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Sato Theory and Transformation Groups. A Unified Approach to Integrable Systems

  • Ralph Willox
  • Junkichi Satsuma
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 644)

Abstract

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.

Keywords

Darboux Transformation Grassmann Manifold Cyclic Vector Bilinear Equation Bilinear Identity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Authors and Affiliations

  • Ralph Willox
    • 1
    • 2
  • Junkichi Satsuma
    • 1
  1. 1.Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, 153-8914 TokyoJapan
  2. 2.Theoretical Physics, Free University of Brussels (VUB), Pleinlaan 2, 1050 BrusselsBelgium

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