Tau-functions and Dressing Transformations for Zero-Curvature Affine Integrable Equations

The solutions of a large class of hierarchies of zero-curvature equations that includes Toda and KdV type hierarchies are investigated. All these hierarchies are constructed from affine (twisted or untwisted) Kac-Moody algebras~$\ggg$. Their common feature is that they have some special ``vacuum solutions'' corresponding to Lax operators lying in some abelian (up to the central term) subalgebra of~$\ggg$; in some interesting cases such subalgebras are of the Heisenberg type. Using the dressing transformation method, the solutions in the orbit of those vacuum solutions are constructed in a uniform way. Then, the generalized tau-functions for those hierarchies are defined as an alternative set of variables corresponding to certain matrix elements evaluated in the integrable highest-weight representations of~$\ggg$. Such definition of tau-functions applies for any level of the representation, and it is independent of its realization (vertex operator or not). The particular important cases of generalized mKdV and KdV hierarchies as well as the abelian and non abelian affine Toda theories are discussed in detail.Comment: 27 pages, plain Te

Comparing omnidirectional reflection from periodic and quasiperiodic one-dimensional photonic crystals

We determine the range of thicknesses and refractive indices for which omnidirectional reflection from quasiperiodic multilayers occurs. By resorting to the notion of area under the transmittance curve, we assess in a systematic way the performance of the different quasiperiodic Fibonacci multilayers.Comment: 5 pages, 4 color figures. Comments welcome

Simple quantum model for light depolarization

Depolarization of quantum fields is handled through a master equation of the Lindblad type. The specific feature of the proposed model is that it couples dispersively the field modes to a randomly distributed atomic reservoir, much in the classical spirit of dealing with this problem. The depolarizing dynamics resulting from this model is analyzed for relevant states.Comment: Improved version. Accepted for publication in the Journal of the Optical Society of America

The transfer matrix: a geometrical perspective

We present a comprehensive and self-contained discussion of the use of the transfer matrix to study propagation in one-dimensional lossless systems, including a variety of examples, such as superlattices, photonic crystals, and optical resonators. In all these cases, the transfer matrix has the same algebraic properties as the Lorentz group in a (2+1)-dimensional spacetime, as well as the group of unimodular real matrices underlying the structure of the abcd law, which explains many subtle details. We elaborate on the geometrical interpretation of the transfer-matrix action as a mapping on the unit disk and apply a simple trace criterion to classify the systems into three types with very different geometrical and physical properties. This approach is applied to some practical examples and, in particular, an alternative framework to deal with periodic (and quasiperiodic) systems is proposed.Comment: 50 pages, 24 figure

Symmetric and Asymmetric Coalescence of Drops on a Substrate

The coalescence of viscous drops on a substrate is studied experimentally and theoretically. We consider cases where the drops can have different contact angles, leading to a very asymmetric coalescence process. Side view experiments reveal that the "bridge" connecting the drops evolves with self-similar dynamics, providing a new perspective on the coalescence of sessile drops. We show that the universal shape of the bridge is accurately described by similarity solutions of the one-dimensional lubrication equation. Our theory predicts a bridge that grows linearly in time and stresses the strong dependence on the contact angles. Without any adjustable parameters, we find quantitative agreement with all experimental observations.Comment: 5 pages, 4 figure

High Gain Amplifier with Enhanced Cascoded Compensation

A two-stage CMOS operational amplifier with both, gain-boosting and indirect current feedback frequency compensation performed by means of regulated cascode amplifiers, is presented. By using quasi-floating-gate transistors (QFGT) the supply requirements, the number of capacitors and the size of the compensation capacitors respect to other Miller schemes are reduced. A prototype was fabricated using a 0.5 μm technology, resulting, for a load of 45 pF and supply voltage of 1.65 V, in open-loop-gain of 129 dB, 23 MHz of gain-bandwidth product, 60o phase margin, 675 μW power consumption and 1% settling time of 28 ns