Wound-up phase turbulence in the complex Ginzburg-Landau equation

Abstract

We consider phase turbulent regimes with nonzero winding number in the one-dimensional complex Ginzburg-Landau equation. We find that phase turbulent states with winding number larger than a critical one are only transients and decay to states within a range of allowed winding numbers. The analogy with the Eckhaus instability for nonturbulent waves is stressed. The transition from phase to defect turbulence is interpreted as an ergodicity breaking transition that occurs when the range of allowed winding numbers vanishes. We explain the states reached at long times in terms of three basic states, namely, quasiperiodic states, frozen turbulence states, and riding turbulence states. Justification and some insight into them are obtained from an analysis of a phase equation for nonzero winding number: Rigidly moving solutions of this equation, which correspond to quasiperiodic and frozen turbulence states, are understood in terms of periodic and chaotic solutions of an associated system of ordinary differential equations. A short report of some of our results has already been published [R. Montagne et al., Phys. Rev. Lett. 77, 267 (1996)].