Wednesdays at 4 PM in Eckhart 206.
In this talk I will present some recent results on NLS equations with isotropic and anisotropic fourth-order dispersion. Applications of these results include analysis of discretizarion effects in simulations of NLS singular solutions, propagation in fiber arrays, and the effect of fourth-order dispersion on spatiotemporal collapse of ultrashort pulses.
We describe LLANO (Lazy Linear Algebraic Numerical Objects), a C++ package which allows operator syntax for dense matrix computations. LLANO uses operator templates, which are essentially a combination of lazy evaluation and compiler technology, to approach the effeciency of more conventional, less aesthetic code. Basic principles and some examples are presented, as well as the implications for some numerical methods for PDE.
I will show how new developments in a perturbative technique introduced by A. Soffer and M.I. Weinstein can quantify the effects of time dependent perturbations on autonomous system supporting both natural oscillations (bound states) and dispersive waves (radiation modes). I will focus on an important model, the Schroedinger equation with an attractive time independent potential perturbed by either almost periodic in time potentials or by sequences of short lived pulses with random time gaps.
The coarsening properties of a faceted crystal surface annealing (in equilibrium) with it's melt is known experimentally to change dramatically if, instead, the crystal is subject to net growth. We present a theory which explains this transition, and exhibit the main ideas in the setting where attachment kinetics is the dominant mass transfer mechanism. Here, the faceting-Cahn-Hilliard ( FCH) equation is the annealing model, while the faceting-eikonal-Cahn-Hilliard (FECH) equation is the associated growth model. We identify the sharp-interface theory of FECH through a matched asymptotic analysis. The result is a novel coarsening dynamical system (CDS) for the edge network of the faceted surface, where coarsening occurs through the merging and annihilation of facets. In the case of square symmetry, we prove for CDS that
Direct numerical simulations of FECH also yield this scaling law and confirm the stability-instability dichotomy through the absence of anti-pyramids in the surface evolution. Hence, our results capture the annealing to growth transition of the coarsening dynamics since, for FCH, L(t) ~ t1/4 and also both pyramids and anti-pyramids appear in the evolution.
The one-dimensional FECH, which is equivalent to the convective Cahn-Hilliard CCH equation, displays unusual coarsening mechanisms. We prove that
These properties stand in marked contrast with with the generic binary coalescence of the 1-d Cahn-Hilliard equation.
Motivated by a linear stability analysis we also propose a length-scale-doubling coarsening ansatz for the FECH evolution, from which we identify the scaling constant C of the t1/2 scaling regime; e.g., L(t) = C t1/2 . This work is, in part, joint with Felix Otto (Bonn) and Stephen H. Davis (Northwestern
The interface between a heavy fluid resting above a light fluid in a porous medium is unstable to small perturbations. In this talk I will describe two scaling laws for the long time evolution of such flows with miscible fluids. The main goal is to understand how the formation of microstructure -- an intricate network of "fingers" -- limits the mixing. The tools are simple ideas from the theory of 1-D conservation laws and the calculus of variations. This is work in progress with Felix Otto (Univ. of Bonn).
We shall survey a few partial differential equation whose formulation is motivated by problem in optics. The first equation is essentially hyperblic, and arises in the process of determining optical surfaces from the deflection of rays. We then consider two kinds of elliptic equations that arise in optical design problems. One of them is related to a general design question, and the other one is related to the design of progressive power lenses.
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