On the Symmetries of Integrable Systems

  • P. G. Grinevich
  • A. Yu. Orlov
  • E. I. Schulman
Part of the Springer Series in Nonlinear Dynamics book series (SSNONLINEAR)


Symmetries play a very important role in mathematics as well as in physics. Many problems were solved by studying their symmetry properties. For example Gauss in his first and beloved work found how to construct a 17-sided polygon with compasses and ruler. The essential part of his method was studying the invariants of the equation z17 - 1 = O. Other significant results based on the symmetry approach were obtained by Abel and Galois in the theory of algebraic equations. Abel proved that the solution of a general 5th degree algebraic equation cannot be expressed via radicals. Galois succeeded to elicit conditions of solvability of an algebraic equation in radicals using its symmetry group. The analog of Galois theory for ordinary differential equations was developed in our century.


Riemann Surface Vertex Operator Riemann Problem Solvable Equation Recursion Operator 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • P. G. Grinevich
    • 1
  • A. Yu. Orlov
    • 2
  • E. I. Schulman
    • 2
  1. 1.Landau Institute for Theoretical PhysicsMoscowRussia
  2. 2.Oceanology InstituteMoscowRussia

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