On non-multiaffine consistent-around-the-cube lattice equations

Kassotakis, Pavlos; Nieszporski, Maciej

doi:10.1016/j.physleta.2012.10.009

Cited by 36 publications

(71 citation statements)

References 48 publications

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“…Entwining maps associated with the H B III Yang-Baxter map The invariants that were derived in [44,45,47,56],…”

Section: Entwining Maps Associated With the H A III Yang-baxter Map Tmentioning

confidence: 99%

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Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps

Kassotakis¹

2019

SIGMA

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We present the explicit form of a family of Liouville integrable maps in 3 variables, the so-called triad family of maps and we propose a multi-field generalisation of the latter. We show that by imposing separability of variables to the invariants of this family of maps, the H I , H II and H A III Yang-Baxter maps in general position of singularities emerge. Two different methods to obtain entwining Yang-Baxter maps are also presented. The outcomes of the first method are entwining maps associated with the H I , H II and H A III Yang-Baxter maps, whereas by the second method we obtain non-periodic entwining maps associated with the whole F and H-list of quadrirational Yang-Baxter maps. Finally, we show how the transfer maps associated with the H-list of Yang-Baxter maps can be considered as the (k − 1)-iteration of some maps of simpler form. We refer to these maps as extended transfer maps and in turn they lead to k-point alternating recurrences which can be considered as alternating versions of some hierarchies of discrete Painlevé equations.

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“…Entwining maps associated with the H B III Yang-Baxter map The invariants that were derived in [44,45,47,56],…”

Section: Entwining Maps Associated With the H A III Yang-baxter Map Tmentioning

confidence: 99%

“…The invariants in separated variables that appear inTable 10, were firstly introduced, in a different context, in[44,45,47,56]. Note that the invariants H1, H2 for tH A III (k) iwere also given in[50].…”

mentioning

confidence: 99%

Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps

Kassotakis¹

2019

SIGMA

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“…The six bottom members of the list, namely (Q1 δ , A2, A1 δ , H3 δ , H2, H1) admit a Lie point symmetry group. The existence of the Lie point symmetry group allows one to reformulate these six members of the list as systems of difference equations defined on the edges of a quadrangle of the Z 2 graph [40], we refer to these systems shortly as bond systems [19,25,24,28,27]. The hallmark of integrability of the bond systems is the 3-dimensional compatibility in analogy with the consistency-around-the-cube [37] of the corresponding quad-equations.…”

Section: Introductionmentioning

confidence: 99%

Integrable two-component systems of difference equations

et al. 2020

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We present two lists of two-component systems of integrable difference equations defined on the edges of the Z 2 graph. The integrability of these systems is manifested by their Lax formulation which is a consequence of the multi-dimensional compatibility of these systems. Imposing constraints consistent with the systems of difference equations, we recover known integrable quad-equations including the discrete version of the Krichever–Novikov equation. The systems of difference equations give us in turn quadrirational Yang–Baxter maps.

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“…System (28) is integrable with Lax representation L a (p 01 , q 01 , q 11 , r 11 , τ 01 , θ 11 )L b (p, q, q 01 , r 01 , τ, θ 01 ) = L b (p 10 , q 10 , q 11 , r 11 , τ 10 , θ 11 )L a (p, q, q 10 , r 10 , τ, θ 10 ), (29) where L a (p, q, q 10 , r 10 , τ, θ 10 ) :=     −q 10 1 0 0 −r 10 0 1 0 a − pq 10 − qr 10 − τ θ 10 − λ p q τ −θ 10 0 0 1…”

Section: Step III : Squeeze Down To Grassmann Extension Of the Boussimentioning

confidence: 99%

On the 3D consistency of a Grassmann extended lattice Boussinesq system

Konstantinou-Rizos

2020

Nuclear Physics B

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In this paper, we formulate a "Grassmann extension" scheme for constructing noncommutative (Grassmann) extensions of Yang-Baxter maps together with their associated systems of P∆Es, based on the ideas presented in [11]. Using this scheme, we first construct a Grassmann extension of a Yang-Baxter map which constitutes a lift of a lattice Boussinesq system. The Grassmann-extended Yang-Baxter map can be squeezed down to a novel, integrable, Grassmann lattice Boussinesq system, and we derive its 3D-consistent limit. We show that some systems retain their 3D-consistency property in their Grassmann extension. The formulation of the ideas presented in [11] into a Grassmann extension scheme;2. The derivation of a new Boussinesq-type Yang-Baxter map together with its Grassmann extension;3. The construction of an integrable, noncommutative (Grassmann) extension of a discrete Boussinesq system and its 3D-consistent limit. The latter gives rise to the following important point.4. We show that, for some systems, the 3D-consistency property does not break in their noncommutative extension.The paper is organised as follows: The next section provides with preliminary knowledge, essential for the text to be self-contained. In particular, we fix the notation we use throughout the text, and we give the basic definitions of systems on quad-graphs and Yang-Baxter maps. Furthermore, we demonstrate the relation between the former and the latter and the relation between the 3D consistency property and the Yang-Baxter equation. We also explain what a Lax representation is for both equations on quad graphs and Yang-Baxter maps. Finally, we present the basic steps of a simple scheme for constructing Grassmann extensions of discrete integrable systems and their associated Yang-Baxter maps; the related ideas were presented in [11]. In section 3, we apply the aforementioned scheme to system (1). Specifically, we consider the associated Yang-Baxter lift of (1), for which we construct a Grassmann extension. Then, we show that the latter can be squeezed down to a novel integrable system of lattice equations which can be considered as the Grassmann extension of system (1). Finally, in section 4, we present a Boussinesq-type system associated via a conservation law to the one obtained in section 3, and we prove the integrability-in the sense of 3D-consistency-for a limit of this system. Finally, the last section deals with some concluding remarks and thoughts for future work. Preliminaries NotationHere, we explain the notation we shall be using throughout the text.

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On non-multiaffine consistent-around-the-cube lattice equations

Cited by 36 publications

References 48 publications

Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps

Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps

Integrable two-component systems of difference equations

On the 3D consistency of a Grassmann extended lattice Boussinesq system

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