Title

Snake Solitons In The Cubic-Quintic Ginzburg-Landau Equation

Keywords

Dissipative solitons; Ginzburg-Landau equation; Nonlinear PDE; Nonlinear waves; Snaking solitons

Abstract

Comprehensive numerical simulations of pulse solutions of the cubic-quintic Ginzburg-Landau equation (CGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal various intriguing and entirely novel classes of solutions. In particular, there are five new classes of pulse or solitary waves solutions, viz. pulsating, creeping, snake, erupting, and chaotic solitons. In contrast to the regular solitary waves investigated in numerous integrable and non-integrable systems over the last three decades, these dissipative solitons [C.I. Christov, M.G. Velarde, Dissipative solitons, Physica D 86 (1995) 323] are not stationary in time. Rather, they are spatially confined pulse-type structures whose envelopes exhibit complicated temporal dynamics. The numerical simulations also reveal very interesting bifurcations sequences of these pulses as the parameters of the CGLE are varied. In this paper, we address the issues of central interest in the area, i.e., the conditions for the occurrence of the five categories of dissipative solitons, as well the dependence of both their shape and their stability on the various parameters of the CGLE, viz. the nonlinearity, dispersion, linear and nonlinear gain, loss and spectral filtering parameters. Our predictions on the variation of the soliton amplitudes, widths and periods with the CGLE parameters agree with simulation results. First, we develop and discuss a variational formalism within which to explore the various classes of dissipative solitons. Given the complex dynamics of the various dissipative solutions, this formulation is, of necessity, significantly generalized overall earlier approaches in several crucial ways. Firstly, the starting formulation for the Lagrangian is recent and not well explored. Also, the trial functions have been generalized considerably over conventional ones to keep the shape relatively simple (and the trial function integrable!) while allowing arbitrary temporal variation of the amplitude, width, position, speed and phase of the pulses. In addition, the resulting Euler-Lagrange equations are treated in a completely novel way. Rather than consider the stable fixed points which correspond to the well-known stationary solitons or plain pulses, we use dynamical systems theory to focus on more complex attractors, viz. periodic, quasiperiodic, and chaotic ones. Periodic evolution of the trial function parameters on stable periodic attractors yield solitons whose amplitudes and widths are non-stationary or time dependent. In particular, pulsating and snake dissipative solitons may be treated in this manner. Detailed results are presented here for the pulsating solitary waves; their regimes of occurrence, bifurcations, and the parameter dependences of the amplitudes, widths, and periods agree with simulation results [1,26]. © 2009 IMACS.

Publication Date

9-1-2009

Publication Title

Mathematics and Computers in Simulation

Volume

80

Issue

1

Number of Pages

73-82

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.matcom.2009.06.017

Socpus ID

70349163597 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/70349163597

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