Integrability criterion in terms of coprime property for the discrete Toda equation

Kanki, Masataka; Mada, Jun; Tokihiro, Tetsuji

doi:10.1063/1.4908109

Cited by 10 publications

(11 citation statements)

References 22 publications

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“…However, x 8 and x 9 are co-prime with each other, which leads to a contradiction. In the same manner, we conclude that x 9 must not be a factor of x 10 . Therefore x 10 is irreducible, and is trivially not a unit.…”

Section: Proof Of Theorem 22supporting

confidence: 62%

“…The coprimeness of the discrete KdV equation was formulated and proved in [9]. In [10], Mada and two of the authors investigated initial value dependence of the discrete Toda lattice equation with several boundary conditions, and showed that the discrete Toda lattice has the coprimeness property. In the subsequent paper [11], we introduced an extension of the bilinear two-dimensional discrete Toda equation:…”

Section: Definition 11mentioning

confidence: 99%

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Nonlinear forms of coprimeness preserving extensions to the Somos-4 recurrence and the two-dimensional Toda lattice equation—investigation into their extended Laurent properties

Kamiya¹,

Kanki²,

Mase³

et al. 2018

J. Phys. A: Math. Theor.

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Coprimeness property was introduced to study the singularity structure of discrete dynamical systems. In this paper we shall extend the coprimeness property and the Laurent property to further investigate discrete equations with complicated pattern of singularities. As examples we study extensions to the Somos-4 recurrence and the two-dimensional discrete Toda equation. By considering their non-autonomous polynomial forms, we prove that their tau function analogues possess the extended Laurent property with respect to their initial variables and some extra factors related to the non-autonomous terms. Using this Laurent property, we prove that these equations satisfy the extended coprimeness property. This coprimeness property reflects the singularities that trivially arise from the equations.

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Section: Proof Of Theorem 22supporting

confidence: 62%

Section: Definition 11mentioning

confidence: 99%

Nonlinear forms of coprimeness preserving extensions to the Somos-4 recurrence and the two-dimensional Toda lattice equation—investigation into their extended Laurent properties

Kamiya¹,

Kanki²,

Mase³

et al. 2018

J. Phys. A: Math. Theor.

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“…and the relation (11). Let us denote the exponent of p ′ n−j (0 ≤ j ≤ n) in the numerator p n as I n−j .…”

Section: Lemmamentioning

confidence: 99%

Algebraic entropy of an extended Hietarinta–Viallet equation

Kanki¹,

Mase²,

Tokihiro³

2015

J. Phys. A: Math. Theor.

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We introduce a series of discrete mappings, which is considered to be an extension of the Hietarinta-Viallet mapping with one parameter. We obtain the algebraic entropy for this mapping by obtaining the recurrence relation for the degrees of the iterated mapping. For some parameter values the mapping has a confined singularity, in which case the mapping is equivalent to a recurrence relation between six irreducible polynomials. For other parameter values, the mapping does not pass the singularity confinement test. The properties of irreducibility and co-primeness of the terms play crucial roles in the discussion.MSC: 37K10, 39A20

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“…The irreducibility and co-primeness are found out to be useful in formulating the integrability of discrete equations defined over the lattice of dimension more than one. For example, we have proved that these two properties can also be formulated for the discrete Toda equation (both the τ -function form and the nonlinear form) with various boundary conditions: i.e., open, the Dirichlet, and the periodic boundaries [13].…”

Section: Introductionmentioning

confidence: 95%

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

Kamiya

Kanki

Mase

et al. 2017

Journal of Mathematical Physics

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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.

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Integrability criterion in terms of coprime property for the discrete Toda equation

Cited by 10 publications

References 22 publications

Nonlinear forms of coprimeness preserving extensions to the Somos-4 recurrence and the two-dimensional Toda lattice equation—investigation into their extended Laurent properties

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 an extended Hietarinta–Viallet equation

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

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