

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

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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 KricheverNovikov 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 375:39603963 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:10.1088/09517715/24/7/009
\newblock arXiv:1009.2403v1.
\end{thebibliography}
Language: English

