Geometric discretization of the Bianchi system

We introduce the dual Koenigs lattices, which are the integrable discrete analogues of conjugate nets with equal tangential invariants, and we find the corresponding reduction of the fundamental transformation. We also introduce the notion of discrete normal congruences. Finally, considering quadrilateral lattices "with equal tangential invariants" which allow for harmonic normal congruences we obtain, in complete analogy with the continuous case, the integrable discrete analogue of the Bianchi system together with its geometric meaning. To obtain this geometric meaning we also make use of the novel characterization of the circular lattice as a quadrilateral lattice whose coordinate lines intersect orthogonally in the mean.Comment: 26 pages, 7 postscript figure

Discrete asymptotic nets and W-congruences in Plucker line geometry

The asymptotic lattices and their transformations are studied within the line geometry approach. It is shown that the discrete asymptotic nets are represented by isotropic congruences in the Plucker quadric. On the basis of the Lelieuvre-type representation of asymptotic lattices and of the discrete analog of the Moutard transformation, it is constructed the discrete analog of the W-congruences, which provide the Darboux-Backlund type transformation of asymptotic lattices.The permutability theorems for the discrete Moutard transformation and for the corresponding transformation of asymptotic lattices are established as well. Moreover, it is proven that the discrete W-congruences are represented by quadrilateral lattices in the quadric of Plucker. These results generalize to a discrete level the classical line-geometric approach to asymptotic nets and W-congruences, and incorporate the theory of asymptotic lattices into more general theory of quadrilateral lattices and their reductions.Comment: 28 pages, 4 figures; expanded Introduction, new Section, added reference

Integrable Systems and Discrete Geometry

This is an expository article for Elsevier's Encyclopedia of Mathematical Physics on the subject in the title. Comments/corrections welcome.Comment: 22 pages, 7 figure

Geometric discretization of the Koenigs nets

We introduce the Koenigs lattice, which is a new integrable reduction of the quadrilateral lattice (discrete conjugate net) and provides natural integrable discrete analogue of the Koenigs net. We construct the Darboux-type transformations of the Koenigs lattice and we show permutability of superpositions of such transformations, thus proving integrability of the Koenigs lattice. We also investigate the geometry of the discrete Koenigs transformation. In particular we characterize the Koenigs transformation in terms of an involution determined by a congruence conjugate to the lattice.Comment: 17 pages, 2 figures; some spelling and typing errors correcte

Quadratic reductions of quadrilateral lattices

It is shown that quadratic constraints are compatible with the geometric integrability scheme of the multidimensional quadrilateral lattice equation. The corresponding Ribaucour reduction of the fundamental transformation of quadrilateral lattices is found as well, and superposition of the Ribaucour transformations is presented in the vectorial framework. Finally, the quadratic reduction approach is illustrated on the example of multidimensional circular lattices.Comment: 24 page

On τ\tau-function of conjugate nets

We study a potential introduced by Darboux to describe conjugate nets, which within the modern theory of integrable systems can be interpreted as a $\tau$-function. We investigate the potential using the non-local $\bar\partial$ dressing method of Manakov and Zakharov, and we show that it can be interpreted as the Fredholm determinant of an integral equation which naturally appears within that approach. Finally, we give some arguments extending that interpretation to multicomponent Kadomtsev-Petviashvili hierarchy.Comment: 8 page

Geometric Discretisation of the Toda System

The Laplace sequence of the discrete conjugate nets is constructed. The invariants of the nets satisfy, in full analogy to the continuous case, the system of difference equations equivalent to the discrete version of the generalized Toda equation.Comment: 12 pages, LaTeX, 2 Postscript figure

The B-quadrilateral lattice, its transformations and the algebro-geometric construction

The B-quadrilateral lattice (BQL) provides geometric interpretation of Miwa's discrete BKP equation within the quadrialteral lattice (QL) theory. After discussing the projective-geometric properties of the lattice we give the algebro-geometric construction of the BQL ephasizing the role of Prym varieties and the corresponding theta functions. We also present the reduction of the vectorial fundamental transformation of the QL to the BQL case.Comment: 23 pages, 3 figures; presentation improved, some typos correcte