Sato Theory and Transformation Groups. A Unified Approach to Integrable Systems
Sato Theory and Transformation Groups. A Unified Approach to Integrable Systems
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DOI：
10.1007/978-3-540-40357-9_2
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
R. Willox;J. Satsuma
中科院分区：
文献类型：
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作者：
R. Willox;J. Satsuma
20多年前，人们发现Kadomtsev-Petviashvili(KP)方程的解构成了无限维Grassmann流形，并且这个Grassman方程的Plücker关系具有Hirota双线性恒等式的形式。正如在这篇文章中所解释的，由此产生的统一的可积性方法，通常被称为佐藤理论，提供了对具有无限多个自由度的可积系统的深刻的代数和几何理解。从简单地介绍Sato理论开始，然后用无限维Clifford代数及其表示来解释它，该理论的范围逐渐扩展到包括多分量系统、可积格子方程和全离散系统。特别强调了该理论所描述的可积方程的对称性，特别是这些方程的达布变换和初等Bäck lund变换。最后，讨论了对低维系统的约化，以及最终对可积常微分方程组的约化。作为一个例子，详细解释了第四个Painlevé方程及其在KP谱系中的Bäck lund变换的起源。
More than 20 years ago, it was discovered that the solutions of the Kadomtsev-Petviashvili (KP) hierarchy constitute an infinite-dimensional Grassmann manifold and that the Plücker relations for this Grassmannian take the form of Hirota bilinear identities. As is explained in this contribution, the resulting unified approach to integrability, commonly known as Sato theory, offers a deep algebraic and geometric understanding of integrable systems with infinitely many degrees of freedom. Starting with an elementary introduction to Sato theory, followed by an exposé of its interpretation in terms of infinite-dimensional Clifford algebras and their representations, the scope of the theory is gradually extended to include multi-component systems, integrable lattice equations and fully discrete systems. Special emphasis is placed on the symmetries of the integrable equations described by the theory and especially on the Darboux transformations and elementary Bäcklund transformations for these equations. Finally, reductions to lower dimensional systems and eventually to integrable ordinary differential equations are discussed. As an example, the origins of the fourth Painlevé equation and of its Bäcklund transformations in the KP hierarchy are explained in detail.