Desargues maps and the Hirota-Miwa equation

We study the Desargues maps \phi:\ZZ^N\to\PP^M, which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multidimensional compatibility of the map is equivalent to the Desargues theorem and its higher-dimensional generalizations. The nonlinear counterpart of the map is the non-commutative (in general) Hirota--Miwa system. In the commutative case of the complex field we apply the nonlocal $\bar\partial$-dressing method to construct Desargues maps and the corresponding solutions of the equation. In particular, we identify the Fredholm determinant of the integral equation inverting the nonlocal $\bar\partial$-dressing problem with the $\tau$-function. Finally, we establish equivalence between the Desargues maps and quadrilateral lattices provided we take into consideration also their Laplace transforms.Comment: 17 pages, 5 figures; v2 - presentation improve

Darboux transformations for linear operators on two dimensional regular lattices

Darboux transformations for linear operators on regular two dimensional lattices are reviewed. The six point scheme is considered as the master linear problem, whose various specifications, reductions, and their sublattice combinations lead to other linear operators together with the corresponding Darboux transformations. The second part of the review deals with multidimensional aspects of (basic reductions of) the four point scheme, as well as the three point scheme.Comment: 23 pages, 3 figures, presentation improve