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

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 dis- crete Schrodinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Backlund 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 Backlund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.