### Abstract

We present new time-averaged and time-asymptotic bounds on various norms of solutions to the 2-d complex Ginzburg-Landau (CGL) equation, ∂A ∂t = RA + (1 + iv)ΔA - (1 + iμ)|A|^{2}A. These bounds establish the existence of a finite dimensional global attractor and inertial manifolds, so that the dynamics on the attractor are those of a finite dimensional dynamical system. The CGL equation is known to have chaotic solutions characterized by relatively long length-scale, low-modal dynamics, also known as soft turbulence. Our new estimates suggest that near the nonlinear Schrödinger (NLS) limit (|v|, |μ| →∞, if and only it vμ<0), solutions of the CGL equation are dominated by the remnants of the blow-up solutions of the NLS equation, resulting in asymptotic dynamics marked by large intermittent spikes in space and time, i.e., hard turbulence.

Original language | English (US) |
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Pages (from-to) | 349-356 |

Number of pages | 8 |

Journal | Physics Letters A |

Volume | 142 |

Issue number | 6-7 |

DOIs | |

State | Published - Dec 18 1989 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

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## Cite this

*Physics Letters A*,

*142*(6-7), 349-356. https://doi.org/10.1016/0375-9601(89)90380-0