On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation

We present analytical methods which predict the occurrence of both soft (weak) and hard (strong) turbulence in the complex Ginzburg-Landau (CGL) equation: A_t=RA+(1+iν)δA-(1+iμ)A|A|² on a periodic domain [0,1] ^D in D spatial dimensions. Hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales. This form of hard turbulence appears to occur not in 1D but only in 2D and 3D in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the μ-ν planes where the system is modulationally unstable (ε{lunate}=1+μν<0). This region goes out to the dissipationless limit (μ,ν→±∞,∓∞) where the CGL equation becomes the NLS equation. When D≥2 this latter equation blows up in finite time and it is clear from our results that this finite time singularity is fundamental in causing strong turbulent behaviour. The CGL equation has an attractor consisting of C^∞ functions for all finite μ and ν when D=1 and 2. When D=3 we have the same regularity in part of the μ-ν plane, which covers some of the predicted hard turbulent area. The CGL equation also possesses inertial manifolds when D=1 and 2. Our results are based on a new method where we consider an infinity of Lyapunov functionals of rank 2n: F_n=∫(|∇^n-1A|²+α_n|A|²ⁿ)dx, for α_n > 0. For large times and large R, upper bounds exist for the infinite set of F_n's, constructed from the hierarchy of differential inequalities F_n≤(2nR+c_n{norm of matrix}A{norm of matrix}²_∞)F _n-b_nF²_n/F_n-1, for c_n, b_n > 0 (F₀≡1). Estimates for the "bottom rung" F₂ give upper bounds for the whole ladder. Long time upper bounds on F₂ and {norm of matrix}A{norm of matrix}²_∞ (and hence all F_n) are well controlled in the soft region but become much larger in the hard region, whereas spati al and temporal averages remain comparatively small. When the nonlinearity is A|A|^2q, the critical case qD=2 gives parallel results.