Slowly modulated oscillations in nonlinear diffusion processes

It is shown here that certain systems of nonlinear (parabolic) reaction-diffusion equations have solutions which are approximated by oscillatory functions in the form R(ξ - cτ)P(t^*) where P(t^*) represents a sinusoidal oscillation on a fast time scale t* and R(ξ - cτ) represents a slowly-varying modulating amplitude on slow space (ξ) and slow time (τ) scales. Such solutions describe phenomena in chemical reactors, chemical and biological reactions, and in other media where a stable oscillation at each point (or site) undergoes a slow amplitude change due to diffusion

Smallest small-world network

Efficiency in passage times is an important issue in designing networks, such as transportation or computer networks. The small-world networks have structures that yield high efficiency, while keeping the network highly clustered. We show that among all networks with the small-world structure, the most efficient ones have a single ``center'', from which all shortcuts are connected to uniformly distributed nodes over the network. The networks with several centers and a connected subnetwork of shortcuts are shown to be ``almost'' as efficient. Genetic-algorithm simulations further support our results.Comment: 5 pages, 6 figures, REVTeX

Kinetics of photoinduced ordering in azo-dye films: two-state and diffusion models

We study the kinetics of photoinduced ordering in the azo-dye SD1 photoaligning layers and present the results of modeling performed using two different phenomenological approaches. A phenomenological two state model is deduced from the master equation for an ensemble of two-level molecular systems. Using an alternative approach, we formulate the two-dimensional (2D) diffusion model as the free energy Fokker-Planck equation simplified for the limiting regime of purely in-plane reorientation. The models are employed to interpret the irradiation time dependence of the absorption order parameters extracted from the available experimental data by using the exact solution to the light transmission problem for a biaxially anisotropic absorbing layer. The transient photoinduced structures are found to be biaxially anisotropic whereas the photosteady and the initial states are uniaxial.Comment: revtex4, 34 pages, 9 figure

Photoinduced reordering in thin azo-dye films and light-induced reorientation dynamics of nematic liquid-crystal easy axis

We theoretically study the kinetics of photoinduced reordering triggered by linearly polarized (LP) reorienting light in thin azo-dye films that were initially illuminated with LP ultraviolet (UV) pumping beam. The process of reordering is treated as a rotational diffusion of molecules in the light intensity-dependent mean-field potential. The two dimensional diffusion model which is based on the free energy rotational Fokker-Planck equation and describes the regime of in-plane reorientation is generalized to analyze the dynamics of the azo-dye order parameter tensor at varying polarization azimuth of the reorienting light. It is found that, in the photosteady state, the intensity of LP reorienting light determines the scalar order parameter (the largest eigenvalue of the order parameter tensor), whereas the steady state orientation of the corresponding eigenvector (the in-plane principal axis) depends solely on the polarization azimuth. We show that, under certain conditions, reorientation takes place only if the reorienting light intensity exceeds its critical value. Such threshold behavior is predicted to occur in the bistability region provided that the initial principal axis lies in the polarization plane of reorienting light. The model is used to interpret the experimental data on the light-induced azimuthal gliding of liquid-crystal easy axis on photoaligned azo-dye substrates.Comment: 27 pages, 11 fugure

Quasi-static state analysis of differential, difference, integral, and gradient systems

This book is based on a course on advanced topics in differential equations given in Spring 2010 at the Courant Institute of Mathematical Sciences. It describes aspects of mathematical modeling, analysis, computer simulation, and visualization in the mathematical sciences and engineering that involve singular perturbations. There is a large literature devoted to singular perturbation methods for ordinary and partial differential equations, but there are not many studies that deal with difference equations, Volterra integral equations, and purely nonlinear gradient systems where there is no do