2,115 research outputs found
Integrable Discrete Geometry: the Quadrilateral Lattice, its Transformations and Reductions
We review recent results on Integrable Discrete Geometry. It turns out that
most of the known (continuous and/or discrete) integrable systems are
particular symmetries of the quadrilateral lattice, a multidimensional lattice
characterized by the planarity of its elementary quadrilaterals. Therefore the
linear property of planarity seems to be a basic geometric property underlying
integrability. We present the geometric meaning of its tau-function, as the
potential connecting its forward and backward data. We present the theory of
transformations of the quadrilateral lattice, which is based on the discrete
analogue of the theory of rectilinear congruences. In particular, we discuss
the discrete analogues of the Laplace, Combescure, Levy, radial and fundamental
transformations and their interrelations. We also show how the sequence of
Laplace transformations of a quadrilateral surface is described by the discrete
Toda system. We finally show that these classical transformations are strictly
related to the basic operators associated with the quantum field theoretical
formulation of the multicomponent Kadomtsev-Petviashvilii hierarchy. We review
the properties of quadrilateral hyperplane lattices, which play an interesting
role in the reduction theory, when the introduction of additional geometric
structures allows to establish a connection between point and hyperplane
lattices. We present and fully characterize some geometrically distinguished
reductions of the quadrilateral lattice, like the symmetric, circular and
Egorov lattices; we review also basic geometric results of the theory of
quadrilateral lattices in quadrics, and the corresponding analogue of the
Ribaucour reduction of the fundamental transformation.Comment: 27 pages, 9 figures, to appear in Proceedings from the Conference
"Symmetries and Integrability of Difference Equations III", Sabaudia, 199
The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice
We present a detailed study of the geometric and algebraic properties of the
multidimensional quadrilateral lattice (a lattice whose elementary
quadrilaterals are planar; the discrete analogue of a conjugate net) and of its
basic reductions. To make this study, we introduce the notions of forward and
backward data, which allow us to give a geometric meaning to the tau-function
of the lattice, defined as the potential connecting these data. Together with
the known circular lattice (a lattice whose elementary quadrilaterals can be
inscribed in circles; the discrete analogue of an orthogonal conjugate net) we
introduce and study two other basic reductions of the quadrilateral lattice:
the symmetric lattice, for which the forward and backward data coincide, and
the D-invariant lattice, characterized by the invariance of a certain natural
frame along the main diagonal. We finally discuss the Egorov lattice, which is,
at the same time, symmetric, circular and D-invariant. The integrability
properties of all these lattices are established using geometric, algebraic and
analytic means; in particular we present a D-bar formalism to construct large
classes of such lattices. We also discuss quadrilateral hyperplane lattices and
the interplay between quadrilateral point and hyperplane lattices in all the
above reductions.Comment: 48 pages, 6 figures; 1 section added, to appear in J. Geom. & Phy
Multidimensional Quadrilateral Lattices are Integrable
The notion of multidimensional quadrilateral lattice is introduced. It is
shown that such a lattice is characterized by a system of integrable discrete
nonlinear equations. Different useful formulations of the system are given. The
geometric construction of the lattice is also discussed and, in particular, it
is clarified the number of initial--boundary data which define the lattice
uniquely.Comment: 18 pages, LaTeX, 6 Postscript figure
Integrable Systems and Discrete Geometry
This is an expository article for Elsevier's Encyclopedia of Mathematical
Physics on the subject in the title. Comments/corrections welcome.Comment: 22 pages, 7 figure
Quadratic reductions of quadrilateral lattices
It is shown that quadratic constraints are compatible with the geometric
integrability scheme of the multidimensional quadrilateral lattice equation.
The corresponding Ribaucour reduction of the fundamental transformation of
quadrilateral lattices is found as well, and superposition of the Ribaucour
transformations is presented in the vectorial framework. Finally, the quadratic
reduction approach is illustrated on the example of multidimensional circular
lattices.Comment: 24 page
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