228 research outputs found
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: for very short times and
for longer times, where near . 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 as well as on the root lattice . 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 which generalize
Q-nets (maps with planar elementary
quadrilaterals) and Darboux nets (-valued maps defined on the
edges of 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
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