Spectroscopy of drums and quantum billiards: perturbative and non-perturbative results

We develop powerful numerical and analytical techniques for the solution of the Helmholtz equation on general domains. We prove two theorems: the first theorem provides an exact formula for the ground state of an arbirtrary membrane, while the second theorem generalizes this result to any excited state of the membrane. We also develop a systematic perturbative scheme which can be used to study the small deformations of a membrane of circular or square shapes. We discuss several applications, obtaining numerical and analytical results.Comment: 29 pages, 12 figures, 7 tabl

Scale-Free topologies and Activatory-Inhibitory interactions

A simple model of activatory-inhibitory interactions controlling the activity of agents (substrates) through a "saturated response" dynamical rule in a scale-free network is thoroughly studied. After discussing the most remarkable dynamical features of the model, namely fragmentation and multistability, we present a characterization of the temporal (periodic and chaotic) fluctuations of the quasi-stasis asymptotic states of network activity. The double (both structural and dynamical) source of entangled complexity of the system temporal fluctuations, as an important partial aspect of the Correlation Structure-Function problem, is further discussed to the light of the numerical results, with a view on potential applications of these general results.Comment: Revtex style, 12 pages and 12 figures. Enlarged manuscript with major revision and new results incorporated. To appear in Chaos (2006

Systematic derivation of a rotationally covariant extension of the 2-dimensional Newell-Whitehead-Segel equation

An extension of the Newell-Whitehead-Segel amplitude equation covariant under abritrary rotations is derived systematically by the renormalization group method.Comment: 8 pages, to appear in Phys. Rev. Letters, March 18, 199

Kink Arrays and Solitary Structures in Optically Biased Phase Transition

An interphase boundary may be immobilized due to nonlinear diffractional interactions in a feedback optical device. This effect reminds of the Turing mechanism, with the optical field playing the role of a diffusive inhibitor. Two examples of pattern formation are considered in detail: arrays of kinks in 1d, and solitary spots in 2d. In both cases, a large number of equilibrium solutions is possible due to the oscillatory character of diffractional interaction.Comment: RevTeX 13 pages, 3 PS-figure

Shear induced grain boundary motion for lamellar phases in the weakly nonlinear regime

We study the effect of an externally imposed oscillatory shear on the motion of a grain boundary that separates differently oriented domains of the lamellar phase of a diblock copolymer. A direct numerical solution of the Swift-Hohenberg equation in shear flow is used for the case of a transverse/parallel grain boundary in the limits of weak nonlinearity and low shear frequency. We focus on the region of parameters in which both transverse and parallel lamellae are linearly stable. Shearing leads to excess free energy in the transverse region relative to the parallel region, which is in turn dissipated by net motion of the boundary toward the transverse region. The observed boundary motion is a combination of rigid advection by the flow and order parameter diffusion. The latter includes break up and reconnection of lamellae, as well as a weak Eckhaus instability in the boundary region for sufficiently large strain amplitude that leads to slow wavenumber readjustment. The net average velocity is seen to increase with frequency and strain amplitude, and can be obtained by a multiple scale expansion of the governing equations

Thermodynamic Limit Of The Ginzburg-Landau Equations

We investigate the existence of a global semiflow for the complex Ginzburg-Landau equation on the space of bounded functions in unbounded domain. This semiflow is proven to exist in dimension 1 and 2 for any parameter values of the standard cubic Ginzburg-Landau equation. In dimension 3 we need some restrictions on the parameters but cover nevertheless some part of the Benjamin-Feijer unstable domain.Comment: uuencoded dvi file (email: [email protected]

Unstable decay and state selection II

The decay of unstable states when several metastable states are available for occupation is investigated using path-integral techniques. Specifically, a method is described which allows the probabilities with which the metastable states are occupied to be calculated by finding optimal paths, and fluctuations about them, in the weak noise limit. The method is illustrated on a system described by two coupled Langevin equations, which are found in the study of instabilities in fluid dynamics and superconductivity. The problem involves a subtle interplay between non-linearities and noise, and a naive approximation scheme which does not take this into account is shown to be unsatisfactory. The use of optimal paths is briefly reviewed and then applied to finding the conditional probability of ending up in one of the metastable states, having begun in the unstable state. There are several aspects of the calculation which distinguish it from most others involving optimal paths: (i) the paths do not begin and end on an attractor, and moreover, the final point is to a large extent arbitrary, (ii) the interplay between the fluctuations and the leading order contribution are at the heart of the method, and (iii) the final result involves quantities which are not exponentially small in the noise strength. This final result, which gives the probability of a particular state being selected in terms of the parameters of the dynamics, is remarkably simple and agrees well with the results of numerical simulations. The method should be applicable to similar problems in a number of other areas such as state selection in lasers, activationless chemical reactions and population dynamics in fluctuating environments.Comment: 28 pages, 6 figures. Accepted for publication in Phys. Rev.

Efficiency of primary saliva secretion: an analysis of parameter dependence in dynamic single-cell and acinus models, with application to aquaporin knockout studies

Secretion from the salivary glands is driven by osmosis following the establishment of osmotic gradients between the lumen, the cell and the interstitium by active ion transport. We consider a dynamic model of osmotically driven primary saliva secretion and use singular perturbation approaches and scaling assumptions to reduce the model. Our analysis shows that isosmotic secretion is the most efficient secretion regime and that this holds for single isolated cells and for multiple cells assembled into an acinus. For typical parameter variations, we rule out any significant synergistic effect on total water secretion of an acinar arrangement of cells about a single shared lumen. Conditions for the attainment of isosmotic secretion are considered, and we derive an expression for how the concentration gradient between the interstitium and the lumen scales with water- and chloride-transport parameters. Aquaporin knockout studies are interpreted in the context of our analysis and further investigated using simulations of transport efficiency with different membrane water permeabilities. We conclude that recent claims that aquaporin knockout studies can be interpreted as evidence against a simple osmotic mechanism are not supported by our work. Many of the results that we obtain are independent of specific transporter details, and our analysis can be easily extended to apply to models that use other proposed ionic mechanisms of saliva secretion

Passive Scalar: Scaling Exponents and Realizability

An isotropic passive scalar field $T$ advected by a rapidly-varying velocity field is studied. The tail of the probability distribution $P(\theta,r)$ for the difference $\theta$ in $T$ across an inertial-range distance $r$ is found to be Gaussian. Scaling exponents of moments of $\theta$ increase as $\sqrt{n}$ or faster at large order $n$, if a mean dissipation conditioned on $\theta$ is a nondecreasing function of $|\theta|$. The $P(\theta,r)$ computed numerically under the so-called linear ansatz is found to be realizable. Some classes of gentle modifications of the linear ansatz are not realizable.Comment: Substantially revised to conform with published version. Revtex (4 pages) with 2 postscript figures. Send email to [email protected]