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Register a new userNonlinear Analysis of the Eckhaus Instability: Modulated Amplitude Waves and Phase Chaos with Non-zero Average Phase Gradient
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We analyze the Eckhaus instability of plane waves in the one-dimensional complex Ginzburg-Landau equation (CGLE) and describe the nonlinear effects arising in the Eckhaus unstable regime. Modulated amplitude waves (MAWs) are quasi-periodic solutions of the CGLE that emerge near the Eckhaus instability of plane waves and cease to exist due to saddle-node bifurcations (SN). These MAWs can be characterized by their average phase gradient $ u$ and by the spatial period P of the periodic amplitude modulation. A numerical bifurcation analysis reveals the existence and stability properties of MAWs with arbitrary $ u$ and P. MAWs are found to be stable for large enough $ u$ and intermediate values of P. For different parameter values they are unstable to splitting and attractive interaction between subsequent extrema of the amplitude. Defects form from perturbed plane waves for parameter values above the SN of the corresponding MAWs. The break-down of phase chaos with average phase gradient $ u$ > 0 (``wound-up phase chaos) is thus related to these SNs. A lower bound for the break-down of wound-up phase chaos is given by the necessary presence of SNs and an upper bound by the absence of the splitting instability of MAWs.
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The mechanism for transitions from phase to defect chaos in the one-dimensional complex Ginzburg-Landau equation (CGLE) is presented. We introduce and describe periodic coherent structures of the CGLE, called Modulated Amplitude Waves (MAWs). MAWs of various period P occur naturally in phase chaotic states. A bifurcation study of the MAWs reveals that for sufficiently large period P, pairs of MAWs cease to exist via a saddle-node bifurcation. For periods beyond this bifurcation, incoherent near-MAW structures occur which evolve toward defects. This leads to our main result: the transition from phase to defect chaos takes place when the periods of MAWs in phase chaos are driven beyond their saddle-node bifurcation.
The transition from phase chaos to defect chaos in the complex Ginzburg-Landau equation (CGLE) is related to saddle-node bifurcations of modulated amplitude waves (MAWs). First, the spatial period P of MAWs is shown to be limited by a maximum P_SN which depends on the CGLE coefficients; MAW-like structures with period larger than P_SN evolve to defects. Second, slowly evolving near-MAWs with average phase gradients $ u approx 0$ and various periods occur naturally in phase chaotic states of the CGLE. As a measure for these periods, we study the distributions of spacings p between neighboring peaks of the phase gradient. A systematic comparison of p and P_SN as a function of coefficients of the CGLE shows that defects are generated at locations where p becomes larger than P_SN. In other words, MAWs with period P_SN represent ``critical nuclei for the formation of defects in phase chaos and may trigger the transition to defect chaos. Since rare events where p becomes sufficiently large to lead to defect formation may only occur after a long transient, the coefficients where the transition to defect chaos seems to occur depend on system size and integration time. We conjecture that in the regime where the maximum period P_SN has diverged, phase chaos persists in the thermodynamic limit.
Do nonlinear waves destroy Anderson localization? Computational and experimental studies yield subdiffusive nonequilibrium wave packet spreading. Chaotic dynamics and phase decoherence assumptions are used for explaining the data. We perform a quantitative analysis of the nonequilibrium chaos assumption, and compute the time dependence of main chaos indicators - Lyapunov exponents and deviation vector distributions. We find a slowing down of chaotic dynamics, which does not cross over into regular dynamics up to the largest observed time scales, still being fast enough to allow for a thermalization of the spreading wave packet. Strongly localized chaotic spots meander through the system as time evolves. Our findings confirm for the first time that nonequilibrium chaos and phase decoherence persist, fueling the prediction of a complete delocalization.
We probe the limits of nonlinear wave spreading in disordered chains which are known to localize linear waves. We particularly extend recent studies on the regimes of strong and weak chaos during subdiffusive spreading of wave packets [EPL {bf 91}, 30001 (2010)] and consider strong disorder, which favors Anderson localization. We probe the limit of infinite disorder strength and study Frohlich-Spencer-Wayne models. We find that the assumption of chaotic wave packet dynamics and its impact on spreading is in accord with all studied cases. Spreading appears to be asymptotic, without any observable slowing down. We also consider chains with spatially inhomogeneous nonlinearity which give further support to our findings and conclusions.
We study the chaotic behavior of multidimensional Hamiltonian systems in the presence of nonlinearity and disorder. It is known that any localized initial excitation in a large enough linear disordered system spreads for a finite amount of time and then halts forever. This phenomenon is called Anderson localization (AL). What happens to AL when nonlinearity is introduced is an interesting question which has been considered in several studies over the past decades. However, the characteristics and the asymptotic fate of such evolutions still remain an issue of intense debate due to their computational difficulty, especially in systems of more than one spatial dimension. As the spreading of initially localized wave packets is a non-equilibrium thermalization process related to the ergodic and chaotic properties of the system, in our work we investigate the properties of chaos studying the behavior of observables related to the systems tangent dynamics. In particular, we consider the disordered discrete nonlinear Schrodinger (DDNLS) equation of one (1D) and two (2D) spatial dimensions. We present detailed computations of the time evolution of the systems maximum Lyapunov exponent (MLE--$Lambda$), and the related deviation vector distribution (DVD). We find that although the systems MLE decreases in time following a power law $t^{alpha_Lambda}$ with $alpha_Lambda <0$ for both the weak and strong chaos regimes, no crossover to the behavior $Lambda propto t^{-1}$ (which is indicative of regular motion) is observed. In addition, the analysis of the DVDs reveals the existence of random fluctuations of chaotic hotspots with increasing amplitudes inside the excited part of the wave packet, which assist in homogenizing chaos and contribute to the thermalization of more lattice sites.
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