Coprimeness-preserving non-integrable extension to the two-dimensional discrete Toda lattice equation

Abstract

We introduce a so-called `coprimeness-preserving non-integrable' extension (another terminology is `quasi-integrable' extension) to the two-dimensional Toda lattice equation. We believe that this equation is the first example of such discrete equation defined over a three-dimensional lattice. We prove that all the iterates of the equation are irreducible Laurent polynomials of the initial data and that every pair of two iterates is co-prime, which indicate confined singularities of the equation. By reducing the equation to two- or one-dimensional lattices, we obtain coprimeness-preserving non-integrable extensions to the one-dimensional Toda lattice equation and the Somos-4 recurrence.

arXiv comment18 pages, 1 figure

References in corpus (8)

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• Singularity confinement and chaos in discrete systems
• Discrete integrable systems and Poisson algebras from cluster maps
• Cluster mutation-periodic quivers and associated Laurent sequences
• Investigation into the role of the Laurent property in integrability
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Cited by in corpus (5)

• On the Coprimeness Property of Discrete Systems without the Irreducibility Condition
• Nonlinear forms of coprimeness preserving extensions to the Somos-$4$ recurrence and the two-dimensional Toda lattice equation --investigation into their extended Laurent properties--
• Algebraic entropy of a multi-term recurrence of the Hietarinta-Viallet type
• A two dimensional lattice equation as an extension of the Heideman-Hogan recurrence
• Toda type equations over multi-dimensional lattices