Symmetries as integrability criteria for differential difference equations

Abstract

In this paper we review the results obtained by the generalized symmetry method in the case of differential difference equations during the last 20 years. Together with general theory of the method, classification results are discussed for classes of equations which include the Volterra, Toda and relativistic Toda lattice equations.

On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations

“…equation (29) has been found by Yamilov in [30], discussed in [21,4], and in most detailed form in [31]. Its continuous limit goes into the Krichever-Novikov equation [15].…”

“…The Volterra equation corresponds to f (u 1 , u 0 , u −1 ) = u 0 (u 1 −u −1 ). An exhaustive list of differentialdifference integrable equations of the form (28) has been obtained in [30] (details can be found in [31]). All three-point generalized symmetries of the ABS equations, with no explicit dependence on n, m, have the same structure as equation (19) (see details in Section 4.4 below) and are particular cases of the YdKN equation…”

“…In the case of equations (H1) and (H3) with δ = 0, the corresponding differential-difference equations have a different form, and the resulting KdV-type equations differ from equation (31). In [5] it is shown that the continuous limit of equation (Q4) …”

“…Let us just mention the classification scheme introduced by Shabat, using the formal symmetry approach (see [21] for a review). This approach has been successfully extended to the differential-difference case by Yamilov [30,31,20]. In the completely discrete case the situation turns out to be quite different.…”

Рекурсионные Операторы, Законы Сохранения И Условия Интегрируемости Для Разностных Уравнений

“…Именно этот подход позволил полу-чить наиболее полные результаты классификации интегрируемых уравнений, осно-ванные на этом свойстве [4]- [10]. В работах [11], [12] было показано, что подход, базирующийся на симметриях, эффективен в задаче о классификации интегрируе-мых дифференциально-разностных уравнений.…”