Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-24T22:28:02.505Z Has data issue: false hasContentIssue false
  • English
  • Français

Nearly real fronts in a Ginzburg–Landau equation

Published online by Cambridge University Press:  14 November 2011

Christopher K. R. T. Jones ,
Todd M. Kapitula  and
James A. Powell
Christopher K. R. T. Jones
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742, U.S.A.
Todd M. Kapitula
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742, U.S.A.
James A. Powell
Affiliation:
Department of Mathematics, University of Arizona, Tucson, Arizona 85721, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Subcritical fronts are shown to exist in a quintic version of the well-known complex Ginzburg–Landau equation, which has a subcritical pitchfork as well as a supercritical saddle-node bifurcation. The fronts connect a finite amplitude plane wave state to a stable zero solution. The unstable manifold at finite amplitude and stable manifold of vanishing amplitude solutions are shown to intersect transversely on an invariant zero-wavenumber manifold with parameters set to be real. By the persistence of transverse intersection, frontal connections exist for a continuum of nearly real fronts parametrised by appropriate variables that exhibit some interesting changes in dimension.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 116 , Issue 3-4 , 1990 , pp. 193 - 206
Copyright
Copyright © Royal Society of Edinburgh 1990

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

1Ahlers, G., Cannell, D. and Heinrichs, R.. Convection in a binary mixture, Phys. Rev. A (3) 35 (1987), 2757–2760.Google Scholar
2Arnol'd, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations (Heidelberg: Springer, 1983).CrossRefGoogle Scholar
3Bernoff, A.. Slowly varying fully nonlinear wavetrains in the Ginsburg-Landau equation. Phys. D 30 (1988), 363–381.CrossRefGoogle Scholar
4Brachet, M., Coulett, P. and Fauve, S.. Propagative phase dynamics in temporally intermittent systems. Europhys. Lett. 4 (1987), 1017–1022.CrossRefGoogle Scholar
5Brand, H., Lomdahl, P. and Newell, A.. Benjamin–Feir turbulence in convective binary fluid mixtures. Phys. D 23 (1986), 345–361.CrossRefGoogle Scholar
6Brand, H., Lomdahl, P. and Newell, A.. Evolution of the order parameter in situations with broken rotational symmetry. Phys. Lett. A 118 (16), 67–73.CrossRefGoogle Scholar
7Dee, G.. Dynamical properties of propagating front solutions. Phys. D 15 (1985), 295–304.CrossRefGoogle Scholar
8Dee, G. and Langer, J. S.. Propagating pattern selection. Phys. Rev. Lett. 50 (1983), 383–386.CrossRefGoogle Scholar
9Howard, L. N. and Kopell, N.. Slowly varying waves and shock structures in reaction-diffusion equations. Stud. Appl. Math. 56 (1977), 95–145.CrossRefGoogle Scholar
10Kopell, N. and Howard, L. N.. Plane wave solutions to reaction-diffusion equations. Stud. Appl. Math. 52 (1973), 291–328.CrossRefGoogle Scholar
11Landman, M.. Solutions of the Ginsburg–Landau equation of interest in shear flow transition. Stud. Appl. Math. 76 (1987), 187–237.CrossRefGoogle Scholar
12Moses, E., Fineberg, J. and Steinberg, V.. Multistability and confined travelling wave patterns in a convecting binary mixture. Phys. Rev. A (3) 35 (1987), 2761–2764.CrossRefGoogle Scholar
13Newell, A.. Envelope equations. Lect. Appl. Math. 15 (1974), 157–163.Google Scholar
14Nozaki, K. and Bekki, N.. Pattern selection and spatiotemporal transition to chaos in the Ginsburg–Landau equation. Phys. Rev. Lett. 51 (1983), 2171–2174.CrossRefGoogle Scholar
15Powell, J. and Bernoff, A.. Saddle-node bifurcation of slowly varying nonlinear traveling waves. Phys. D (submitted).Google Scholar
16Schopf, W. and Zimmermann, W.. Multicritical behavior in binary fluid convection. Europhys. Lett. 8 (1989), 41–46.CrossRefGoogle Scholar
17Steinberg, V. and Moses, E.. Experiments in binary mixtures. In Patterns, Defects and Microstructures in Nonequilibrium Systems, ed. Walgraef, D., pp. 309–335. (Dordrecht: Martin Nijhoff, 1987).CrossRefGoogle Scholar
18Van Saarloos, W.. Front propagation into unstable states II: linear vs. nonlinear marginal stability and rates of convergence (preprint).Google Scholar