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
Article

Abstract

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 formut=uxx+f(u, v, uxvx), −vt=vxx+g(u, v, ux,vx), 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.

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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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