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Algebraic and differential geometric aspects of the integrability of nonlinear dynamical systems on infinite-dimensional functional manifolds

  • Anatoliy K. Prykarpatsky
  • Ihor V. Mykytiuk
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 443)

Abstract

Lie algebraic ideals, as is well known, are widely used in the modern theory of smooth integrable dynamical systems on manifolds. Using them one has succeeded in the classification of many nonlinear dynamical systems whose complete integrability was stated before by means of different mathematical techniques (Fushchych et al., 1990). For the last few years the special attention has been paid to nonlinear dynamical systems that also possess the Lax type isospectrality property (Novikov, 1980), (Holod et al., 1992), (Takhtadjian et al., 1987), (Newell, 1985). We focus on their description below, although our treatment will not be entirely complete. Here our aim is to single out the main mathametical content of the theory of the integrability of nonlinear dynamical systems, which is universal in the sense that it applies to almost all such systems.

Keywords

Poisson Bracket Nonlinear Dynamical System Geometric Quantization Infinite Hierarchy Integrable Dynamical System 
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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© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  • Anatoliy K. Prykarpatsky
    • 1
    • 2
  • Ihor V. Mykytiuk
    • 2
    • 3
  1. 1.Institute of MathematicsUniversity of Mining and MetallurgyCracowPoland
  2. 2.Institute for Applied Problems of Mechanics and Mathematics of the NASLvivUkraine
  3. 3.Lviv Polytechnic State UniversityLvivUkraine

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