Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-27T19:43:45.644Z Has data issue: false hasContentIssue false
  • English
  • Français

Instabilities and Dynamics of Weakly SubcriticalPatterns

Published online by Cambridge University Press:  17 September 2013

H.-C Kao  and
E. Knobloch
Get access

Abstract

The bifurcation to one-dimensional weakly subcritical periodic patterns is described bythe cubic-quintic Ginzburg-Landau equation

       At = µA + Axx + i(a1|A|2Ax + a2A2Ax*) + b|A|2A - |A|4A.

These periodic patterns may in turn become unstable through one of two differentmechanisms, an Eckhaus instability or an oscillatory instability. We study the dynamicsnear the instability threshold in each of these cases using the corresponding modulationequations and compare the results with those obtained from direct numerical simulation ofthe equation. We also study the stability properties and dynamical evolution of differenttypes of fronts present in the protosnaking region of this equation. The results providenew predictions for the dynamical properties of generic systems in the weakly subcriticalregime.

Keywords

bifurcationcubic-quintic Ginzburg-Landau equationweakly subcritical periodic pattern
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 5: Bifurcations , 2013 , pp. 131 - 154
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Kao, H.-C., Knobloch, E.. Weakly subcritical stationary patterns: Eckhaus stability and homoclinic snaking, Phys. Rev. E 85 (2012), 026207. CrossRefGoogle Scholar
Eckhaus, W., Iooss, G.. Strong selection or rejection of spatially periodic patterns in degenerate bifurcations, Physica D 39 (1989), pp. 124–146. CrossRefGoogle Scholar
Shepeleva, A.. On the validity of the degenerate Ginzburg-Landau equation, Math. Method Appl. Sci. 20 (1997), pp. 1239–1256. 3.0.CO;2-O>CrossRefGoogle Scholar
Brand, H., Deissler, R.. Eckhaus and Benjamin-Feir instabilities near a weakly inverted bifurcation, Phys. Rev. A 45 (1992), pp. 3732–3736. CrossRefGoogle Scholar
Burke, J., Knobloch, E.. Localized states in the generalized Swift-Hohenberg equation, Phys. Rev. E 73 (2006), 056211. CrossRefGoogle ScholarPubMed
Batiste, O., Knobloch, E., Alonso, A., Mercader, I.. Spatially localized binary-fluid convection, J. Fluid Mech. 560 (2006), pp. 149–158. CrossRefGoogle Scholar
Beaume, C., Bergeon, A., Knobloch, E.. Homoclinic snaking of localized states in doubly diffusive convection, Phys. Fluids 23 (2011), 094102.
Doelman, A. and Eckhaus, W.. Periodic and quasi-periodic solutions of degenerate modulation equations, Physica D 53 (1991), pp. 249–266. CrossRefGoogle Scholar
Duan, J., Holmes, P.. Fronts, domain walls and pulses in a generalized Ginzburg-Landau equation, Proc. Edinburgh Math. Soc. 38 (1995), pp. 77–97. CrossRefGoogle Scholar
Shepeleva, A.. Modulated modulations approach to the loss of stability of periodic solutions for the degenerate Ginzburg-Landau equation, Nonlinearity 11 (1998), pp. 409–429. CrossRefGoogle Scholar
D. Henry. Geometric theory of semilinear parabolic equations, Springer-Verlag, Berlin, 1981.
R. Hoyle. Pattern Formation, Cambridge University Press, Cambridge, 2006.
Duan, J., Holmes, P., Titi, E. S.. Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity 5 (1992), pp. 1303–1314. CrossRefGoogle Scholar
Cox, S. M., Matthews, P. C.. Exponential time differencing for stiff systems, J. Comp. Phys. 176 (2002), pp. 430–455. CrossRefGoogle Scholar
Kramer, L., Zimmermann, W.. On the Eckhaus instability for spatially periodic patterns, Physica D 16 (1985), pp. 221–232. CrossRefGoogle Scholar
Ben-Jacob, E., Brand, H., Dee, G., Kramer, L., Langer, J. S.. Pattern propagation in nonlinear dissipative systems, Physica D 14 (1985), pp. 348–364. CrossRefGoogle Scholar
van Saarloos, W.. Front propagation into unstable states. II. Linear versus nonlinear marginal stability and rate of convergence, Phys. Rev. A 39 (1989), pp. 6367–6390. CrossRefGoogle ScholarPubMed
Burke, J., Dawes, J. H. P.. Localised states in an extended Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst. 11 (2012), pp. 261–284. CrossRefGoogle Scholar
Boya, L. J.. Supersymmetric quantum mechanics: two simple examples, Eur. J. Phys. 9 (1988), pp. 139–144. CrossRefGoogle Scholar
Elliott, C. M., Zheng, S.. On the Cahn-Hilliard equation, Arch. Rational Mech. Anal. 96 (1986), pp. 339–357. CrossRefGoogle Scholar
Beaume, C., Bergeon, A., Kao, H.-C., Knobloch, E.. Convectons in a rotating fluid layer, J. Fluid Mech. 717 (2013), pp. 417–448. CrossRefGoogle Scholar
Knobloch, E., DeLuca, J.. Amplitude equations for travelling wave convection, Nonlinearity 3 (1990), pp. 975–980. CrossRefGoogle Scholar