Multidimensional quadrilateral lattices are integrable

Doliwa, Adam; Santini, P.

doi:10.1016/s0375-9601(97)00456-8

Cited by 137 publications

(279 citation statements)

References 19 publications

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“…It turns out that, with the help of the celebrated Pascal theorem on six points on a conic, the discrete Koenigs constraint can be formulated in a linear way. (22) and L xx (12) intersect in a single point.…”

Section: The Koenigs Latticementioning

confidence: 99%

Geometric discretization of the Koenigs nets

Doliwa

2003

Journal of Mathematical Physics

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We introduce the Koenigs lattice, which is a new integrable reduction of the quadrilateral lattice (discrete conjugate net) and provides natural integrable discrete analogue of the Koenigs net. We construct the Darboux-type transformation of the Koenigs lattice and we show permutability of superpositions of such transformations, thus proving integrability of the Koenigs lattice. We also investigate the geometry of the discrete Koenigs transformation. In particular we characterize the Koenigs transformation in terms of an involution determined by a congruence conjugate to the lattice.

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Section: The Koenigs Latticementioning

confidence: 99%

Geometric discretization of the Koenigs nets

Doliwa

2003

Journal of Mathematical Physics

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“…N ≤ M , whose elementary quadrilaterals are planar (i. e., a quadrilateral lattice) is the correct discrete analogue of a conjugate net [34,10]. The planarity condition can be expressed by the following linear equation (compare with (18))…”

Section: Conjugate Nets and Quadrilateral Laticesmentioning

confidence: 99%

“…Let us also mention that conjugate nets are connected with the description of the three wave resonant interaction. The discrete analogues of the conjugate nets, quadrilateral lattices, are central objects of the integrable discrete geometry which is developing nowadays [10,11]. Finally, Egoroff systems (a particular type of conjugate nets) have recently found application in topological field theory; namely, in the resolution of indescomposable Witten-Dickgraff-Verlinde-Verlinde associativity equations [12].…”

Section: Introductionmentioning

confidence: 99%

Charged free fermions, vertex operators and the classical theory of conjugate nets

Doliwa

Mañas²,

Alonso

et al. 1999

J. Phys. A: Math. Gen.

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We show that the quantum field theoretical formulation of the τ -function theory has a geometrical interpretation within the classical transformation theory of conjugate nets. In particular, we prove that i) the partial charge transformations preserving the neutral sector are Laplace transformations, ii) the basic vertex operators are Lévy and adjoint Lévy transformations and iii) the diagonal soliton vertex operators generate fundamental transformations. We also show that the bilinear identity for the multicomponent Kadomtsev-Petviashvili hierarchy becomes, through a generalized Miwa map, a bilinear identity for the multidimensional quadrilateral lattice equations.

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“…32,33,1 The planarity condition can be expressed by the following linear equation for suitably renormalized tangent vectors…”

Section: Quadrilateral Latticesmentioning

confidence: 99%

From integrable nets to integrable lattices

Mañas

2002

Journal of Mathematical Physics

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Inspired by the results of Jonas, Einsenhart, Demoulin, and Bianchi on the permutability property of classical geometrical transformations of conjugate nets and its reductions-of pseudo-orthogonal, pseudo-symmetric, and pseudo-Egorov typesdressing transformations of the N-component KP hierarchy ͑described within the Grassmannian͒ are used to generate quadrilateral lattices and its corresponding reductions. As a byproduct we get the corresponding discrete dressing transformations; in particular, we characterize the vectorial fundamental discrete transformations preserving the symmetric lattice.

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Multidimensional quadrilateral lattices are integrable

Cited by 137 publications

References 19 publications

Geometric discretization of the Koenigs nets

Geometric discretization of the Koenigs nets

Charged free fermions, vertex operators and the classical theory of conjugate nets

From integrable nets to integrable lattices

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