Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation. (English) Zbl 0759.35045

Summary: The dynamical behavior of a large one-dimensional system obeying the cubic complex Ginzburg-Landau equation is studied numerically as a function of parameters near a supercritical bifurcation. Two types of chaotic behavior can be distinguished beyond the Benjamin-Feir instability, a phase turbulence regime with a conserved phase winding number and no phase dislocations (space-time defects), and a defect regime with a nonzero density of defects. The transition between the two can either be continuous or discontinuous (hysteretic), depending on parameters. The spatial decay of the phase correlation function is inferred to be exponential in both regimes, with a sharp decrease of the correlation length upon entering the defect phase. The temporal decay of correlations is exponential in the defect regime.


35Q58 Other completely integrable PDE (MSC2000)
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
76F20 Dynamical systems approach to turbulence
Full Text: DOI Link


[1] Hao, B.-L., Chaos (1984), World Scientific: World Scientific Singapore · Zbl 0559.58012
[2] Hohenberg, P. C.; Shraiman, B. I., Physica D, 37, 109 (1989)
[4] Bretherton, C. S.; Spiegel, E. A., Phys. Lett., 96A, 152 (1983)
[5] Schöpf, W.; Kramer, L., Phys. Rev. Lett., 66, 2316 (1991)
[6] Stuart, J. T.; DiPrima, R. C., Proc. Roy. Soc. London, A362, 27 (1972)
[7] Janiaud, B.; Pumir, A.; Bensimon, D.; Croquette, V.; Richter, H.; Kramer, L., Physica D, 55, 269 (1992)
[8] Sivashinsky, G. I., Acta Astronaut., 4, 1177 (1977)
[9] Sakaguchi, H., Prog. Theor. Phys., 84, 792 (1990)
[11] Pomeau, Y.; Pumir, A.; Pelce, P., J. Stat. Phys., 37, 39 (1984)
[12] Pumir, A., J. Phys. (Paris), 46, 511 (1985)
[13] Zaleski, S., Physica D, 34, 427 (1989)
[14] Rice, T. M., Phys. Rev., 140, 1889 (1965)
[15] Bunimovich, L. A.; Sinai, Y. G., Nonlinearity, 1, 491 (1988)
[16] Pumir, A.; Shraiman, B. I.; van Saarloos, W.; Hohenberg, P. C.; Chaté, H.; Holen, M., (Andereck, C. D.; Hayot, F., Ordered and Turbulent Patterns in Taylor-Couette Flow (1992), Plenum: Plenum New York), See · Zbl 0759.35045
[17] Bartucelli, M.; Constantin, P.; Doering, C. R.; Gibbon, J. D.; Gisselfält, M., Physica D, 44, 421 (1990)
[18] Kolodner, P.; Glazier, J. A.; Williams, H., Phys. Rev. Lett., 65, 1579 (1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.