Desargues maps and the Hirota–Miwa equation

摘要

We study the Desargues maps φ : ZN →PM, 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 non-local -dressing method to construct Desargues maps and the corresponding solutions of the system. In particular, we identify the Fredholm determinant of the integral equation inverting the non-local -dressing problem with the τ -function. Finally, we establish equivalence between the Desargues maps and quadrilateral lattices provided we take into consideration also their Laplace transforms.