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

Kamiya, Ryo; Kanki, Masataka; Mase, Takafumi; Tokihiro, Tetsuji

doi:10.1088/1751-8121/aaad47

Cited by 2 publications

(27 citation statements)

References 19 publications

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“…which is the relation for an SL 3 frieze on each of the sublattices obtained by restricting s + t to have odd/even parity, and can be put in the standard form (18) by a linear change of coordinates. A further application of (19) shows that the corresponding 4 × 4 determinant vanishes for the 2-frieze relation, while in [23] it is shown that there is also a constant 3 × 3 determinant and a vanishing 4 × 4 determinant associated with (14), and in the sequel we prove an analogous result for the new lattice equation (15).…”

Section: Introductionmentioning

confidence: 55%

“…corresponding to a wave moving on the lattice with constant velocity −q/p ∈ Q. Similarly, it was noted in [23] that the 5-point lattice equation…”

Section: Introductionmentioning

confidence: 72%

“…which is the relation for a 2-frieze [27], reduces to (5) by substituting in (12) with the replacement p → p − k, q → k, to obtain the (p − k, −k) travelling wave reduction. The authors of [23] also considered the Little Pi family (8) as a reduction of the 6-point lattice equation…”

Section: Introductionmentioning

confidence: 99%

“…(Note that, compared with [23], we have switched the order of the independent variables and introduced the parameter a.) By obtaining linear relations for the above lattice equation, they deduced linear recurrences with periodic coefficients for its (l, −k) travelling wave reduction (8) (cf.…”

Section: Introductionmentioning

confidence: 99%

“…In the next section we give a very brief introduction to LP algebras, and explain how nonlinear recurrences of the form (1) can arise in that setting, giving full details for the particular case of the Little Pi family (8). Section 3 is devoted to an independent derivation of the linear recurrences with periodic coefficients found for the Little Pi family in [23], which we then use in Section 4 to derive a constant coefficient relation of order 6kl, of the form (9) or (10), for each pair of coprime positive integers k, l. Section 5 is concerned with the new 6-point lattice equation (15), including the proof of linearization both for the lattice equation and all its travelling wave reductions (16). Finally, in Section 6 we show that the new lattice equation has the Laurent property for suitable band sets of initial values in Z 2 , of the kind described by van der Kamp in [24], and we use this to infer the Laurent property for its reductions (16).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 55%

“…corresponding to a wave moving on the lattice with constant velocity −q/p ∈ Q. Similarly, it was noted in [23] that the 5-point lattice equation…”

Section: Introductionmentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Linear relations for Laurent polynomials and lattice equations

Hone,

Pallister

2019

Preprint

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show abstract

Linear relations for Laurent polynomials and lattice equations

Hone

Pallister

2020

Nonlinearity

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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. Recurrences with this property appear in diverse areas of mathematics and physics, ranging from Lie theory and supersymmetric gauge theories to Teichmüller theory and dimer models. In many cases where such recurrences appear, there is a common structural thread running between these different areas, in the form of Fomin and Zelevinsky’s theory of cluster algebras. Laurent phenomenon algebras, as defined by Lam and Pylyavskyy, are an extension of cluster algebras, and share with them the feature that all the generators of the algebra are Laurent polynomials in any initial set of generators (seed). Here we consider a family of nonlinear recurrences with the Laurent property, referred to as ‘Little Pi’, which was derived by Alman et al via a construction of periodic seeds in Laurent phenomenon algebras, and generalizes the Heideman–Hogan family of recurrences. Each member of the family is shown to be linearizable, in the sense that the iterates satisfy linear recurrence relations with constant coefficients. We derive the latter 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 six-point stencil. By making use of the periodic coefficients, we further show that the birational maps corresponding to the Little Pi family are maximally superintegrable. We also introduce another linearizable lattice equation on the same six-point stencil, and present the corresponding linearization for its travelling wave reductions. Finally, for both of the six-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 two-dimensional lattice equation as an extension of the Heideman–Hogan recurrence

Cited by 2 publications

References 19 publications

Linear relations for Laurent polynomials and lattice equations

Linear relations for Laurent polynomials and lattice equations

Linear relations for Laurent polynomials and lattice equations

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