R. Montagne et al., WOUND-UP PHASE TURBULENCE IN THE COMPLEX GINZBURG-LANDAU-EQUATION, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 56(1), 1997, pp. 151-167
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 on
ly transients and decay to states within a range of allowed winding nu
mbers. The analogy with the Eckhaus instability for nonturbulent waves
is stressed. The transition from phase to defect turbulence is interp
reted 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 state
s, frozen turbulence states, and riding turbulence states. Justificati
on 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 sta
tes, are understood in terms of periodic and chaotic solutions of an a
ssociated system of ordinary differential equations. A short report of
some of our results has already been published [R. Montagne et al., P
hys. Rev. Lett. 77, 267 (1996)].