Cubic-quintic Complex Ginzburg-Landau Equation Research Articles

Stability conditions for moving dissipative solitons in one- and multidimensional systems with a linear potential

We analyze stability of moving dissipative solitons in the one-, two, and three-dimensional cubic-quintic complex Ginzburg-Landau equations in the presence of a linear potential (linear refractive index modulation). The expressions of stability conditions and propagation trajectory of solitons are derived by means of a generalized variational approximation. Predictions of the variational analysis are fully confirmed by direct numerical simulations. The results have potential applications to using spatial dissipative solitons in optics as individually addressable and shift registers of the all-optical data processing systems.

Unusual stability of a one-parameter family of dissipative solitons due to spectral filtering and nonlinearity saturation

The stability of a one-parameter family of dissipative solitons seen in the cubic-quintic complex Ginzburg-Landau equation is studied. It is found that an unusually strong stability occurs for solitons controlled by the spectral filtering and nonlinearity saturation simultaneously, consistently with the linear stability analysis and confirmed by large-perturbation numerical simulations. Two universal types of bifurcations in the spectrum structure are demonstrated.

Influence of Dirichlet boundary conditions on dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation

We investigate the influence of Dirichlet boundary conditions on various types of localized solutions of the cubic-quintic complex Ginzburg-Landau equation as it arises as an envelope equation near the weakly inverted onset of traveling waves. We find that various types of nonmoving pulses and holes can accommodate Dirichlet boundary conditions by having, for holes, two halves of a pi hole at each end of the box. Moving pulses of fixed shape as they arise for periodic boundary conditions are replaced by a nonmoving asymmetric pulse, which has half a pi hole at the end of the box in the original moving direction to guarantee that Dirichlet boundary conditions are met. Moving breathing pulses as they arise for periodic boundary conditions propagate toward one end of the container and stop moving while the breathing persists indefinitely. Finally breathing and moving holes are replaced by two (nonbreathing) half pi holes at each end of the container and one hump in the bulk.

Chirped dissipative solitons of the complex cubic-quintic nonlinear Ginzburg-Landau equation

Approximate analytical chirped solitary pulse (chirped dissipative soliton) solutions of the one-dimensional complex cubic-quintic nonlinear Ginzburg-Landau equation are obtained. These solutions are stable and highly accurate under condition of domination of a normal dispersion over a spectral dissipation. The parametric space of the solitons is three-dimensional, that makes theirs to be easily traceable within a whole range of the equation parameters. Scaling properties of the chirped dissipative solitons are highly interesting for applications in the field of high-energy ultrafast laser physics.

Moving breathing pulses in the one-dimensional complex cubic-quintic Ginzburg-Landau equation

We show and characterize numerically moving breathing pulses in the one-dimensional complex cubic-quintic Ginzburg-Landau equation. This class of stable moving breathing pulses has not been described before for this prototype envelope equation as it arises near the weakly hysteretic onset of traveling waves.

Bound states of one-, two-, and three-dimensional solitons in complex Ginzburg–Landau equations with a linear potential

We analyze interactions between moving dissipative solitons in one- and multidimensional cubic-quintic complex Ginzburg-Landau equations with a linear potential and effective viscosity. The interactions between the solitons are analyzed by using balance equations for the energy and momentum. We demonstrate that the separation between two solitons forming a bound state decreases with the increase of the slope of the linear potential.

DISSIPATIVE SOLITONS: PRESENT UNDERSTANDING, APPLICATIONS AND NEW DEVELOPMENTS

Dissipative solitons form a new paradigm for the investigation of phenomena involving stable structures in nonlinear systems far from equilibrium. Basic principles can be applied to a wide range of phenomena in science. Recent results involving solitons and soliton complexes of the complex cubic-quintic Ginzburg–Landau equation are presented.

Effect of an external periodic potential on pairs of dissipative solitons

We study dissipative soliton pair solutions of the complex cubic-quintic Ginzburg-Landau equation with periodic phase modulation term. The external modulation changes the soliton-pair time separation as well as their phase difference, without destroying the existing two-soliton bound state. Transitions between different stable forms of the pair occur in the form of bifurcations. Quite remarkably, two types of bound states may coexist, which leads to hysteresis loops when the modulation depth is varied.

Multi-hump pulses in systems with reflection and phase invariance

We investigate the existence and stability of standing and travelling multi-hump waves in partial differential equations with reflection and phase symmetries. We focus on 2- and 3-pulse solutions that arise near bi-foci and apply our results to the complex cubic-quintic Ginzburg–Landau equation.

Discrete family of dissipative soliton pairs in mode-locked fiber lasers

We numerically investigate the formation of soliton pairs (bound states) in mode-locked fiber ring lasers. In the distributed model (complex cubic-quintic Ginzburg-Landau equation) we observe a discrete family of soliton pairs with equidistantly increasing peak separation. This family was identified by two alternative numerical schemes and the bound state instability was disclosed by a linear stability analysis. Moreover, similar families of unstable bound state solutions have been found in a more realistic lumped laser model with an idealized saturable absorber (instantaneous response). We show that a stabilization of these bound states can be achieved when the finite relaxation time of the saturable absorber is taken into account. The domain of stability can be controlled by varying this relaxation time.

Noise Induces Partial Annihilation of Colliding Dissipative Solitons

Partial annihilation of two counterpropagating dissipative solitons, with only one pulse surviving the collision, has been widely observed in different experimental contexts, over a large range of parameters, from hydrodynamics to chemical reactions. However, a generic picture accounting for partial annihilation is missing. Based on our results for coupled complex cubic-quintic Ginzburg-Landau equations as well as for the FitzHugh-Nagumo equation we conjecture that noise induces partial annihilation of colliding dissipative solitons in many systems.

Dynamics of modulated waves in electrical lines with dissipative elements

By a means of a method based on the reductive perturbation method, we show that the amplitude of waves on the nonlinear electrical transmission lines (NLTLs) is described by the cubic-quintic complex Ginzburg-Landau (CGL) equation. Then, we revisit analytically and numerically the processes of modulational instability (MI). The evolution of dissipative modulated waves through the network is also examined, and we show that solitonlike excitations can be induced by MI. Analytical results, illustrating the nature of MI of plane-wave solution, are also found to be in good agreement with numerical findings.

Dissipative soliton resonances in the anomalous dispersion regime

We have numerically calculated the resonance curve and the region of existence of high-energy dissipative solitons in systems governed by the complex cubic-quintic Ginzburg-Landau equation. The calculations are carried out for negative reactive quintic nonlinearity. This choice allows the resonance curve to be continued into the region of positive net dispersion, thus showing that high-energy pulses can also be generated by lasers operating in the anomalous dispersion regime.

Dissipative ring solitons with vorticity

We study dissipative ring solitons with vorticity in the frame of the (2+1)-dimensional cubic-quintic complex Ginzburg-Landau equation. In dissipative media, radially symmetric ring structures with any vorticity m can be stable in a finite range of parameters. Beyond the region of stability, the solitons lose the radial symmetry but may remain stable, keeping the same value of the topological charge. We have found bifurcations into solitons with n-fold bending symmetry, with n independent on m. Solitons without circular symmetry can also display (m + 1)-fold modulation behaviour. A sequence of bifurcations can transform the ring soliton into a pulsating or chaotic state which keeps the same value of the topological charge as the original ring.

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.

Collisions between counter-rotating solitary vortices in the three-dimensional Ginzburg-Landau equation

We report results of collisions between coaxial vortex solitons with topological charges +/-S in the complex cubic-quintic Ginzburg-Landau equation. With the increase of the collision momentum, merger of the vortices into one or two dipole or quadrupole clusters of fundamental solitons (for S=1 and 2, respectively) is followed by the appearance of pairs of counter-rotating "unfinished vortices," in combination with a soliton cluster or without it. Finally, the collisions become elastic. The clusters generated by the collisions are very robust, while the "unfinished vortices," eventually split into soliton pairs.

Comparison of Lagrangian approach and method of moments for reducing dimensionality of soliton dynamical systems

For equations that cannot be solved exactly, the trial function approach to modelling soliton solutions represents a useful approximate technique. It has to be supplemented with the Lagrangian technique or the method of moments to obtain a finite dimensional dynamical system which can be analyzed more easily than the original partial differential equation. We compare these two approaches. Using the cubic-quintic complex Ginzburg-Landau equation as an example, we show that, for a wide class of plausible trial functions, the same system of equations will be obtained. We also explain where the two methods differ.

Theory of dissipative solitons in complex Ginzburg-Landau systems

Using soliton amplitude and phase ansatzes, a theory is proposed for searching for stationary soliton solutions to the cubic-quintic complex Ginzburg-Landau (CGL) equation. For arbitrary combinations of system parameters, our approach allows the existence of dissipative solitons together with their specific soliton characteristics to be determined, and we demonstrate this explicitly for the case of a pulsed fiber laser system. The regimes of existence of dissipative solitons and their rules of evolution in a complicated five-dimensional parameter space are also analyzed. This work may open other research opportunities in diverse areas of nonlinear dynamics governed by the CGL equation, and may impact significantly on experimental design.

Dissipative soliton resonances

We have found a dissipative soliton resonance which applies to nonlinear dynamical systems governed by the complex cubic-quintic Ginzburg-Landau equation. Specifically, for particular values of the equation parameters, the soliton energy increases indefinitely. These equation parameters can easily be found using approximate methods, and the results agree very well with numerical ones. The phenomenon can be very useful in the design of high-power passively mode-locked lasers.

Influence of Boundary Conditions on Localized Solutions of the Cubic-Quintic Complex Ginzburg-Landau Equation

We investigate the influence of the boundary conditions and the box size on the existence and stability of various types of localized solutions (particles and holes) of the cubic-quintic complex Ginzburg-Landau equation as it arises as a prototype envelope equation near the weakly hysteretic onset of traveling waves. Two types of boundary conditions are considered for one spatial dimension, both of which can be realized experimentally: periodic boundary conditions, which can be achieved for an annulus and Neumann boundary conditions, which correspond to zero flux, for example in hydrodynamics. We find that qualitative differences between the two types of boundary conditions arise in particular for propagating and breathing localized solutions. While an asymmetry in the localized state is always connected to motion for periodic boundary conditions, this no longer applies for Neumann boundary conditions. In the case of Neumann boundary conditions we observe that breathing localized states can no longer exist below a certain box size, which is comparable to the ‘width’of the localized state.