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

TL;DR

This paper establishes Miura transformations linking ABS lattice equations to Volterra-type equations, revealing their Bäcklund transformations and constructing new symmetries that confirm their integrability.

Contribution

It introduces Miura transformations connecting ABS equations to the YdKN equation, and constructs new generalized symmetries demonstrating their integrability.

Findings

Miura transformations map ABS spectral problems to discrete Schrödinger problems.

ABS equations correspond to Bäcklund transformations for YdKN.

New generalized symmetries confirm the integrability of these 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 discrete Schr\"odinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to B\"acklund 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 B\"acklund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.