Discrete Hirota reductions associated with the lattice KdV equation

Hone, Andrew N. W.; Kouloukas, Theodoros E.

doi:10.1088/1751-8121/aba1b8

Cited by 2 publications

(2 citation statements)

References 27 publications

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“…In the present investigation we are proposing the algebraic setting for constructing 'spacetime' discrete integrable systems. The study of fully discrete systems has been a particularly active field in recent decades, especially after the prototypical Hirota's works [30] on non-linear partial difference equation, leading also to intriguing connections with quantum integrable systems [32,55], (see also [28] and references therein). A fundamental frame for describing such integrable systems and the associated partial difference equations is the so-called consistency approach [5,29,45].…”

Section: Introductionmentioning

confidence: 99%

An algebraic approach to discrete time integrability

Doikou¹,

Findlay²

2021

J. Phys. A: Math. Theor.

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We propose the systematic construction of classical and quantum two-dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r-matrices and underlying spatial and temporal classical and quantum algebras. This is a novel construction that leads to the derivation of fully discrete integrable systems governed by sets of consistent integrable non-linear space-time difference equations. To illustrate the proposed methodology, we derive two versions of the fully discrete non-linear Schrödinger type system. The first one is based on the existence of a rational r-matrix, whereas the second one is the fully discrete Ablowitz–Ladik model and is associated to a trigonometric r-matrix. The Darboux-dressing method is also applied for the first discretization scheme, mostly as a consistency check, and solitonic as well as general solutions, in terms of solutions of the fully discrete heat equation, are also derived. The quantization of the fully discrete systems is then quite natural in this context and the two-dimensional quantum lattice is thus also examined.

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Section: Introductionmentioning

confidence: 99%

An algebraic approach to discrete time integrability

Doikou¹,

Findlay²

2021

J. Phys. A: Math. Theor.

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“…Kakei [1] investigates lattice solutions of the KdV and the Boussinesq family via the Toda Lattice hierarchy and produces a unified construction of solutions. Hone and Kouloukas [2] look at the reductions of the discrete Hirota equation proving Liouville integrability for some particular cases. A paper by Um, Willox, Grammaticos and Ramani [3] looks at singularity confinement of the discrete KdV equation, comparing singularities arising from integrable and non-integrable generalisations they discover some interesting subtleties in the systems.…”

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confidence: 99%