From integrable nets to integrable lattices

Abstract

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.

B\"{a}cklund transformations as integrable discretization. The geometric approach

“…The Darboux-Bäcklund transformations of the discrete Darboux equations were formulated first on the algebraic level in [79], and then in [38] the full geometric flavour of the theory was presented together with the interpretation of the transformations on the nonlocal ∂-dressing method level, see also [73,75,40,77,78]. In particular it was pointed out in [38], referring to [68] and other similar works, that for discrete conjugate nets there is no essential difference between transformations and generation of new dimensions of the lattice.…”

Darboux transformations for linear operators on two-dimensional regular lattices

“…In this Section we present the Darboux transformations for the four point scheme (the discrete Laplace equation) from the point of view of systems of such equations, and the corresponding permutability theorems. To keep the paper of reasonable size and in order to present the results from a simple algebraic perspective we do not discuss important relations of the subject to incidence and difference geometry [20,31,7,13,21,25,26,27,35,32,54,55,57,58,100,101,39,38] (see also [33,8] and earlier works [97,98]), application of analytic [10,34,35,37,107,106,108,36,32,70,29] and algebrogeometric [59,60,4,22,23,5,39,26,47,27] techniques of the integrable systems theory to construct large classes of solutions of the linear systems in question and solutions of the corresponding nonlinear discrete equations.…”