Perturbative Symmetry Approach for Differential–Difference Equations

Abstract

We propose a new method to tackle the integrability problem for evolutionary differential–difference equations of arbitrary order. It enables us to produce necessary integrability conditions, to determine whether a given equation is integrable or not, and to advance in classification of integrable equations. We define and develop symbolic representation for the difference polynomial ring, difference operators and formal series. In order to formulate necessary integrability conditions, we introduce a novel quasi-local extension of the difference ring. We apply the developed formalism to solve the classification problem of integrable equations for anti-symmetric quasi-linear equations of order $$(-3,3)$$ ( - 3 , 3 ) and produce a list of 17 equations satisfying the necessary integrability conditions. For every equation from the list we present an infinite family of integrable higher order relatives. Some of the equations obtained are new.

Hamiltonian and recursion operators for a discrete analogue of the Kaup-Kupershmidt equation

“…In principle, the methods developed in this paper can be used to construct a recursion operator for the entire family of integrable equations (1). For any fixed p, we can write down the recursion operator although it is formed by more parts comparing to (9).…”

“…In a recent paper [1], Mikhailov, Novikov and Wang formulated necessary integrability conditions for evolutionary differential-difference equations (D∆Es) using the perturbative symmetry approach and obtained a family of new integrable D∆Es:…”

“…The authors of [1] have proved the integrability of (1) for any p ∈ N by constructing its Lax representation and provided the r matrix approach to produce the hierarchy of commuting symmetries. They showed that equation (1) possesses a Lax representation L t = [A, L] with a rational (pseudo-difference) operator L = Q −1 P , where…”

“…which was first obtained in the symmetry classification of a class of five-point differentialdifference equations [3]. Its integrability was established in [4], where Garifullin and Yamilov constructed a Lax representation for (7), which is in a different form from (4) when p = 2 (see Remark 3 in [1] for the comparison). As shown also in [4], the continuous limit of ( 7) is the Kaup-Kupershmidt equation using the substitution…”

On the integrability of three two-component bi-Hamiltonian systems

“…Such a problem constitutes the classification problem which in general is one of the most important topics in mathematics. Different approaches, such as the symmetry approach [1,2], the Hirota bilinear method [3] and the consistency around cube approach [4], have been proposed to resolve the problem.…”