Communications in Mathematical Physics

, Volume 115, Issue 1, pp 1–19 | Cite as

Extension of the module of invertible transformations. Classification of integrable systems

  • A. V. Mikhailov
  • A. B. Shabat
  • R. I. Yamilov


We demonstrate that for the systems of equations, which are invariant under a point group or possess conservation laws of the zeroth or first order, a nontrivial extension of the module of invertible transformations is possible. That simplifies greatly a classification of the integrable systems of equations. Here we present an exhaustive list and a classification of the second order systems of the formu t =u xx +f(u, v, u x v x ), −vt=v xx +g(u, v, u x ,v x ), which possess the conservation laws of higher order. The reduction group approach allows us to define the Lax type representations for some new equations of our list.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Integrable System 
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Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • A. V. Mikhailov
    • 1
  • A. B. Shabat
    • 2
  • R. I. Yamilov
    • 2
  1. 1.L. D. Landau Institute for Theoretical PhysicsAcademy of Sciences of USSRMoscowUSSR
  2. 2.Department of Physics and MathematicsBashkirian Branch of the Academy of Sciences of the USSRUSSR

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