Wednesdays at 4 PM in Eckhart 202.
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.
The classical theory of domain coarsening during phase transitions in materials science involves a conservation law for the size distribution of a family of particles. In joint work with Barbara Niethammer, we give a full well-posedness theory for measure-valued size distributions of compact support, using a physically natural topology given by a Wasserstein distance between size distributions. We also analyze long-time behavior of solutions. Universal self-similar behavior was predicted classically based on physical stability arguments. Rigorous analysis shows, however, that the classical model does not yield the predicted behavior. Instead, long-time behavior depends sensitively on the initial distribution of the largest particles. E.g., for a dense set of initial data, convergence to any self-similar solution is impossible.
We will discuss the following remarkable property of solutions u(x,t) of wave equations in random media (or ergodic billiards) with localized initial data. Let us record u(x,T) at some time T, restrict it to a finite domain, transform it linearly in some fashion and use the resulting signal as a new initial data for the wave equation. It turns out that the new re-propagated solution will concentrate at the original source location at the same time T for a very large class of signal processing. In particular this explains the time-reversal experiments. In such experiments a signal is emitted by a localized source, propagated through a medium and recorded on a small array of receivers-transducers. The signal is re-emitted into the medium reversed in time, that is, the part of the signal recorded first is re-emitted last and vice versa. The re-propagated signal approximately refocuses back on the original source. This is somewhat surprising since the recording array has a small finite size. It is also observed that refocusing is significantly better in a random medium. We will give an explanation for refocusing and explain why random media are good for refocusing, as are ergodic billiards.
At eash site (i,j) in 2-dimensional lattice, we attach an identically distributed, independent random variable. We regard, for example, the random variable at each site as the time required to pass through that site. Then the interest is the last passage time to go from the site (1,1) to the site (M,N) along directed paths, as M,N tend to infinity. This last passage percolation problem have applications to random growth models, queueing theory and interacting particle systems. In recent years, the limiting distribution of the last passage time, after suitable centering and scaling, has been found for a few special cases of random variables. We also discuss the symmetrized versions of the problem.
In many problems, it would be desirable to build representation systems which combine the ideas of multiscale analysis with ideas of geometric features and structures. This talk will introduce newly developed multiscale systems like curvelets and ridgelets which are very different form wavelet-like systems. In particular they have very distinct geometric features. Curvelets and ridgelets take the the form of basis elements which exhibit very high directional sensitivity and are highly anisotropic. In two-dimensions, they are localized along curves, in three dimensions along sheets, etc. In two dimensions for instance, curvelets are in some sense provably optimal for representing or extracting information along curved singularities. This is unlike any other system in current use.
Responses of cortical neurons to sensory stimuli are extremely variable. This variability arises because cortical neurons receive large quantities of random background input in addition to the stimulus-related signals that drive their responses. Background input has conventionally been considered a constant and unavoidable source of noise. Using a combination of analytic calculations concerning first-passage times of random walks and experimental methods involving cortical neurons in slice preparations, we have shown that background synaptic input is not merely a source of noise. Rather, the level of background input act as a "volume control" determining the gain of neuronal responses to sensory input. Thus, the advantage of having high levels of noise in cortical circuits is that it allows individual neurons to act somewhat like transistors, that is, input-output devices with dynamically adjustable gain.
We introduce an iterative grid redistribution method based on the variational approach. The iterative procedure enables us to gain more precise control of the grid distribution near the regions of large solution variations. The method is particularly effective for solving PDE's with singular solutions (e.g. blow up solutions). Examples of various applications in two and three dimensions will be given.
We consider 3D Euler and Navier-Stokes equations with a large Coriolis force and periodic boundary conditions. Such equations are important in geophysics. When the rotation rate tends to zero, as a singular limit we obtain reduced equations. We describe properties of the equations, they depend strongly on the aspect ratios of the spatial periodic box. The structure of the limit equations and large-time regularity of their solutions are related to geometry of 3-wave resonant curves in the aspect ratios plane. In the viscous case solutions of the 3D Navier-Stokes equations are regular for all aspect ratios, but the dependence of the smoothness on the viscosity depends on the ratios. In a contrast to the viscous case, regularity for the Euler equations is proven only when 3D part splits into finite-dimensional subsystems (it is a generic case). Regularity of solutions of the original Navier-Stokes and Euler equations for fluids with a finite large Coriolis force follows as a consequence.
The magnetization of ferromagnetic materials forms complex structures of different dimensionality and on a broad range of length scales: domains, walls of different internal structure, Bloch lines and vortices. The cross--tie wall is a wall--type (transition layer) which occurs in moderately thin films of ferromagnetic material. The magnetization lies entirely in the film plane and is constant in the direction of the film normal. But curiously, the wall has an internal structure in tangential direction: If one looks closely, the transition layer consists of a periodic arrangement of narrow Neel walls separeted by Bloch lines. Does the well--accepted micromagnetic model predicts this interesting pattern? Does it at least predict how the distance w between two Bloch lines (hence the period) scales in the material parameters? The material parameters are the exchange length d, the film thickness t and the non--dimensional anisotropy parameter Q. Surprisingly, this question has not been answered in the applied literature. We give the answer w/d=O(d/Qt), in an appropriate parameter regime. This answer is in qualitative agreement with the experiments. The derivation of this scaling law is based on the rigorous analysis of an interesting cross--over of the energy scaling law for a Neel wall. This is joint work with A. DeSimone, R. V. Kohn and S. Mueller.
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