Abstract
We present a detailed study of exploding solitons of the complex cubic-quintic Ginzburg-Landau equation. We show that exploding solitons occur in a vast regions of the parameter space. These are related to the areas where eigenvalues in the linear stability analysis for the ground-state stationary solitons have positive real parts. We also show that this behavior is universal, and that it occurs when we use models with parameter management or add new terms (such as third-order dispersion). The stationary soliton appears to be unstable in these regions and it explodes intermittently, but it attracts the chaotic localized structures around it, thus acting as a ‘strange attractor’.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H. Haus, Theory of mode locking with a fast saturable absorber, J. Appl. Phys., 46, 3049 (1975).
W. van Saarlos and P. C. Hohenberg, Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation, Phys. Rev. Lett., 64, 749 (1990).
R. J. Deissler and H. Brand, Periodic, quasiperiodic and chaotic localized solitons of the quintic complex Ginzburg-Landau equation, Phys. Rev. Lett., 72, 478 (1994).
P. Kolodner, Drift, shape, and intrinsic destabilization of pulses of traveling-wave convection, Phys. Rev. A 44, 6448 (1991).
M. Dennin, G. Ahlers and D. S. Cannell, Chaotic Localized States near the Onset of Electroconvection, Phys. Rev. Lett., 77, 2475 (1996).
K. G. Müller, Structures at the electrodes of gas discharges, Phys. Rev. A, 37, 4836 (1988).
Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, Berlin, 1984).
J. M. Soto-Crespo, N. Akhmediev and A. Ankiewicz, Pulsating, creeping and erupting solitons in dissipative systems, Phys. Rev. Lett. 85, 2937 (2000).
N. Akhmediev, J. M. Soto-Crespo and G. Town, Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: CGLE approach, Phys. Rev. E, 63 056602 (2001).
Steven T. Cundiff, J.M. Soto-Crespo and N. Akhmediev, Experimental Evidence for Soliton Explosions, Phys. Rev. Lett., 88, 073903 (2002).
J. M. Soto-Crespo, N. Akhmediev and V. V. Afanasjev, Stability of the pulse like solutions of the quintic complex Ginzburg-Landau equation, JOSA B, 13, No 7, 1439–1449, (1996).
N. Akhmediev and J. M. Soto-Crespo, Exploding solitons and Shiľnikov’s Theorem, Phys. Lett. A, 317, 287–292 (2003).
Kapitula T. Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrödinger equation, Physica D 116, 95–120 (1998).
Kapitula T. and Sandstede B. Stability of bright solitary wave solutions to perturbed nonlinear Schrödinger equations, Physica D 124, 58–103 (1998)
N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and beams, Chapman and Hall, London, 1997.
J. M. Soto-Crespo, N. Akhmediev and K. Chiang, Simultaneous existence of a multiplicity of stable and unstable solitons in dissipative systems, Phys. Lett. A., 291, 115–123 (2001)
L. P. Shiľnikov, A case of existence of a countable number of periodic motions, Sov. Math. Doklady, 6, 163–166 (1965)
L. P. Shiľnikov, A contribution to the problem of the structure of an extended neighborhood of rough equilibrium state of saddle — focus type, Math. USSR — Sbornik, 10, 91–102 (1970).
C. P. Silva, Shiľnikov’s theorem — a tutorial, IEEE Transactions on circuits and systems, I: Fundamental theory and applications, 40, No 10, 675–682 (1993).
P. Glendinning, Stability, instability and chaos, Cambridge University Press, 1999, p.324.
P. Bergé Y. Pomeau, C. Vidal, Order within chaos, towards a deterministic approach to turbulence, John Wiley & Sons, NY, 1984.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Kluwer Academic Publishers
About this paper
Cite this paper
Akhmediev, N., Soto-Crespo, J.M., Ankiewicz, A. (2004). Solitons as Strange Attractors. In: Abdullaev, F.K., Konotop, V.V. (eds) Nonlinear Waves: Classical and Quantum Aspects. NATO Science Series II: Mathematics, Physics and Chemistry, vol 153. Springer, Dordrecht. https://doi.org/10.1007/1-4020-2190-9_4
Download citation
DOI: https://doi.org/10.1007/1-4020-2190-9_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-2188-6
Online ISBN: 978-1-4020-2190-9
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)