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International conference "Geometrical Methods in Mathematical Physics"
December 14, 2011 10:00–10:45, Moscow, Lomonosov Moscow State University

Integrability conditions for quadrilateral partial difference equations

A. V. Mikhailov

University of Leeds

Abstract: 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].