Multiscale expansion and integrability properties of the lattice potential KdV equation

We apply the discrete multiscale expansion to the Lax pair and to the first few symmetries of the lattice potential Korteweg-de Vries equation. From these calculations we show that, like the lowest order secularity conditions give a nonlinear Schroedinger equation, the Lax pair gives at the same order the Zakharov and Shabat spectral problem and the symmetries the hierarchy of point and generalized symmetries of the nonlinear Schroedinger equation.Comment: 10 pages, contribution to the proceedings of the NEEDS 2007 Conferenc

Expanding perfect fluid generalizations of the C-metric

We reexamine Petrov type D gravitational fields generated by a perfect fluid with spatially homogeneous energy density and in which the flow lines form a timelike non-shearing and non-rotating congruence. It is shown that the anisotropic such spacetimes, which comprise the vacuum C-metric as a limit case, can have \emph{non-zero} expansion, contrary to the conclusion in the original investigation by Barnes (Gen. Rel. Grav. 4, 105 (1973)). This class consists of cosmological models with generically one and at most two Killing vectors. We construct their line element and discuss some important properties. The methods used in this investigation incite to deduce testable criteria regarding shearfree normality and staticity op Petrov type $D$ spacetimes in general, which we add in an appendix.Comment: 16 pages, extended and amended versio

A very low temperature STM for the local spectroscopy of mesoscopic structures

We present the design and operation of a very-low temperature Scanning Tunneling Microscope (STM) working at $60 mK$ in a dilution refrigerator. The STM features both atomic resolution and micron-sized scanning range at low temperature. This work is the first experimental realization of a local spectroscopy of mesoscopic structures at very low temperature. We present high-resolution current-voltage characteristics of tunnel contacts and the deduced local density of states of hybrid Superconductor-Normal metal systems.Comment: 5 pages, 5 figures, slightly corrected versio