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

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

Hopping in a Supercooled Lennard-Jones Liquid: Metabasins, Waiting Time Distribution, and Diffusion

We investigate the jump motion among potential energy minima of a Lennard-Jones model glass former by extensive computer simulation. From the time series of minima energies, it becomes clear that the energy landscape is organized in superstructures, called metabasins. We show that diffusion can be pictured as a random walk among metabasins, and that the whole temperature dependence resides in the distribution of waiting times. The waiting time distribution exhibits algebraic decays: $\tau^{-1/2}$ for very short times and $\tau^{-\alpha}$ for longer times, where $\alpha\approx2$ near $T_c$. We demonstrate that solely the waiting times in the very stable basins account for the temperature dependence of the diffusion constant.Comment: to be published in Phys. Rev.

On the variational interpretation of the discrete KP equation

We study the variational structure of the discrete Kadomtsev-Petviashvili (dKP) equation by means of its pluri-Lagrangian formulation. We consider the dKP equation and its variational formulation on the cubic lattice ${\mathbb Z}^{N}$ as well as on the root lattice $Q(A_{N})$. We prove that, on a lattice of dimension at least four, the corresponding Euler-Lagrange equations are equivalent to the dKP equation.Comment: 24 page

Integrable discrete nets in Grassmannians

We consider discrete nets in Grassmannians $\mathbb{G}^d_r$ which generalize Q-nets (maps $\mathbb{Z}^N\to\mathbb{P}^d$ with planar elementary quadrilaterals) and Darboux nets ($\mathbb{P}^d$-valued maps defined on the edges of $\mathbb{Z}^N$ such that quadruples of points corresponding to elementary squares are all collinear). We give a geometric proof of integrability (multidimensional consistency) of these novel nets, and show that they are analytically described by the noncommutative discrete Darboux system.Comment: 10 p

Discrete Laplace Cycles of Period Four

We study discrete conjugate nets whose Laplace sequence is of period four. Corresponding points of opposite nets in this cyclic sequence have equal osculating planes in different net directions, that is, they correspond in an asymptotic transformation. We show that this implies that the connecting lines of corresponding points form a discrete W-congruence. We derive some properties of discrete Laplace cycles of period four and describe two explicit methods for their construction

Darboux transformations for 5-point and 7-point self-adjoint schemes and an integrable discretization of the 2D Schrodinger operator

With this paper we begin an investigation of difference schemes that possess Darboux transformations and can be regarded as natural discretizations of elliptic partial differential equations. We construct, in particular, the Darboux transformations for the general self adjoint schemes with five and seven neighbouring points. We also introduce a distinguished discretization of the two-dimensional stationary Schrodinger equation, described by a 5-point difference scheme involving two potentials, which admits a Darboux transformation.Comment: 15 pages, 1 figur