Stability of plane wave solutions of complex Ginzburg-Landau equations

Matkowsky, B. J.; Volpert, Vladimir A.

doi:10.1090/qam/1218368

“…Due to the higher order terms (HOT), the diagram looses the q → −q symmetry for non-zero ǫ. The curves drawn here represents the Eckhaus small-wavenumber instability limit, but we also verified that the system is stable against modulations at any finite wavenumber [49].…”

Section: Uniform Hydrothermal Waves (Uhw) In Annular Geometry and Higsupporting

confidence: 61%

“…Another possibility is the occurrence of short-wavelength modulational instability [49,29], which may perhaps describe better the pattern: when the intermittency changes a chaotic state into a single wave pattern, we again notice this pattern to be modulated at K M = 2π/L p with a fast decaying modulation. Thus, metastable UHW at very low k in the intermittent region seems to be stable with respect to long-wavelength modulational instability, i.e., the classical Eckhaus instability.…”

Section: Defect Chaos Patterns Far From K Cmentioning

confidence: 78%

Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. II. Convective/absolute transitions

Garnier¹,

Chiffaudel²,

Daviaud³

2003

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

We present experimental results on hydrothermal traveling-waves dynamics in long and narrow 1D channels. The onset of primary traveling-wave patterns is briefly presented for different fluid heights and for annular or bounded channels, i.e., within periodic or non-periodic boundary conditions. For periodic boundary conditions, by increasing the control parameter or changing the discrete mean-wavenumber of the waves, we produce modulated waves patterns. These patterns range from stable periodic phase-solutions, due to supercritical Eckhaus instability, to spatio-temporal defect-chaos involving traveling holes and/or counter-propagating-waves competition, i.e., traveling sources and sinks. The transition from non-linearly saturated Eckhaus modulations to transient pattern-breaks by traveling holes and spatiotemporal defects is documented. Our observations are presented in the framework of coupled complex Ginzburg-Landau equations with additional fourth and fifth order terms which account for the reflection symmetry breaking at high wave-amplitude far from onset. The second part of this paper [1] extends this study to spatially non-periodic patterns observed in both annular and bounded channel.

show abstract

“…Due to the higher order terms (HOT), the diagram looses the q → −q symmetry for non-zero ǫ. The curves drawn here represents the Eckhaus small-wavenumber instability limit, but we also verified that the system is stable against modulations at any finite wavenumber [49]. a given ǫ, the system is locally equivalent to a CGL model with appropriate c 1 (ǫ) and c 2 (ǫ) (differing from the HOCGL constant c 1 and c 2 ).…”

Section: Uniform Hydrothermal Waves (Uhw) In Annular Geometry and Higsupporting

confidence: 57%

Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. I. General presentation and periodic solutions

Garnier

¹

,

Chiffaudel

²

,

Daviaud

³

et al. 2003

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

We present experimental results on hydrothermal traveling-waves dynamics in long and narrow 1D channels. The onset of primary traveling-wave patterns is briefly presented for different fluid heights and for annular or bounded channels, i.e., within periodic or non-periodic boundary conditions. For periodic boundary conditions, by increasing the control parameter or changing the discrete mean-wavenumber of the waves, we produce modulated waves patterns. These patterns range from stable periodic phase-solutions, due to supercritical Eckhaus instability, to spatio-temporal defect-chaos involving traveling holes and/or counter-propagating-waves competition, i.e., traveling sources and sinks. The transition from non-linearly saturated Eckhaus modulations to transient pattern-breaks by traveling holes and spatiotemporal defects is documented. Our observations are presented in the framework of coupled complex Ginzburg-Landau equations with additional fourth and fifth order terms which account for the reflection symmetry breaking at high wave-amplitude far from onset. The second part of this paper [1] extends this study to spatially non-periodic patterns observed in both annular and bounded channel.

show abstract

“…There is much literature on this. We refer to Matkovsky and Volpert [20] where the stability of purely periodic patterns to systems like (1.4), and thus (1.5), has been studied. The same ideas can be used to study corresponding solutions to (1.6).…”

Section: Discussionmentioning

confidence: 99%

“…In deriving (1.8) we chose 1 = 2 = 0: This only simplifies the analysis of the four-dimensional system. Apart from other solutions, both 'most stable' (see [20]) Stokes-wave solutions, (A = const., B ≡ 0) and (B = const., A ≡ 0), satisfy 1 = 2 = 0 and are thus described by (1.8). This four-dimensional system can be analysed (for instance) by the geometric theory for singularly perturbed systems, originally developed by Fenichel [11]; see also the contribution of Jones to [1].…”

Section: 5)mentioning

confidence: 99%

Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms

Doelman

¹

,

Rottschäfer

²

1997

View full text Add to dashboard Cite

Summary. Two related systems of coupled modulation equations are studied and compared in this paper. The modulation equations are derived for a certain class of basic systems which are subject to two distinct, interacting, destabilising mechanisms. We assume that, near criticality, the ratio of the widths of the unstable wavenumber-intervals of the two (weakly) unstable modes is small-as, for instance, can be the case in doublelayer convection. Based on these assumptions we first derive a singularly perturbed modulation equation and then a modulation equation with a nonlocal term. The reduction of the singularly perturbed system to the nonlocal system can be interpreted as a limit in which the width of the smallest unstable interval vanishes. We study and compare the behaviour of the stationary solutions of both systems. It is found that spatially periodic stationary solutions of the nonlocal system exist under the same conditions as spatially periodic stationary solutions of the singularly perturbed system. Moreover, these solutions can be interpreted as representing the same quasi-periodic patterns in the underlying basic system. Thus, the 'Landau reduction' to the nonlocal system has no significant influence on the stationary quasi-periodic patterns. However, a large variety of intricate heteroclinic and homoclinic connections is found for the singularly perturbed system. These orbits all correspond to so-called 'localised structures' in the underlying system: They connect simple periodic patterns at x → ±∞. None of these patterns can be described by the nonlocal system. So, one may conclude that the reduction to the nonlocal system destroys a rich and important set of patterns.

show abstract

Stability of plane wave solutions of complex Ginzburg-Landau equations

Cited by 34 publications

References 24 publications

Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. II. Convective/absolute transitions

Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. II. Convective/absolute transitions

Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. I. General presentation and periodic solutions

Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms

Contact Info

Product

Resources

About