Linear integral equations and nonlinear difference-difference equations

Quispel, G.; Nijhoff, F.W.; Capel, H.W.; Linden, J.G.M. van der

doi:10.1016/0378-4371(84)90059-1

Cited by 182 publications

(168 citation statements)

References 34 publications

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“…which follow from (30). By construction, this function has the property (R) and is, therefore, a candidate for the three-dimensional consistency with the tetrahedron property.…”

Section: Classification: Synthesismentioning

confidence: 99%

“…Another way of relating continuous and discrete systems, connected with the idea of compatibility, is based on the notion of Bäcklund transformations and the Bianchi permutability theorem for them [4]. This notion, born in the classical differential geometry, found its place in the modern theory of discrete integrable systems [30]. A sort of the synthesis of the analytic and the geometric approach was achieved in [5] and is being actively developed since then, see a review in [6].…”

Section: Introductionmentioning

confidence: 99%

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Classification of Integrable Equations on Quad-Graphs. The Consistency Approach

Адлер¹,

Bobenko²,

Suris³

2003

Commun. Math. Phys.

533

2,020

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A classification of discrete integrable systems on quad-graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the threedimensional consistency. This property yields, among other features, the existence of the discrete zero curvature with a spectral parameter. For all integrable systems of the obtained exhaustive list, the so called three-leg forms are found. This establishes Lagrangian and symplectic structures for these systems, and the connection to discrete systems of the Toda type on arbitrary graphs. Generalizations of these ideas to the three-dimensional integrable systems and to the quantum context are also discussed.

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“…which follow from (30). By construction, this function has the property (R) and is, therefore, a candidate for the three-dimensional consistency with the tetrahedron property.…”

Section: Classification: Synthesismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Classification of Integrable Equations on Quad-Graphs. The Consistency Approach

Адлер¹,

Bobenko²,

Suris³

2003

Commun. Math. Phys.

533

2,020

View full text Add to dashboard Cite

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“…Furthermore, the dependence on these parameters can be exploited to obtain continuum limits of the lattice equations to semi-continuous and fully continuous equations that are compatible with the original fully discrete equation, cf. [17,18].…”

Section: Introductionmentioning

confidence: 99%

Lax pair for the Adler (lattice Krichever–Novikov) system

2002

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“…Якобиева форма (Q 4 ), представленная в теореме 4, была найдена в [14]. Уравнения (Q 1 ) и (Q 3 | δ=0 ) восходят к [21]. Уравнения (Q 2 ) и (Q 3 | δ=1 ) в явном виде появились впервые в [3].…”

Section: § 1 введениеunclassified