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

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 tr...

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Bibliographic Details
Published inSymmetry, integrability and geometry, methods and applications Vol. 4; p. 077
Main Author Levi, Decio
Format Journal Article
LanguageEnglish
Published Kiev National Academy of Sciences of Ukraine 01.01.2008
National Academy of Science of Ukraine
Subjects
ABS lattice equations
generalized symmetries
Miura transformations
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Summary: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.
ISSN:1815-0659
1815-0659
DOI:10.3842/SIGMA.2008.077