SCIENTIFIC PROGRAMS AND ACTIVITIES

2010-11 Schedule - Future talks to be held at the Fields Institute

Wednesday, June 1, 2011
3:10PM
Fields Institute, Room 230

Erwan Faou (University of Rennes)
Resonances and long time integration of nonlinear Schroedinger equations

In this talk, we will review some recent advances in long time simulation of Hamiltonian PDE, by focusing on the special case of the nonlinear Schrödinger equation with cubic nonlinearity and without potential (the resonant case). After discussing some results concerning the long time behavior of the exact solution (preservation of the actions in dimension 1, energy cascade in dimension 2), we will study the persistence of such qualitative behaviors by fully discrete splitting schemes. In particular, we will show how the choice of the number of grid points or the stepsize can lead to numerical instabilities, and on the other hand how implicit schemes are in general unable to reproduce correctly the energy exchanges in such a resonant situation. The main tool to analyze these phenomena is the use of backward error analysis for splitting methods under CFL applied to Hamiltonian PDEs, as stated in a recent common work with B. Grébert.

Past Talks

Nicolas Perinet, UOIT
Numerical simulation of Faraday waves

When two superposed fluids are vertically shaken the initially plane interface forms patterns if the oscil- lation is of suffcient amplitude. This phenomenon, called the Faraday instability, constitutes an amazingly rich macroscopic model for the study of pattern formation. In addition to the classically observed crystalline patterns (stripes, squares or hexagons) very singular structures have been seen: quasipatterns, oscillons and superlattices.

In an attempt to survey the patterns supplied by the Faraday experiment, we solve the complete nonlinear Navier–Stokes equations by a finite-difference projection method coupled to a Front-Tracking technique for the calculation of the surface tension forces and advection of the interface.

In the linear regime the instability thresholds and temporal eigenmodes calculated numerically are com- pared to those obtained from Floquet theory. In the nonlinear regime we compare with experimental work which provides quantitative features of squares and hexagons arising at saturation for several forcing amplitudes. The evolution of the nonlinear spatial modes and the spatiotemporal spectra are in good agreement with experimental results.

However, experiments and early numerical simulations highlight that the hexagonal symmetry may only be transient. The alternate emergence of hexagonal structures and patterns with other symmetries suggests that the hexagonal regime may be a fixed point belonging to a homoclinic orbit. We have developed an algorithm which forces the hexagonal symmetry in order to calculate the fixed state that will be the starting point for the exploration of this orbit.

Finally, we have carried out a numerical study of the drift instability in the Faraday experiment in an annular configuration. An azimuthal displacement of initially stationary patterns has been experimentally observed when the oscillation amplitude exceeds a secondary threshold. Our numerical simulations have con?rmed this result. Bifurcation diagrams displaying additional instabilities have been constructed, as well as a complementary spatio-temporal spectral analysis.

This is joint work with Damir Juric (Laboratoire d’Informatique pour la Mecanique et les Sciences de l’Ingenieur, CNRS, Orsay) and Laurette Tuckerman (Laboratoire de Physique et Mecanique des Milieux Heterogenes, ESPCI-CNRS, Paris)

Wednesday, April 20, 2011
2:10PM
Fields Institute, Room 230

Greg Lewis, UOIT
Mixed-mode solutions in the differentially heated rotating annulus

The differentially heated rotating annulus experiment has long been regarded as a useful tool for studying baroclinic waves. These waves can be generated in rotating fluids with an imposed horizontal temperature gradient, and thus, may play an important role in the poleward transport of heat and momentum in the atmosphere.

We present a bifurcation analysis of a mathematical model that uses the (three-dimensional) Navier-Stokes equations in the Boussinesq approximation to describe the flow of a near unity Prandtl number fluid (i.e. air) in the differentially heated rotating annulus. In particular, we study the Hopf bifurcations that correspond to the transition from axisymmetric to nonaxisymmetric flow, where the axisymmetric flow loses stability to an azimuthal mode of integer wave number, and rotating waves may be observed. Of particular interest are the double Hopf (Hopf-Hopf) bifurcations that occur along the transition, where thereis an interaction of two modes with azimuthal wave numbers differing by one.

The analysis shows that in certain regions in parameter space, stable quasiperiodic mixed-azimuthal mode solutions result from the mode-interaction. These flows have been called wave dispersion and interference vacillation. The results differ from similar studies of the annulus with a higher Prandtl number fluid (e.g.water). In particular, we show that a decrease in Prandtl number can stabilize these mixed-mode solutions. We also discuss the mode-interaction with 1:2 spatial resonance, which indicates another mechanism by which a mixed-mode solution may arise.

Wednesday, March 16, 2011
3:10PM
BA6183, Bahen Center, 40 St. George St.

Professor Charles Fefferman, Princeton University
Breakdown of Smoothness in the Muskat Problem

The problem concerns the evolution of the interfaces between two or more fluids in a porous medium. The talk presents new phenomena arising when at least three fluids are present. (Joint work with several coauthors.)

Ever since its introduction by Kenneth G. Wilson in the seventies, the renormalization group has been the main conceptual tool used by physicists in order to make meaningful calculations with functional integrals. The latter are, largely conjectural, infinite-dimensional probability measures over spaces of functions which one can try to construct rigorously using a scaling limit of similar measures where the continuum is discretized by finer and finer grids.

The renormalization group is a dynamical system corresponding to averaging over the short distance fluctuations of the random function and zooming out by a fixed scale ratio. Fixed points of this dynamical system correspond to the possible scaling limits one can achieve. The renormalization group provides a far reaching generalization of the familiar central limit theorem, in a situation where the random variables are dependent, in a way which is subordinated to the geometry of the space labeling these variables. In this nontechnical presentation, we will provide an introduction to the basic ideas of the renormalization group.

Optimal transport and geodesics for H1 metrics on diffeomorphism groups

We describe the Wasserstein space for the homogeneous H1 metric which turns out to be isometric to (a piece of) an infinite-dimensional sphere. The corresponding geodesic flow turns out to be integrable, and it is a generalization of the Hunter-Saxton equation. The corresponding optimal transport can be used for the "size-recognition", as opposed to the "shape recognition". This is a joint work with J. Lenells, G. Misiolek, and S. Preston.

New global methods for solving complex, nonlinear, continuous and discrete,deterministic and stochastic,differential or integral, and combined functional equations, will be presented and compared. This talk will outline the Adomian Decomposition Method,and the Homotopy Perturbation Techniques, all offering solutions as convergent infinite functional series.In this talk , the Cherruault Alienor transformations based on a generalization of the space-filling curves theory(for quasi-lossless dimensionality compression, and for functional Global Optimization ) will be outlined and applied to real-world problems.

Chris King (Northeastern University)
The classical capacity of a quantum channel

Quantum channels describe the dynamical evolution of open quantum systems. From the point of view of information theory, a quantum channel is also the simplest quantum analog of the discrete memoryless channel whose capacity for information transmission was analyzed by Shannon more than sixty years ago. This analogy has led to many interesting questions and conjectures concerning the capacities of quantum channels. In this talk I will describe recent progress regarding the transmission of classical information through a quantum channel, and in particular discuss recent counterexamples to the additivity conjecture.