Desargues maps and the Hirota-Miwa equation

Abstract

We study the Desargues maps $ϕ:\ZZ^N\to\PP^M$, 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.

17 pages, 5 figures; v2 - presentation improved

References in corpus (5)

• Quantum geometry of 3-dimensional lattices
• The B-quadrilateral lattice, its transformations and the algebro-geometric construction
• Isothermic surfaces in sphere geometries as Moutard nets
• The tangential map and associated integrable equations
• On $\tau$-function of the quadrilateral lattice

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• Difference systems in bond and face variables and non-potential versions of discrete integrable systems
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