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

Abstract

Defect turbulence described by the one-dimensional complex Ginzburg-Landau equation is investigated and analyzed via a birthdeath 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 are fully described by a non-Markovian master equation.

Anomalous velocity fluctuation in one-dimensional defect turbulence

“…(1) were chosen to produce the defect turbulence with (ci,C2) = (1.5, -1.2). Note that this change of numerical scheme and system size does not affect the qualitative statistical law of the defect turbulence previously reported [13]. Figure 1 shows the pseudocolor plots for spatiotemporal profiles of amplitude |A| and phase arg(A).…”

“…Indeed, as was presented in our previous paper [13], with ri being a real parameter. In addition, another time scale is expected to exist since each hole displayed particlelike motion with a faster time scale.…”

“…It is plausible that such behaviors of the hole velocities, as shown in Fig. 3, are caused by energy and/or momentum exchanges of the holes with their surroundings in the birth-death process of the local structures [13].…”

“…In our previous work [13], we proposed appropriate identification methods of the two distinct holes, carried out discrimination between them, and tagged the BN hole and the homoclininc hole as a defect and a hole, respectively, in the defect turbulence. Based on the identification methods, stochastic dynamics of birth-death processes of both holes and modulated amplitude waves in defect turbulence have been unveiled to show Poisson processes with long memory, which is described successfully by a nonstationary master equation.…”

“…In inclined layer thermal fluid convection, non-Gaussian defect velocity distributions have been observed experimentally [10], and have been iden tified by the theoretical framework of nonextensive statistical mechanics [11]. On the other hand, a non-Gaussian velocity distribution have been derived from a one-dimensional (ID) transport equation [12], We have introduced appropriate identification methods for local structures, which are localized nonlinear waves in 1D systems, such as the defect, the hole, and the modulated amplitude wave, and have investigated their statistical characteristics [13]. Using the method, we have tracked successfully each hole in the defect turbulence and have speculated that the hole velocity distribution can be described by a generalized Cauchy distribution [14], While the anomalous velocity distributions at steady states have been investigated in many systems, their dynamical properties, such as autocorrelation function (ACF) and meansquare displacement (MSD), have yet to be described by a unified theoretical model.…”