Birth–death process of local structures in defect turbulence described by the one-dimensional complex Ginzburg–Landau equation

• Defect turbulence with composite local structures is studied in the 1D CGLE.
• The local structures are identified as defects, holes, and MAWs.
• Number fluctuations of the local structures are subjected to the Poisson statistics.
• Interarrival times of the local structures exhibit power-laws with some peaks.
• A non-Markovian master equation mimics successfully the stochastic dynamics.

Defect turbulence described by the one-dimensional complex Ginzburg–Landau equation is investigated and analyzed via a birth–death process of the local structures composed of defects, holes, and modulated amplitude waves (MAWs). All the number statistics of each local structure, in its stationary state, are subjected to Poisson statistics. In addition, the probability density functions of interarrival times of defects, lifetimes of holes, and MAWs show the existence of long-memory and some characteristic time scales caused by zigzag motions of oscillating traveling holes. The corresponding stochastic process for these observations is fully described by a non-Markovian master equation.