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

Abstract

We introduce a so-called coprimeness-preserving non-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 threedimensional 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.

A two-dimensional lattice equation as an extension of the Heideman–Hogan recurrence

“…The equation, through a reduction (i.e., a projection of the domain of definition onto a line), generates a family of nonlinear mappings with positive algebraic entropy, including the Hietarinta-Viallet equation [6]. Two and three-dimensional lattice equations with similar properties have also been constructed by the deformation of the discrete Toda lattice equation [7]. Another class of exceptional second order mappings are so called the linearizable mappings [8], which do not have the singularity confinement property but have zero algebraic entropy.…”

Toda type equations over multi-dimensional lattices

“…When a = b = 1, equation (2) is equivalent to the coprimeness-preserving extensions to the two-dimensional Toda lattice equation (1). The degree growth of the iterates τ t,n,m of (1) is proved to be exponential unless k 1 = k 2 = l 1 = l 2 = 1 in our previous work [14,17]. Similarly, it is easy to prove that the degrees deg τ t,n of (2) grow exponentially with respect to t, unless (a, b) = (1, 1) and k 1 = k 2 = l 1 = l 2 = 1.…”

“…In fact, the coprimeness property is considered to be an algebraic re-interpretation (and in some cases the refinement) of the SC test. In previous works, we have studied coprimeness-preserving extensions to the discrete Toda lattice equation and the two-dimensional discrete Toda lattice equation [14]:…”

On the Coprimeness Property of Discrete Systems without the Irreducibility Condition

“…It is called the coprimeness property and is defined over the field of rational functions of the intial variables [16]. The coprimeness property is one type of singularity analysis of a discrete equation, which is quite similar to the singularity confinement test and is proved to be satisfied for many of the known discrete integrable systems [14,15]. The Laurent and the irreducibility properties played important roles in proving the coprimeness property of the given equations.…”

“…where M , L, K are some positive integers [14]. When (M, L, K) = (1, 1, 2) the equation (3.1) is the one-dimensional discrete Toda equation.…”