Theoretical and Mathematical Physics

, Volume 177, Issue 3, pp 1655–1679 | Cite as

Formal diagonalization of a discrete lax operator and conservation laws and symmetries of dynamical systems

  • I. T. HabibullinEmail author
  • M. V. Yangubaeva


We consider the problem of constructing a formal asymptotic expansion in the spectral parameter for an eigenfunction of a discrete linear operator. We propose a method for constructing an expansion that allows obtaining conservation laws of discrete dynamical systems associated with a given linear operator. As illustrative examples, we consider known nonlinear models such as the discrete potential Kortewegde Vries equation, the discrete version of the derivative nonlinear Schrödinger equation, the Veselov-Shabat dressing chain, and others. We describe the infinite set of conservation laws for the discrete Toda chain corresponding to the Lie algebra A 1 (1) . We find new examples of integrable systems of equations on a square lattice.


Lax pair asymptotic expansion conservation law symmetry equations on a quad graph discrete nonlinear Schrödinger equation dressing method 


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© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Institute of Mathematics with Computing Center, Ufa Science CenterRASUfaRussia

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