## Abstract

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-1}A|^{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_{n}F^{2}_{n}/F_{n-1}, for c_{n}, b_{n} > 0 (F_{0}≡1). Estimates for the "bottom rung" F_{2} give upper bounds for the whole ladder. Long time upper bounds on F_{2} 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.

Original language | English (US) |
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Pages (from-to) | 421-444 |

Number of pages | 24 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 44 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 1990 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics