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

Abstract

Abstract. We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Bäcklund transformations for some particular cases of the discrete Krichever-Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate Bäcklund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.

Classification of five-point differential-difference equations

“…More recent results on this line of research can be found in [6,12,17]. All equations obtained by ABS and their extensions have generalized symmetries which are integrable D∆Es, belonging to the classification presented in [25,39] and given, in general, by D∆Es defined on three-point lattices [18,19,24].…”

Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations

“…the equation (1) is assumed to be autonomous. It is known [12,30,[32][33][34] that the generalized symmetries of quad-equations are given by differential-difference equations of the form: du n,m dt 1 = ϕ n,m u n+k 1 ,m , . .…”

“…In this sense the equations in (2) are a k 1 +k ′ 1 +1-point differential-difference equation and a k 2 + k ′ 2 + 1-point differential-difference equation respectively. The vast majority of quad-equations known in literature admits as generalized symmetries three-point differential-difference equations in both directions [24,30,33,36,42,43]. When the given quad-equation admits three-point generalized symmetries it can be interpreted as a Bäcklund transformation for these differential-difference equantions [29,30].…”

Classification of five-point differential–difference equations II

“…Almost all integrable known P∆Es have the lowest order associated generalized symmetries given by integrable evolutionary D∆Es which are defined on three-point lattices [21,22,27,30,41] and belong to the classification presented in [28,43]. This is the classification of Volterra type equationṡ u n = Φ(u n+1 , u n , u n−1 ) (1) presented in [42], and the resulting list of equations is quite big, see the details in the review article [43].…”