A family of linearizable recurrences with the Laurent property

Hone, Andrew N. W.; Ward, Chloe

doi:10.1112/blms/bdu004

Cited by 10 publications

(23 citation statements)

References 22 publications

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“…and this formula coincides with the trace formula of the (different) monodromy that is given by Veselov, Shabat in [27]. Equation ( 55) is obtained by a corresponding trace formula in Lemma 4.3 of [15] for the product p i=1 T i , where T i = (P L(g i , λ i )P −1 ) T , with P = 0 1 1 0 .…”

Section: Connection With the Dressing Chain And Integrabilitysupporting

confidence: 75%

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Discrete Hirota reductions associated with the lattice KdV equation

Hone¹,

Kouloukas²

2020

Preprint

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We study the integrability of a family of birational maps obtained as reductions of the discrete Hirota equation, which are related to travelling wave solutions of the lattice KdV equation. In particular, for reductions corresponding to waves moving with rational speed N/M on the lattice, where N, M are coprime integers, we prove the Liouville integrability of the maps when N + M is odd, and prove various properties of the general case. There are two main ingredients to our construction: the cluster algebra associated with each of the Hirota bilinear equations, which provides invariant (pre)symplectic and Poisson structures; and the connection of the monodromy matrices of the dressing chain with those of the KdV travelling wave reductions.

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Section: Connection With the Dressing Chain And Integrabilitysupporting

confidence: 75%

“…, u 2d ). The birational map (15) defines the U-system associated with the T-system that is specified by the matrix B. It preserves a symplectic form ω which is log-canonical in the coordinates (u i ), and pulls back to the presymplectic form corresponding to B, so that ω = i<j Bi,j…”

Section: T-systems and U-systemsmentioning

confidence: 99%

Discrete Hirota reductions associated with the lattice KdV equation

Hone¹,

Kouloukas²

2020

Preprint

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“…For instance, see [11,12,17,20,22,23,24,25,26,31,34]. More recently, a generalization of cluster algebras called Laurent phenomenon algebras [39] was also introduced in order to study the Laurent phenomenon [1,36].…”

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confidence: 99%

Hankel Determinant Solutions to Several Discrete Integrable Systems and the Laurent Property

Chang¹,

Hu²,

Xin³

2015

SIAM J. Discrete Math.

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Abstract. Many discrete integrable systems exhibit the Laurent phenomenon. In this paper, we investigate three integrable systems: the Somos-4 recurrence, the Somos-5 recurrence and a system related to so-called A 1 Q-system, whose general solutions are derived in terms of Hankel determinant. As a result, we directly confirm that they satisfy the Laurent property. Additionally, it is shown that the Somos-5 recurrence can be viewed as a specified Bäcklund transformation of the Somos-4 recurrence. The related topics about Somos polynomials are also studied.Key words. Hankel determinant solution, Laurent property, discrete integrable systems AMS subject classifications. 11B83, 37J35, 15B05, 11Y651. Introduction. Laurent phenomenon is a crucial property behind integrality shared by a class of combinatorial models while integrability is a key feature for a class of what we call integrable systems. An interesting observation is that many discrete integrable systems exhibit the Laurent phenomenon, and many mappings with the Laurent property are proved to be integrable. In the recent decade, a lot of related work were done via the study of a class of commutative algebras, called cluster algebras [19,21], for its success in proving the Laurent phenomenon. For instance, see [11,12,17,20,22,23,24,25,26,31,34]. More recently, a generalization of cluster algebras called Laurent phenomenon algebras [39] was also introduced in order to study the Laurent phenomenon [1,36].One question is whether we can prove the Laurent property (or a stronger property) from other views. In this paper, we shall give one possible choice-the view of explicit determinant solution. That is, the Laurent property of some discrete integrable systems can be proved by their determinant solutions. Of course, how to obtain the desired determinant formulae is another challenging problem. This approach succeeds for the following three integrable systems:The first two systems are the Somos-4 recurrence and the Somos-5 recurrence defined by

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“…where the arbitrary parameter b is an integration constant. The latter family of recurrences was referred to as the "Extreme polynomial" in [1], where it was obtained from another set of period 1 seeds in LP algebras, and for b = 0 it was independently found in [30], where it was also shown to be linearizable and have the Laurent property (see [21] for further details). However, the recurrences (16) lie beyond the setting of LP algebras.…”

Section: Introductionmentioning

confidence: 99%

Linear relations for Laurent polynomials and lattice equations

Hone,

Pallister

2019

Preprint

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A recurrence relation is said to have the Laurent property if all of its iterates are Laurent polynomials in the initial values with integer coefficients. We consider a family of nonlinear recurrences with the Laurent property, which were derived by Alman et al. via a construction of periodic seeds in Laurent phenomenon algebras, and generalize the Heideman-Hogan recurrences. Each member of the family is shown to be linearizable, in the sense that the iterates satisfy linear recurrence relations with constant coefficients. The latter are obtained from linear relations with periodic coefficients, which were found recently by Kamiya et al. from travelling wave reductions of a linearizable lattice equation on a 6-point stencil. We introduce another linearizable lattice equation on the same stencil, and present the corresponding linearization for its travelling wave reductions. Finally, for both of the 6-point lattice equations considered, we use the formalism of van der Kamp to construct a broad class of initial value problems with the Laurent property.

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A family of linearizable recurrences with the Laurent property

Cited by 10 publications

References 22 publications

Discrete Hirota reductions associated with the lattice KdV equation

Discrete Hirota reductions associated with the lattice KdV equation

Hankel Determinant Solutions to Several Discrete Integrable Systems and the Laurent Property

Linear relations for Laurent polynomials and lattice equations

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