Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T21:45:04.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Fronts, domain walls and pulses in a generalized Ginzburg-Landau equation*

Published online by Cambridge University Press:  20 January 2009

Jinqiao Duan  and
Philip Holmes
Jinqiao Duan
Affiliation:
Applied Mechanics 104–44California Institute of TechnologyPasadena, California 91125, U.S.A.
Philip Holmes
Affiliation:
Department of Mechanical and Aerospace Engineering and Program in Applied and Computational MathematicsPrinceton UniversityPrinceton, New Jersey 08544, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We discuss the existence and non-existence of front, domain wall and pulse type traveling wave solutions of a Ginzburg-Landau equation with cubic terms containing spatial derivatives and a fifth order term, in both subcritical and supercritical cases. Our results appear to be the first rigorous existence and non-existence proofs for the full equation with all possible terms derived from second order perturbation theory present.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , Issue 1 , February 1995 , pp. 77 - 97
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

REFERENCES

1.Arnold, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations (Springer-Verlag, Berlin, 2nd ed., 1988).Google Scholar
2.Arnold, V. I., Mathematical Methods of Classical Mechanics (Springer-Verlag, Berlin, 1978).CrossRefGoogle Scholar
3.Bernoff, A. J., Slowly varying fully nonlinear wavetrains in the Ginzburg-Landau equation, Phys. D 23 (1988), 363–381.CrossRefGoogle Scholar
4.Boothby, W. M., An Introduction to Differentiable Manifolds and Riemannian Geometry (Academic Press, New York, 2nd ed., 1986).Google Scholar
5.Campbell, S. and Holmes, P., Bifurcation from O(2) symmetric heteroclinic cycles with three interacting modes, Nonlinearity 4 (1991), 697–726.CrossRefGoogle Scholar
6.Collet, P. and Eckmann, J.-P., The existence of dendritic fronts, Comm. Math. Phys. 107 (1986), 39–92.CrossRefGoogle Scholar
7.Collet, P. and Eckmann, J.-P., Instabilities and Fronts in Extended Systems (Princeton University Press, 1990).CrossRefGoogle Scholar
8.Craig, W., Sulem, C. and Sulem, P. L., Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity 5 (1992), 497–522.CrossRefGoogle Scholar
9.Deissler, R. J. and Brand, H. R., The effect of nonlinear gradient terms on localized states near a weakly inverted bifurcation, Phys. Lett. 146 (1990), 252–255.CrossRefGoogle Scholar
10.Doelman, A., On the nonlinear evolution of patterns (modulation equations and their solutions) (Ph.D. Thesis, University of Utrecht, the Netherlands, 1990).Google Scholar
11.Doelman, A., Slow time-periodic solutions of the Ginzburg-Landau equation, Phys. D 40 (1989), 156–172.CrossRefGoogle Scholar
12.Doelman, A., Traveling waves in the complex Ginzburg-Landau equation, J. Nonlinear Sci. 3 (1993), 225–266.CrossRefGoogle Scholar
13.Doelman, A. and Eckhaus, W., Periodic and quasiperiodic solutions of degenerate modulation equations, Phys. D 53 (1991), 249–266.CrossRefGoogle Scholar
14.Duan, J., Holmes, P. and Titi, E. S., Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity 5 (1992), 1303–1314.CrossRefGoogle Scholar
15.Duan, J. and Holmes, P., On the Cauchy problem for a generalized Ginzburg-Landau equation, in Nonlinear Anal.: Theory, Methods and Applications 22 (1994), 1033–1040.CrossRefGoogle Scholar
16.Eckmann, J.-P. and Galay, Th., Front solutions for the Ginzburg-Landau equation, Comm. Math. Phys. 152 (1993), 221–248.CrossRefGoogle Scholar
17.Eckmann, J.-P. and Wayne, C. E., Propagating fronts and the center manifold theorem, Comm. Math. Phys. 136 (1991), 285–307.CrossRefGoogle Scholar
18.Fauve, S. and Thual, O., Solitary waves generated by subcritical instabilities in dissipative systems, Phys. Rev. Lett. 64 (1990), 282–284.CrossRefGoogle ScholarPubMed
19.Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, 1983 (3rd printing, 1990)).CrossRefGoogle Scholar
20.Guckenheimer, J., Myers, M. R., Wicklin, F. J. and Worfolk, P. A., dstool: Dynamical System Toolkit with Interactive Graphic Interface, User's Manual (Center for Applied Mathematics, Cornell University, USA, 1991).Google Scholar
21.Hakim, V., Jakobsen, P. and Pomeau, Y., Fronts vs. solitary waves in nonequilibrium systems, Europhys. Lett. 11 (1990), 19–24.CrossRefGoogle Scholar
22.Hasimoto, H. and Ono, H., Nonlinear modulation of gravity waves. J Phys. Soc. Japan 33 (1972), 805–811.CrossRefGoogle Scholar
23.Hocking, L. M. and Stewartson, K., On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance. Proc. R. Soc. London A 326 (1972), 289–313.Google Scholar
24.Holmes, P., Spatial structure of time-periodic solutions of the Ginzburg-Landau equation, Phys.D 23 (1986), 84–90.CrossRefGoogle Scholar
25.Jones, C. K. R. T., Kopell, N. and Langer, R., Construction of the Fitzhugh-Nagumo pulse using differential forms, in Patterns and dynamics in reactive media, Aris, G., Aronson, D. and Swinney, H. (eds.), IMA volumes in mathematics and its applications 37, Springer-Verlag, Berlin, 1991, 101–116.CrossRefGoogle Scholar
26.Jones, C. K. R. T., Kapitula, T. and Powell, J., Nearly real fronts in a Ginzburg-Landau equation, Proc. Roy. Soc. Edinburgh A 116 (1990), 193–206.CrossRefGoogle Scholar
27.Landman, M. J., Solutions of the Ginzburg-Landau equation of interest in shear flow transition. Stud. Appl. Math. 76 (1987), 187–237.CrossRefGoogle Scholar
28.Malomed, B. A. and Nepomnyashchy, A. A., Kinks and solitons in the generalized Ginzburg-Landau equations, Phys. Rev. A 42 (1990), 6009–6014.CrossRefGoogle Scholar
29.Van Saarloos, W. and Hohenberg, P. C., Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation, Phys. Rev. Lett. 64 (1990), 749–752.CrossRefGoogle Scholar
30.Van Saarloos, W. and Hohenberg, P. C., Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations, Phys. D 56 (1992), 303–367.CrossRefGoogle Scholar
31.Schopf, W. and Kramer, L., Small amplitude periodic and chaotic solutions of the complex Ginzburg-Landau equation for a subcritical bifurcation, Phys. Rev. Lett. 66 (1991), 2316–2319.CrossRefGoogle ScholarPubMed
32.Schopf, W., and Zimmermann, W., Multicritical behavior in binary fluid convection, Europhys. Lett. 8 (1989), 41–46.CrossRefGoogle Scholar
33.Schopf, W. and Zimmermann, W., Convection in binary fluids: amplitude equations, codimension-2 bifurcation and thermal fluctuations, Phys. Rev. E 47 (1993), 1739–1764.CrossRefGoogle ScholarPubMed
34.Sirovich, L. and Newton, P. K., Periodic solutions of the Ginzburg-Landau equation, Phys. D 21 (1986), 115–125.CrossRefGoogle Scholar
35.Smoller, J., Shock Waves and Reaction-Diffusion Equations (Springer-Verlag, Berlin, 1983).CrossRefGoogle Scholar
36.Thual, O. and Fauve, S., Localized structures generated by subcritical instabilities, J. Phys. France 49 (1988), 1829–1833.CrossRefGoogle Scholar
37.Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, Berlin, 1990).CrossRefGoogle Scholar
38.Weinstein, M. I., Modulational stability of ground states of nonlinear Schrodinger equations, SIAM J. Math. Anal. 16 (1985), 472–490.CrossRefGoogle Scholar