The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice

Details The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice

1999-07-08

arxiv.org/abs/solv-int/9907012v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

48 pages, 6 figures; 1 section added, to appear in J. Geom. & Phys

Scientific paper

10.1016/S0393-0440(00)00011-5

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.

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