date，JIMBO，Kashiwara和Miwa(FM的原著)的文件[5]中了解到这些想法。而Y.Sato似乎只有日语版)。我们将描述一种结构，它将KdV方程的解赋给某个无穷维格拉斯曼数的每个点。以这种方式得到的解的类别，被错误地称为<；(在[5]中的通解，包括Krichever[10，n]的显式代数几何解；其中包括众所周知的c<；n-孤立子解和有理解。我们的主要目的是确定通过该方法得到的解的类别，详细说明Grassman几何如何反映在解的性质中，以及代数几何解如何与图像相适应。我们还试图解释在文[5]中起基础作用的“r-函数”的几何意义。但最重要的是，我们努力对这一理论提出一个清晰而完整的描述，并希望澄清文献中一些模糊的观点。一、引言
The purpose of this paper is to work out some of the implications of recent ideas of M. and Y. Sato about the Korteweg-de Vries (KdV) equation and related non-linear partial differential equations. We learned of these ideas from the papers [5] of Date, Jimbo, Kashiwara and Miwa (the original work ofM. and Y. Sato appears to be available only in Japanese). We shall describe a construction which assigns a solution of the KdV equation to each point of a certain infinite dimensional Grassmannian. The class of solutions obtained in this way, which is misleadingly referred to as<(the general solution" in [5], includes the explicit algebro-geometric solutions ofKrichever [10, n]; among these are the well known c< n-soliton" and rational solutions. Our main aims are to determine what class of solutions is obtained by the method, to illustrate in detail how the geometry of the Grassmannian is reflected in properties of the solutions, and to show how the algebro-geometric solutions fit into the picture. We have also tried to explain the geometric meaning of the" r-function", which plays a fundamental role in the papers [5]. But above all we have endeavoured to present a clear and self-contained account of the theory, and hope to have elucidated a number of points left obscure in the literature. i. Introduction