Integrability conditions for quadrilateral partial difference equations

We propose an algebraic framework for the theory of integrable partial difference equations (P$\Delta$Es) . With a polynomial P$\Delta$E we associate a difference ideal and a difference field of fractions. Symmetries of the equation can be defined as derivations of this difference field of fractions. Existence of an infinite hierarchy of symmetries can be taken as a definition of integrability. We show that for integrable P$\Delta$Es there are recursion operators that generate infinite hierarchies of symmetries and provide with a sequence of canonical conserved densities [1]. Similar to the case of partial differential equations these canonical densities can serve as integrability conditions for difference equations. We have found two recursion operators for the Adler equation (in the Viallet form) satisfying to the elliptic curve equation associated with the equation [2]. These recursion operators have factorisations into Hamiltonian and symplectic operators which have natural applications to Yamilov's discretisation of the Krichever-Novikov equation. We have discovered a new type of factorisation for the recursion operators of difference equations over the field of fractions [3].
\begin{thebibliography}{100}
\bibitem{mwx1} A.V. Mikhailov, J.P. Wang, and P. Xenitidis. \newblock Recursion operators, conservation laws and integrability conditions for difference equations. \newblock Theoretical and Mathematical Physics, 167 ( 2011), no. 1, 421–443. \newblock arXiv:1004.5346.
\bibitem{mw} A.V. Mikhailov, and J.P. Wang, \newblock A new recursion operator for Adler’s equation in the Viallet form, \newblock Physics Letters A 375:3960-3963 21 Sep 2011 \newblock arXiv:1105.1269
\bibitem{mwx2} A.V. Mikhailov, J.P. Wang, and P. Xenitidis. \newblock Cosymmetries and Nijenhuis recursion operators for difference equations \newblock Nonlinearity, 24 (2011) 2079–2097, doi:10.1088/0951-7715/24/7/009 \newblock arXiv:1009.2403v1. \end{thebibliography}