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Theoretical and Mathematical Physics

, Volume 125, Issue 3, pp 1603–1661 | Cite as

Symmetry Approach to the Integrability Problem

  • V. É. Adler
  • A. B. Shabat
  • R. I. Yamilov
Article

Abstract

We review the results of the twenty-year development of the symmetry approach to classifying integrable models in mathematical physics. The generalized Toda chains and the related equations of the nonlinear Schrödinger type, discrete transformations, and hyperbolic systems are central in this approach. Moreover, we consider equations of the Painlevé type, master symmetries, and the problem of integrability criteria for (2+1)-dimensional models. We present the list of canonical forms for (1+1)-dimensional integrable systems. We elaborate the effective tests for integrability and the algorithms for reduction to the canonical form.

Keywords

Mathematical Physic Related Equation Integrable System Integrable Model Canonical Form 
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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Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • V. É. Adler
    • 1
  • A. B. Shabat
    • 2
  • R. I. Yamilov
    • 2
  1. 1.Mathematical Institute, Ufa CenterRASUfaRussia
  2. 2.Landau Institute for Theoretical PhysicsRASMoscowRussia

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