Abstract
In this chapter, we consider a theoretical framework for analyzing the strongly-amplitude modulated numerical pulse solutions recently observed in the complex Ginzburg-Landau Equation, which is a canonical model for dissipative, weakly-nonlinear systems. As such, the chapter also reviews background concepts of relevance to coherent structures in general dissipative systems (i.e. in regimes where such structures are stable and dominate the dynamics). This framework allows a comprehensive analysis of various bifurcations leading to transitions from one type of coherent structure to another as the system parameters are varied. It will also form a basis for future theoretical analysis of the great diversity of numericallyobserved solutions, even including the spatially-coherent structures with temporally quasi-periodic or chaotic envelopes observed in recent simulations.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
REFERENCES
P. G. Drazin and R. S. Johnson, Solitons: an Introduction, (Cambridge, Cambridge, 1989).
J. D. Murray, Mathematical Biology, (Springer-Verlag,Berlin, 1989).
W. van Saarloos and P. C. Hohenberg, { Physica}, D 56, 303 (1992).
N. J. Balmforth, { Ann. Rev. Fluid Mechanics} 27, 335 (1995).
C. Jayaprakash, F. Hayot and R. Pandit, { Phys. Rev. Lett.}, 71, 12 (1993).
C. V. Conrado and T. Bohr, { Phys. Rev. Lett.}, 72, 3522, (1994).
R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, Solitons and Nonlinear Wave Equations, (Academic, London, 1982).
I. S. Aranson and L. Kramer, { Rev. Mod. Phys.}, 74, 99, (2002).
M. C. Cross and P. C. Hohenberg, { Rev. Mod. Phys.}, 65, 851, (1993).
L. Debnath and S. Roy Choudhury (Eds), Nonlinear Instability Analysis, (Computational Mechanics Publishers, Southampton, 1997).
S. Strogatz, Nonlinear Dynamics and Chaos, (Addison-Wesley, Reading, 1994).
N. Akhmediev, J. Soto-Crespo, and G. Town, { Phys. Rev. E} 63, 1 (2001).
R. J. Deissler and H.R. Brand, { Phys. Rev. Lett.} 72, 478 (1994).
S. Roy Choudhury, { Chaos, Solitons and Fractals} 2, 393 (1992).
A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics, (Wiley, New York, 1995).
B. Janiaud, A. Pumir, D. Bensimon V. Croquette, H. Richter, and L. Kramer, { Physica} D 55, 269 (1992).
L. Brusch, A. Torcini, and M. Bar,{ Physica} D 174, 152 (2003).
M. Buttiker and H. Thomas, { Phs. Rev.} A24, 2635 (1981).
A. V. Porubov and M.G. Velarde,{ J. Math. Phys.} 40, 884 (1999).
D. E. Bar and A. A. Nepomnyashchy, { Physica} D 86, 586 (1995);
H. H. Chang, E. A. Demekhin and E. Kalaidin, { SIAM J. Appl. Math.} 58, 1246 (1998).
C. Elphick, E. Meron and E.A. Spiegel, { SIAM J. Appl. Math.} 50, 490 (1990).
M. Or-Guil, I. G. Kevrekidis and M. Bar, { Physica} D135, 154 (2000).
Author information
Authors and Affiliations
Editor information
Rights and permissions
About this chapter
Cite this chapter
Choudhury, S. Bifurcations and Strongly Amplitude-Modulated Pulses of the Complex Ginzburg-Landau Equation. In: Akhmediev, N., Ankiewicz, A. (eds) Dissipative Solitons. Lecture Notes in Physics, vol 661. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10928028_17
Download citation
DOI: https://doi.org/10.1007/10928028_17
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23373-2
Online ISBN: 978-3-540-31528-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)