Fall 2001 Abstracts

We will discuss several situations in which small amplitude waves propagate in a time-dependent potential that is induced by an excitation in a self-consistent nonlinear field. Although the small amplitude waves do not influence the nonlinear field, they are modulated by its presence. This modulation can lead to scattering, resonant amplification, or under certain circumstances, "trapping" or localization of wave energy. The trapping phenomenon is associated with a kind of integrability of the coupled system consisting of the nonlinear field and the modulated linear field. With the help of this integrability, a generalized transform method will be presented for solving the general initial-value problem for the modulated linear waves. Perturbation theory for nearly integrable couplings will be presented, and numerical simulations will be used to illustrate the scattering and resonance effects that are present far from integrability. Applications range from planar waveguide optics to wave propagation in molecular chains.

This talk summarizes joint work with N. N. Akhmediev, J. A. Besley, P. L. Christiansen, S. R. Clarke, A. Soffer, and M. I. Weinstein.

We consider the Navier-Stokes equations in vorticity-streamfunction formulation. The main difficulty associated with this formulation is that there is no boundary condition for the vorticity. We show that every function in L₂ may be decomposed into a sum of a function in the image of the Laplacian operator, having appropriate boundary conditions, and a harmonic function. Thus, we project the vorticity onto the space of functions which reside in the image of the Laplacian operator. The projection is carried out by the biharmonic equation, which relates the streamfunction to the Laplacian of the vorticity. Numerical results for several test problems including the driven cavity are shown.

We also present a new fourth order compact scheme for the time-dependent Navier-Stokes equations. The idea is to extend existing compact schemes for the steady-state equations to the full time-dependent case. This is done by treating the time-derivative term as a source term, and then applying standard time marching schemes to the set of semi-discrete equations. The resulting fourth-order scheme has a nine-point (3x3) stencil.

We will discuss activated escape from a metastable state and nucleation in driven systems. The formulation applies for an arbitrary noise spectrum. We will show that the escape probabilities can be changed very strongly even by a comparatively weak force. In a broad parameter range, the activation energy of escape depends linearly on the force amplitude, even for a high-frequency field. This dependence is described by the logarithmic susceptibility, which has interesting analytical properties. A closed-form explicit expression for the escape rate of an overdamped periodically driven Brownian particle will be presented. It has been quantitatively confirmed by simulations. We will also describe some recent experiments on driven systems and the onset of directed diffusion in spatially periodic non-sinusoidally driven systems.

Tuberculosis (TB) is the leading cause of death due to infectious disease world-wide, with three million deaths per year and as many as two billion latent but infected carriers. A number of factors likely contribute to differences in tuberculosis levels as observed throughout the world. The host immune response plays a role, and evidence suggests correlations between susceptibility to TB and host genes involved in immune responses. We present a modified S-I-R model describing a TB epidemic in a population comprised of individuals who are either inherently more susceptible to TB or neutral. As many of the studies associating susceptibility to active TB have been performed in India, we base model demographics on these studies. Our results indicate that both prevalence and incidence of disease are greatly increased by a subpopulation with inherent susceptibility to active TB. In addition, simulations suggest that there are multiple mechanisms of action of the susceptibility allele at both the host and population level.

Experimental results seem to suggest that the nature of some hydrodynamical phenomena calls for their statistical or stochastic formulation. The main focus of our discussion will be on a 2-dimensional stochastic vorticity equation for an incompressible homogeneous fluid. We consider a signed measure valued stochastic partial differential equation for a vorticity process based on the Skorohod-Ito evolution of a system of N randomly moving point vortices. This approach provides an interesting alternative to some known Navier-Stokes models perturbed by external random forces. We then pose a nonlinear filtering problem associated with the stochastic evolution of vorticity and derive a corresponding Fujisaki-Kallianpur-Kunita stochastic differential equation for the filter.

While the basic atomistic processes that underlie deformation and brittle/ductile fracture are well understood in crystals, the same cannot be said for the corresponding processes in noncrystalline solids. Understanding these processes is of both practical and theoretical interest. Glasses and particularly glassy metals display a host of interesting phenomena including the spontaneous localization of shear, strong dependence of mechanical properties on processing conditions and transitions between brittle and ductile fracture. Particularly intriguing is the fact that many of these physical phenomena such as the brittle-ductile transition are analogous to observations in crystals although the microscopic mechanism must be quite different. This talk will discuss the insights into these phenomena that can be gained through computer simulation as well as some theoretical results that have arisen from these simulations. Recent results relating mechanical properties to the nature of the glass transition will also be discussed.

Experimental information about protein structure as well as energetics (such as binding free energies) are nowadays readily available. However it is difficult to connect structure with energy experimentally. Computer simulations necessarily connect structure with energy and thus can play a crucial role in the scientific quest to connect structure with function. However the role of simulations is limited by the accuracy of the force fields that model energy as a function of conformation, making the search for accurate but efficient force fields an important area of study. Electrostatic interactions represent the largest energy components in current force fields; thus improved modeling of electrostatic interactions is an important first step towards improved force fields. In this talk I will review previous work by our group and others directed at efficient but accurate representation of "long-range" electrostatics, and then discuss possible paths towards improved treatment of "short-range" electrostatics.

Complex spatiotemporal behavior in nonlinear spatially extended systems has received extensive recent study, but is less well-understood than complex dynamics in finite-dimensional dynamical systems. We characterize some aspects of spatiotemporal chaos (STC) in a class of one-dimensional partial differential equations including the Kuramoto-Sivashinsky (KS) equation, focussing in particular on the essential contribution of large spatial scales, which act as a Gaussian ``heat bath'' maintaining the spatiotemporal disorder. We illustrate our conclusions by discussing wavelet-based numerical experiments; the construction of an effective stochastic model for the large scale dynamics; the KS equation in the presence of an additional destabilizing linear term, which displays a transition from STC to a stationary shock-like solution, due to excitation at the large scales; and a sixth-order analogue of the KS equation (the Nikolaevskii model) in which STC is maintained by the coupling to large scales.

There is much current interest in nonlinear periodic structures for applications such as all-optical switching, optical bistability, pulse compression, and the propagation of Bragg solitons. These structures rely on the contradirectional coupling between two modes as well as the intensity-dependent phase changes between them. In this talk we review our work in this area and introduce some novel excitations known as Raman gap solitons.

Networks of inhibitory neurons are thought to play crucial roles in generating and coordinating electrical activity in the brain. For this reason, there has been much interest in trying to understand the mechanisms underlying the behavior that these networks display. Recent findings show that cells in many inhibitory networks are connected by both inhibitory and electrical coupling. However, it is unclear how these two modes of intercellular communication and the intrinsic properties of cells interact to determine the dynamics of the networks. In an attempt to construct a theoretical framework for networks with both electrical and chemical synapses, I formulate an integrate-and-fire model and use the theory of weakly coupled oscillators to examine the influence of coupling kinetics and intrinsic properties of the cells on dynamics of coupled cell pairs.