Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 077, 14 pages      arXiv:0802.1850      http://dx.doi.org/10.3842/SIGMA.2008.077

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

Decio Levi a, Matteo Petrera b, a, Christian Scimiterna b, a and Ravil Yamilov c
a) Dipartimento di Ingegneria Elettronica, Università degli Studi Roma Tre and Sezione INFN, Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
b) Dipartimento di Fisica E. Amaldi, Università degli Studi Roma Tre and Sezione INFN, Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
c) Ufa Institute of Mathematics, 112 Chernyshevsky Str., Ufa 450077, Russia

Received August 29, 2008, in final form October 30, 2008; Published online November 08, 2008

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.

Key words: Miura transformations; generalized symmetries; ABS lattice equations.

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