Direct linearisation of the discrete-time two-dimensional Toda lattices

Fu, Wei

doi:10.1088/1751-8121/aace36

Cited by 11 publications

(23 citation statements)

References 34 publications

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“…The formula (A (1) r ) reduces the semi-discrete equation (3.15) to the (1+1)-dimensional differential-difference system with respect to n 1 and x 1 . However, the obtained integrable system is different from that discussed in [17]. This is because here the discrete dispersion relation described by n 1 relies on the spectral parameter in a fractionally linear way according to (1.3).…”

Section: 2mentioning

confidence: 80%

“…Equation (3.27) is one of the nonlinear forms of the 2D Toda equation (see e.g. [17]), gauge equivalent to the well-known form…”

Section: 2mentioning

confidence: 99%

“…This gives us a strong hint about how to construct the discrete Drinfel'd-Sokolov hierarchies of these types from the DL framework. Notice that the discrete A (1) r -type equations have been worked out from the DL by selecting the plane wave factors (1.2), see [17,18,44]. To search for the rest classes of discrete integrable systems, we need to introduce discrete odd-flow variables, because integrable PDEs of A (2) 2r -, C (1) r -and D (2) r+1 -types are fully described by continuous odd-flow variables x 2 j+1 , see e.g.…”

Section: Introductionmentioning

confidence: 99%

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Integrable semi-discretisation of the Drinfel'd--Sokolov hierarchies

Yin,

2021

Preprint

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We propose a novel semi-discrete Kadomtsev-Petviashvili equation with two discrete and one continuous independent variables, which is integrable in the sense of having the standard and adjoint Lax pairs, from the direct linearisation framework. By performing reductions on the semi-discrete Kadomtsev-Petviashvili equation, new semi-discrete versions of the Drinfel'd-Sokolov hierarchies associated with Kac-Moody Lie algebras A (1) r , A (2) 2r , C (1) r and D (2) r+1 are successfully constructed. A Lax pair involving the fraction of Z N graded matrices is also found for each of the semi-discrete Drinfel'd-Sokolov equations. Furthermore, the direct linearisation construction guarantees the existence of exact solutions of all the semi-discrete equations discussed in the paper, providing more insights into their integrability in addition of the analysis of Lax pairs.

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Section: 2mentioning

confidence: 80%

“…Equation (3.27) is one of the nonlinear forms of the 2D Toda equation (see e.g. [17]), gauge equivalent to the well-known form…”

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Integrable semi-discretisation of the Drinfel'd--Sokolov hierarchies

Yin,

2021

Preprint

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“…The difference is that here the lattice parameters p i and p j act the independent variables; while in the two-dimensional Toda lattice (2DTL), the bilinear derivatives are with respect to the continuous flow variables x 1 and x −1 (cf. [10]), which leads to an autonomous equation.…”

Section: Formal Structure Of the Direct Linearisationmentioning

confidence: 99%

On non-autonomous differential-difference AKP, BKP and CKP equations

Nijhoff

2021

Proc. R. Soc. A.

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Based on the direct linearization framework of the discrete Kadomtsev–Petviashvili-type equations presented in the work of Fu & Nijhoff (Fu W, Nijhoff FW. 2017 Direct linearizing transform for three-dimensional discrete integrable systems: the lattice AKP, BKP and CKP equations. Proc. R. Soc. A 473 , 20160915 ( doi:10.1098/rspa.2016.0915 )), six novel non-autonomous differential-difference equations are established, including three in the AKP class, two in the BKP class and one in the CKP class. In particular, one in the BKP class and the one in the CKP class are both in (2 + 2)-dimensional form. All the six models are integrable in the sense of having the same linear integral equation representations as those of their associated discrete Kadomtsev–Petviashvili-type equations, which guarantees the existence of soliton-type solutions and the multi-dimensional consistency of these new equations from the viewpoint of the direct linearization.

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“…We list some concrete examples for N = 2 and N = 3 within this framework in appendix A for reference. Among these examples, the classes of (α, β) = (0, 0), (α, β) = (0, N − 1) and (α, β) = (N − 1, N − 1) amount to the discrete GD hierarchy [13], the discrete-time two-dimensional Toda lattice (2DTL) equations [31] and the discrete Schwarzian GD hierarchy [34], respectively; while the other classes of equations are the 'new' integrable discrete equations which so far only appeared in [16] and were not considered by others. Our parametrisations/expressions of these equations could sometimes be slightly different from those given by Fordy and Xenitidis, for convenience of constructing their DTs in the forthcoming sections, i.e.…”

Section: 2mentioning

confidence: 99%

$\mathbb{Z}_\mathcal{N}$ graded discrete integrable systems and Darboux transformations

Shi¹

2020

J. Phys. A: Math. Theor.

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We present the Darboux transformations for a novel class of two-dimensional discrete integrable systems named as Z N graded discrete integrable systems, which were firstly proposed by Fordy and Xenitidis within the framework of Z N graded discrete Lax pairs very recently. In this paper, the Z N graded discrete equations in coprime case and their corresponding Lax pairs are derived from the discrete Gel'fand-Dikii hierarchy by applying a transformation of the independent variables. The construction of the Darboux tranformations is realised by considering the associated linear problems in the bilinear formalism for the Z N graded lattice equations. We show that all these Z N graded equations share a unified solution structure in our scheme.Key words and phrases. Z N discrete integrable system, Darboux transformations, tau function, discrete Gel'fand-Dikii hierarchy.1 Apart from the nonlinear forms given here, there also exist other nonlinear forms in the discrete KP family such as the discrete Schwarzian KP equation [24], the (2+1)-dimensional Nijhoff-Quispel-Capel (NQC) equation [25], and the various nonpotential versions of the discrete KP equations, see [26][27][28].

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Direct linearisation of the discrete-time two-dimensional Toda lattices

Cited by 11 publications

References 34 publications

Integrable semi-discretisation of the Drinfel'd--Sokolov hierarchies

Integrable semi-discretisation of the Drinfel'd--Sokolov hierarchies

On non-autonomous differential-difference AKP, BKP and CKP equations

$\mathbb{Z}_\mathcal{N}$ graded discrete integrable systems and Darboux transformations

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