LCDS Seminars 2002

Monday, January 28, 2002

Lefschetz Center for Dynamical Systems Seminar

Abstract :   The Discrete Nonlinear Schr{\"o}dinger equation (DNLS) has been recognized as a model of relevance in nonlinear optics, Bose-Einstein condensation or the local denaturation of DNA. In this talk we present the basic features of its solitary wave solutions in 1+1 and 2+1 dimensions and describe their stability and dynamical properties, using analytical and numerical methods. Some interesting implications for applications and connections to recent experiments will also be briefly discussed.

Monday, February 4, 2002

Lefschetz Center for Dynamical Systems Seminar

Monday, February 11, 2002

Lefschetz Center for Dynamical Systems Seminar

Thursday, February 14, 2002

Special Lefschetz Center for Dynamical Systems Seminar

Abstract :   We discuss the properties of nonlinear photonic crystals including the nonlinearity-induced light localization and the formation of "discrete spatial optical solitons", and the bistable light transmission in straight waveguides and waveguide bends. With the help of the Green's function technique, we derive a discrete NLS-type model with long-range interaction that describes, with a good accuracy, many of the properties of the photonic-crystal circuits such as dispersion, defect states, bound modes in bends, resonant linear and nonlinear transmission, etc. The effective discrete equations may be considered as an analog of the Kirchhoff equations for the electronic circuits which are, however, more complicated and take into account the interference effects and

Monday, February 25, 2002

Lefschetz Center for Dynamical Systems Seminar

Abstract :   We investigate the dynamical system governing travelling wave solutions of a perturbed Boussinesq system. In contrast to classical dynamical system theory, the current system possesses a variety of weak, non-analytic solitary wave solutions that coexist in the neighborhood of the same equilibrium points and are the limits of homoclinic, or heteroclinic orbits. This phenomenon is closely related to the nonlinear dispersion in the original physical problem. In this talk, we shall analyze the nonlinear effect on the existence of non-smooth travelling waves and their singularity formation in the dynamical system.

Monday, March 4, 2002

Lefschetz Center for Dynamical Systems Seminar

Monday, March 11, 2002

Lefschetz Center for Dynamical Systems Seminar

Abstract : Wave turbulence formalism for long internal waves in a stratified fluid is developed, based on a natural Hamiltonian description. A kinetic equation appropriate for the description of spectral energy transfer is derived, and its self-similar stationary solution corresponding to a direct cascade of energy toward the short scales is found. This solution is very close to the high wavenumber limit of the Garrett-Munk spectrum of long internal waves in the ocean. In fact, a small modification of the Garrett-Munk formalism includes a spectrum consistent with the one predicted by wave turbulence.

Monday, March 18, 2002

Lefschetz Center for Dynamical Systems Seminar

Abstract :   In this talk, a geometric treatment of singularly perturbed systems with a certain type of turning points will be discussed. We will provide a result on the existence of semi-global invariant manifolds, which reveals a very crucial geometric structure of the system with the type of turning points. Incorporating the delay of stability loss associated with the turning points, we then establish some exchange lemmas to describe the behavior of invariant manifolds passing through the vicinity of the turning points.

We then illustrate an application to the study of wave fan profiles of some Riemann solutions of hyperbolic systems of conservation law in one space dimension.

Monday, April 1, 2002

Lefschetz Center for Dynamical Systems Seminar

Abstract :   We present a bifurcation theory for dispersion managed solitons. After applying the lens transform to the dispersion-managed nonlinear Schroedinger equation (DMNLS) we derive a Schroedinger-type equation with additional quadratic potential. By analyzing the so-called nonlinear TM-equations and averaging of the lens transformed DMNLS we arrive at qualitatively different equations depending on the sign of the residual dispersion and the value of the local dispersion. In the case of positive residual dispersion and moderate values of the local dispersion the quadratic potential is of trapping type and we are able to apply the global bifurcation theorem of Rabinowitz to the averaged equation.

Monday, April 8, 2002

Lefschetz Center for Dynamical Systems Seminar

Abstract :   In this talk, we will give a brief survey of the problem of the relation between bifurcations and stability of traveling waves. When the equation of travelling waves undergoes a homoclinic/hetroclinic bifurcations, this naturally corresponds to emergence of a new travelling wave. The question is "How the stability of the wave is related to the `geometry' of the bifurcation?" We will give some overview of this problem.

Monday, April 15, 2002

Lefschetz Center for Dynamical Systems Seminar

Abstract :   Solutions of kinetic equations in which a collision kernel appear often converge (in large time) to some equilibrium profile (e.g. Maxwellian or Fermi-Dirac distributions of the velocity variable). On the other hand, in many situations, this equilibrium has also to be independant of the position variable (this is due to the fact that no Maxwellian solution of the equation exists, except those which do not depend on the position variable). In our talk, we propose to give some results on how to quantify the previous effects in order to get estimates on the rate of convergence of the solutions towards the global equilibrium (that is, a Maxwellian which does not depend on the position variable). Logarithmic Sobolev inequalities are the key tool in this study.

Monday, April 22, 2002

Lefschetz Center for Dynamical Systems Seminar

Abstract :   I shall begin by describing the Riemann ellipsoids, which are an exact, self-graviting solution of the incompressible Euler equations with free boundary. This problem has a rich history and I shall outline some of the main participants - from Newton to Riemann - and their contributions.

The non-canonical (Lie-Poisson) formulation of the compressible ideal fluid equations was presented by Morrison and Greene (1980). This work shows how to extend that description to the case of incompressible fluids (with free boundary) by careful application of a Hodge-Weyl decomposition of functional variations.

Using the Hamiltonian formulation and a moment description of incompressible flow, an exact reduction of the full partial differential equations to a 15-dimensional, non-canonical Hamiltonian system is afforded. This dynamical system has 3 Casimirs corresponding to circulation, volume and divergence of the velocity field, respectively. I shall discuss the bracket structure of the non-canonical symplectic structure and its consequences for equilibria and dynamics.

In conclusion, I will show how the Dirac bracket formulation - essentially a projection approach for symplectic structures - yields the same non-canonical structure. This is the first example I have seen of the Dirac Bracket formalism used to generate additional Casimirs of a non-canonical symplectic structure resulting in an exact, finite dimensional reduction of an infinite dimensional problem.

Monday, May 6, 2002

Lefschetz Center for Dynamical Systems Seminar

Abstract :   Diophantine conditions are well-known from time averaging and Kolmogorov-Arnold-Moser theory. In contrast, we consider semilinear reaction diffusion equations with spatially quasiperiodic coefficients in the nonlinearity, rapidly varying on spatial scale \epsilon. Under Diophantine conditions on the spatial frequencies, we derive quantitative homogenization estimates of order \epsilon^{\gamma} on Sobolev spaces H^{\sigma} in the triangle

0 < \gamma < min(\sigma - n/2, 2 - \sigma).


Here n denotes spatial dimension. The estimates measure the distance to a solution of the homogenized equation with the same initial condition, on bounded time intervals. The same estimates hold for C1-convergence of local stable and unstable manifolds of hyperbolic equilibria. Our results apply to homogenization of the Navier-Stokes equations with spatially rapidly varying quasiperiodic forces in space dimensions 2 and 3. Our results also extend to quantitative homogenization of global attractors for near-gradient system. An extension to damped hyperbolic wave equations will be indicated. All results are joint work with Mark I. Vishik. For references see www.math.fu-berlin.de/~Dynamik/

Monday, May 13, 2002

Lefschetz Center for Dynamical Systems Seminar

Monday, October 7, 2002

Lefschetz Center for Dynamical Systems Seminar

Monday, October 21, 2002

Lefschetz Center for Dynamical Systems Seminar

Monday, October 28, 2002

Lefschetz Center for Dynamical Systems Seminar

Monday, November 11, 2002

Lefschetz Center for Dynamical Systems Seminar