zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Modulated amplitude waves and defect formation in the one-dimensional complex Ginzburg-Landau equation. (English) Zbl 0996.35071
Summary: The transition from phase chaos to defect chaos in the complex Ginzburg-Landau equation (CGLE) is related to saddle-node bifurcations of modulated amplitude waves (MAWs). First, the spatial period $P$ of MAWs is shown to be limited by a maximum $P_{\text{SN}}$ which depends on the CGLE coefficients; MAW-like structures with period larger than $P_{\text{SN}}$ evolve to defects. Second, slowly evolving near-MAWs with average phase gradients $\nu\approx 0$ and various periods occur naturally in phase chaotic states of the CGLE. As a measure for these periods, we study the distributions of spacings $p$ between neighbouring peaks of the phase gradient. A systematic comparison of $p$ and $P_{\text{SN}}$ as a function of coefficients of the CGLE shows that defects are generated at locations where $p$ becomes larger than P$_{SN}$. In other words, MAWs with period $P_{\text{SN}}$ represent “critical nuclei” for the formation of defects in phase chaos and may trigger the transition to defect chaos. Since rare events where $p$ becomes sufficiently large to lead to defect formation may only occur after a long transient, the coefficients where the transition to defect chaos seems to occur depend on system size and integration time. We conjecture that in the regime where the maximum period $P_{\text{SN}}$ has diverged, phase chaos persists in the thermodynamic limit.

MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
76F20Dynamical systems approach to turbulence
35B32Bifurcation (PDE)
WorldCat.org
Full Text: DOI arXiv
References:
[1] Brusch, L.; Zimmermann, M. G.; Van Hecke, M.; Bär, M.; Torcini, A.: Phys. rev. Lett.. 85, 86 (2000)
[2] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer, Berlin, 1984. · Zbl 0558.76051
[3] Cross, M. C.; Hohenberg, P. C.: Rev. mod. Phys.. 65, 851 (1993)
[4] T. Bohr, M.H. Jensen, G. Paladin, A. Vulpiani, Dynamical Systems Approach to Turbulence, Cambridge University Press, Cambridge, 1998. · Zbl 0933.76002
[5] I.S. Aranson, L. Kramer, The world of complex Ginzburg--Landau equation, Rev. Mod. Phys., in press. · Zbl 1205.35299
[6] Vince, J. M.; Dubois, M.: Physica D. 102, 93 (1997)
[7] Vallette, D. P.; Jacobs, G.; Gollub, J. P.: Phys. rev. E. 55, 4274 (1997)
[8] Akamatsu, S.; Faivre, G.: Phys. rev. E. 58, 3302 (1998)
[9] Lücke, M.; Barten, W.; Kamps, M.: Physica D. 61, 183 (1992)
[10] Liu, Y.; Ecke, R. E.: Phys. rev. Lett.. 78, 4391 (1997)
[11] Zhou, L. Q.; Ouyang, Q.: Phys. rev. Lett.. 85, 1650 (2000)
[12] Bot, P.; Mutabazi, I.: Eur. phys. J.. 13, 141 (2000)
[13] Burguete, J.; Chaté, H.; Daviaud, F.; Mukolobwiez, N.: Phys. rev. Lett.. 82, 3252 (1999)
[14] Wierschem, A.; Linde, H.; Velarde, M. G.: Phys. rev. E. 62, 6522 (2000)
[15] Janiaud, B.; Pumir, A.; Bensimon, D.; Croquette, V.; Richter, H.; Kramer, L.: Physica D. 55, 269 (1992)
[16] A. Pumir, B.I. Shraiman, W. van Saarloos, P.C. Hohenberg, H. Chaté, M. Holen, in: C.D. Andereck, F. Hayot (Eds.), Ordered and Turbulent Patterns in Taylor--Couette Flow, Plenum Press, New York, 1992, p. 173. · Zbl 0759.35045
[17] Shraiman, B. I.; Pumir, A.; Van Saarloos, W.; Hohenberg, P. C.; Chaté, H.; Holen, M.: Physica D. 57, 241 (1992)
[18] Bazhenov, M. V.; Rabinovich, M. I.; Fabrikant, A. L.: Phys. lett. A. 163, 87 (1994)
[19] P.E. Cladis, Palffy-Muhoray (Eds.), Spatio-Temporal Pattern Formation in Nonequilibrium Complex Systems, Addison Wesley, Reading, 1995, p. 33.
[20] Sakaguchi, H.: Prog. theor. Phys.. 84, 792 (1990)
[21] Egolf, D. A.; Greenside, H. S.: Phys. rev. Lett.. 74, 1751 (1995)
[22] Montagne, R.; Hernández-Garcı\acute{}a, E.; Amengual, A.; Miguel, M. San: Phys. rev. E. 55, 151 (1997)
[23] Torcini, A.; Frauenkron, H.; Grassberger, P.: Phys. rev. E. 55, 5073 (1997)
[24] Van Saarloos, W.; Hohenberg, P. C.: Physica D. 69, 209 (1993)
[25] Van Hecke, M.: Phys. rev. Lett.. 80, 1896 (1998)
[26] Giacomelli, G.; Hegger, R.; Politi, A.; Vassalli, M.: Phys. rev. Lett.. 85, 3616 (2000)
[27] Manneville, P.; Chaté, H.: Physica D. 96, 30 (1996)
[28] Nozaki, K.; Bekki, N.: J. phys. Soc. jpn.. 53, 1581 (1984)
[29] Van Hecke, M.; Howard, M.: Phys. rev. Lett.. 86, 2018 (2001)
[30] G. Hager, Quasiperiodische Lösungen der eindimensionalen komplexen Ginzburg--Landau Gleichung, Diploma Thesis, University of Bayreuth, Germany, 1996.
[31] L. Brusch, A. Torcini, M. Bär, Physica D, submitted for publication.
[32] E.J. Doedel, X.J. Wang, T.F. Fairgrieve, AUTO94: Software for continuation and bifurcation problems in ordinary differential equations, Applied Mathematics Report, California Institute of Technology, 1994.
[33] Kness, M.; Tuckerman, L.; Barkley, D.: Phys. rev. A. 46, 5054 (1992)
[34] G. Iooss, D.D. Joseph, Elementary Stability and Bifurcation Theory, Springer, Berlin, 1980 (Chapter 3). · Zbl 0443.34001
[35] Chang, H. -C; Demekhin, E. A.; Kopelevich, D. I.: Physica D. 63, 299 (1993)
[36] M. Abel, H. Chaté, H. Voss, in preparation.
[37] P. Collet, J.-P. Eckmann, Instabilities and Fronts in Extended Systems, Princeton University Press, 1990. · Zbl 0732.35074
[38] Or-Guil, M.; Kevrekidis, I. G.; Bär, M.: Physica D. 135, 154 (2000)