Structure and turbulence in the complex Ginzburg-Landau equation with a nonlinearity of arbitrary order.

Abstract

Numerical and analytical studies are undertaken for the "inviscid" limit of the complex Ginzburg-Landau (CGL) equation with the objective of studying the applicability of paradigms from finite dimensional dynamical systems and statistical mechanics to the case of an infinite dimensional dynamical system. The analytical results rely on exploiting the structure of this limit, which becomes the nonlinear Schrodinger (NLS) equation. In the NLS limit the CGL equation can exhibit strong spatio-temporal chaos. The initial growth of the bursts closely mimics the blowup solutions of the NLS equation. The study of this turbulent behavior focuses on the inertial range of the time-averaged wavenumber spectrum. Analytical estimates of the decay rate are constructed assuming both structure driven and homogeneous turbulence, and are compared with numerical simulations. The quintic case is observed to have a stronger decay rate than what is predicted by either theory. This reflects the dominance of dissipation in the dynamics. In the septic case, two distinct inertial ranges are observed. This combination suggests that the evolution of a single burst, on average, is predominantly due to the self-focusing mechanism of blowup NLS in the initial stage, and regularization effects of dissipation in the final stage. Because the initial stage is primarily influenced by the NLS structure, the rate of decay for this range is close to the decay predicted for the structure driven turbulence. In a numerical experiment it is observed that some NLS solutions survive the deformation due to a CGL perturbation. In some cases the question of persistence can be addressed analytically using an averaging technique similar to a Melnikov method for pde's.