Reduction of SO(2) symmetry for spatially extended dynamical systems.

Spatially extended systems, such as channel or pipe flows, are often equivariant under continuous symmetry transformations, with each state of the flow having an infinite number of equivalent solutions obtained from it by a translation or a rotation. This multitude of equivalent solutions tends to obscure the dynamics of turbulence. Here we describe the "first Fourier mode slice," a very simple, easy to implement reduction of SO(2) symmetry.

Cartography of high-dimensional flows: a visual guide to sections and slices.

Symmetry reduction by the method of slices quotients the continuous symmetries of chaotic flows by replacing the original state space by a set of charts, each covering a neighborhood of a dynamically important class of solutions, qualitatively captured by a "template." Together these charts provide an atlas of the symmetry-reduced "slice" of state space, charting the regions of the manifold explored by the trajectories of interest. Within the slice, relative equilibria reduce to equilibria and relative periodic orbits reduce to periodic orbits.

Stability of nonlinear Vlasov-Poisson equilibria through spectral deformation and Fourier-Hermite expansion.

We study the stability of spatially periodic, nonlinear Vlasov-Poisson equilibria as an eigenproblem in a Fourier-Hermite basis (in the space and velocity variables, respectively) of finite dimension, N. When the advection term in the Vlasov equation is dominant, the convergence with N of the eigenvalues is rather slow, limiting the applicability of the method. We use the method of spectral deformation introduced by Crawford and Hislop [Ann.

Self-organization and threshold of stimulated Raman scattering.

We derive, both theoretically and using an envelope code, threshold intensities for stimulated Raman scattering, which compare well with results from Vlasov simulations. To do so, we account for the nonlinear decrease of Landau damping and for the detuning induced by both the nonlinear wave number shift δk{p} and the frequency shift δω{p} of the plasma wave. In particular, we show that the effect of δk{p} may cancel out that of δω{p}, but only in that plasma region where the laser intensity decreases along the direction of propagation of the scattered wave.