―∂-Reductions of the Multidimensional Quadrilateral Lattice. The Multidimensional Circular Lattice

Abstract:

We apply a recently introduced [21, 15] reduction method, based on the ―∂ - dressing, to construct a large class of integrable reductions of the equations characterizing the multidimensional quadrilateral lattice (an N-dimensional lattice in ℝ^M, N≤M, whose elementary quadrilaterals are planar and whose continuous limit describes submanifolds parametrized by conjugate lines [11]). We also show that, generically, in the limit of the small lattice parameter, half of these reductions lead to the Darboux equations for symmetric fields and the second half lead to the generalized Lamé equations describing N-dimensional submanifolds of ?^M parametrized by conjugate orthogonal systems of coordinates. We finally show that a distinguished example of the second class of reductions corresponds to the multidimensional circular lattice (an N-dimensional lattice in ?^M, N≤M, whose elementary quadrilaterals are inscribed in circles [7]).