One-dimensional Cubic-quintic Complex Ginzburg-Landau Equation Research Articles

Cascade replication of soliton solutions in the one-dimensional complex cubic-quintic Ginzburg-Landau equation

In the framework of the one-dimensional cubic-quintic complex Ginzburg-Landau equation, we demonstrate two methods to achieve cascade replication of dissipative solitons from a single one by using external forces whose positions and magnitudes are being precisely controlled. The first method combines high amplitude positive currents and negative currents to increase the efficiency of the process significantly and to make the process repeatable. By using two repulsive forces whose locations are time dependent, the second method enables one to obtain multiple dissipative solitons in only one cascade replication process.

Induced waveform transitions of dissipative solitons.

The effect of an externally applied force upon the dynamics of dissipative solitons is analyzed in the framework of the one-dimensional cubic-quintic complex Ginzburg-Landau equation supplemented by a potential term with an explicit coordinate dependence. The potential accounts for the external force manipulations and consists of three symmetrically arranged potential wells whose depth varies along the longitudinal coordinate. It is found out that under an influence of such potential a transition between different soliton waveforms coexisting under the same physical conditions can be achieved. A low-dimensional phase-space analysis is applied in order to demonstrate that by only changing the potential profile, transitions between different soliton waveforms can be performed in a controllable way. In particular, it is shown that by means of a selected potential, stationary dissipative soliton can be transformed into another stationary soliton as well as into periodic, quasi-periodic, and chaotic spatiotemporal dissipative structures.

Cascade replication of dissipative solitons.

We report a new effect of a cascade replication of dissipative solitons from a single one. It is discussed in the framework of a common model based on the one-dimensional cubic-quintic complex Ginzburg-Landau equation in which an additional linear term is introduced to account the perturbation from a particular potential of externally applied force. The effect is demonstrated on the light beams propagating through a planar waveguide. The waveguide consists of a nonlinear layer able to guide dissipative solitons and a magneto-optic substrate. In the waveguide an externally applied force is considered to be an inhomogeneous magnetic field which is induced by modulated electric currents flowing along a set of conducting wires adjusted on the top of the waveguide.

Control of dissipative solitons in a magneto-optic planar waveguide.

We propose a mechanism to control propagation of a group of stable dissipative solitons in a nonlinear magneto-optic planar waveguide. The control is realized by means of a spatially inhomogeneous external magnetic field, which is induced by a set of direct conducting wires placed on the top of the guiding layer. The wires are extended in the direction of soliton propagation, and carry electric currents with particular piecewise constant profiles. In order to describe the soliton evolution the one-dimensional cubic-quintic complex Ginzburg-Landau equation has been adapted by tailoring an additional linear term, which is responsible for the magneto-optic effect.

Nonlinear diffusion control of defect turbulence in cubic-quintic complex Ginzburg-Landau equation

The control of spatiotemporal chaos generated by traveling holes in spatially extended oscillatory media near a subcritical Hopf bifurcation is investigated. Analytical and numerical approaches are proposed for the purpose of this study. This work is done by using the nonlinear diffusion parameter control. We show that the unstable traveling hole in the one-dimensional cubic-quintic complex Ginzburg-Landau equation can be effectively stabilized in the chaotic regime.

Existence Conditions for Stable Stationary Solitons of the Cubic-Quintic Complex Ginzburg-Landau Equation with a Viscosity Term

We report the existence of a new family of stable stationary solitons in the one-dimensional cubicquintic complex Ginzburg-Landau equation with the viscosity term. By applying the paraxial ray approximation, we obtain the relation between the width and the peak amplitude of the stationary soliton in terms of the model parameters. We find the bistable solitons in the presence of the small viscosity term.We verify the analytical results by direct numerical simulations and show the stability of the stationary solitons. We conclude that the existence condition obtained by the paraxial method may serve as physically more reliable initial condition for stable stationary soliton propagation in the optical system modeled by the one-dimensional cubic-quintic complex Ginzburg-Landau equation with the viscosity term, of which analytical solution is not obtainable with all nonzero parameters.

A collective variable approach for optical solitons in the cubic–quintic complex Ginzburg–Landau equation with third-order dispersion

Considering the theory of electromagnetic, especially from the Maxwell equations, a basic equation modeling the propagation of ultrashort optical solitons in optical fibers is derived, namely a cubic–quintic complex Ginzburg–Landau equation (CQGLE) with third-order dispersion (TOD). Considering this one-dimensional CQGLE, we derive the equations of motion of pulse parameters called collective variables (CVs), of a pulse propagating in dispersion-managed (DM) fiber optic-links. Equations obtained are investigated numerically in order to view the evolution of pulse parameters along the propagation distance. A fully numerical simulation of the CQGLE finally tests the results of the CV theory. It appears chaotic pulses, attenuate pulses and stable pulses under some parameter values.

Effect of the Random Field on the Dynamics of Pulsating, Erupting, and Creeping Solitons in the Cubic-Quintic Complex Ginzburg-Landau Equation

It is shown that the dynamics of pulsating, erupting, and creeping (PEC) solitons obtained from the one-dimensional cubic-quintic complex Ginzburg-Landau equation can be drastically modified in the presence of a random background field. It is found that, when the random field is applied to a pulselike initial profile, multiple soliton trains are formed for the parameters of the pulsating and erupting solitons. Furthermore, as the strength of the gain term increases, the multiple pulsating or erupting solitons transform into fixed-shape stable solitons. This may be important for a practical use such as to generate stable femtosecond pulses. For the case of creeping soliton parameters, the presence of the random field does not generate multiple solitons, however, it induces a rapidly twisting or traveling soliton with a fixed-shape, of which stability can be also controlled by the gain term. - PACS numbers: 42.65.Tg, 03.40.Kf, 05.70.Ln, 47.20.Ky.

Dynamics of Pulsating, Erupting, and Creeping Solitons in the Cubic- Quintic Complex Ginzburg-Landau Equation under the Modulated Field

It is shown that the dynamics of the pulsating, erupting, and creeping (PEC) solitons in the one-dimensional cubic-quintic complex Ginzburg-Landau equation can be drastically modified in the presence of a modulated field. We first perform the linear instability analysis of continuous-wave (CW) and obtain the gain by the modulational instability (MI). It is found that the CW states applied by the weakly modulated field always transform into fronts for the parameters of the PEC solitons. We then show that, when the modulated field is applied to the pulse-like initial profile, multiple solitons are formed for the parameters of the pulsating and erupting solitons. Furthermore, as the strength of the gain term increases, the multiple pulsating or erupting solitons transform into fixedshape stable solitons. This may be important for a practical use such as to generate multiple stable femtosecond pulses. For the case of creeping soliton parameters, the presence of a modulated field does not generate multiple solitons, however, the initial profile transforms into an irregularly pulsating soliton or evolves into a fixed-shape soliton as the strength of the gain term is increased. - PACS numbers: 42.65.Tg, 03.40.Kf, 05.70.Ln, 47.20.Ky

Effect of Nonlinear Gradient Terms on the Dynamics of Solitons in the Cubic-Quintic Complex Ginzburg-Landau Equation

The modulational instability of the one-dimensional cubic-quintic complex Ginzburg-Landau equation with the nonlinear gradient terms is investigated. The presence of the nonlinear gradient terms modifies the modulational instability gain spectrum. We numerically investigate the dynamics of modulational instability in the presence of the nonlinear gradient terms. It is found that they introduce more interactions (both elastic and inelastic) dynamics to the solitons generated by the modulational instability. The signs of the gradient terms determine the propagation direction of the soliton. - PACS numbers: 42.65.Tg, 42.81Dp, 42.65.Sf.

Modulation Instability Of Optical Waves In The Cubic-Quintic Complex Ginzburg-Landau Equation With Fourth-Order Dispersion And Gain Terms

The modulation instability of the one-dimensional cubic-quintic complex Ginzburg-Landau equation with fouth-order dispersion and gain terms, a. k. a., the quintic complex Swift-Hohenberg equation, is investgated. The effects of the fourth-order terms to the modulational instability is studied. We numerically investigate the dynamics of the modulational instability in the presence of the fourthorder dispersion and gain terms. - PACS numbers: 42.65.Tg, 42.81DP, 42.65Sf