Quintic Complex Ginzburg-Landau Equation Research Articles

Dissipative solitons: The finite bandwidth of gain as a viscous friction

We consider the effect on the motion of a dissipative soliton of the diffusion term in the quintic complex Ginzburg-Landau (CGL) equation, which accounts for the finite bandwidth of the gain when this model describes light pulse evolution in a fiber laser. We show analytically that, if the velocity is small enough, this effect can be modeled by a viscous friction force acting on the soliton. Numerical resolution of the CGL equation shows that this analytical approximation is valid with good accuracy in the case of anomalous dispersion.

Non-unique results of collisions of quasi-one-dimensional dissipative solitons.

We investigate collisions of quasi-one-dimensional dissipative solitons (DSs) for a large class of initial conditions, which are not temporally asymptotic quasi-one-dimensional DSs. For the case of sufficiently small approach velocity and sufficiently large values of the dissipative cross-coupling between the counter-propagating DSs, we find non-unique results for the outcome of collisions. We demonstrate that these non-unique results are intrinsically related to a modulation instability along the crest of the quasi-one-dimensional objects. As a model, we use coupled cubic-quintic complex Ginzburg-Landau equations. Among the final results found are stationary and oscillatory compound states as well as more complex assemblies consisting of quasi-one-dimensional and localized states. We analyse to what extent the final results can be described by the solutions of one cubic-quintic complex Ginzburg-Landau equation with effective parameters.

Spatiotemporal lattice Boltzmann model for the three-dimensional cubic–quintic complex Ginzburg–Landau equation

A spatiotemporal lattice Boltzmann model for solving the three-dimensional cubic–quintic complex Ginzburg–Landau equation (CQCGLE) is proposed. Different from the classic lattice Boltzmann models, this lattice Boltzmann model is based on uniformly distributed lattice points in a three-dimensional spatiotemporal space, and the evolution of the model is about a spatial axis rather than time. The algorithm possesses advantages similar to the lattice Boltzmann method in that it is easily adapted to complex Ginzburg–Landau equations. Examples show that the model reproduces the phenomena in the CQCGLE accurately.

Bifurcation to traveling waves in the cubic–quintic complex Ginzburg–Landau equation

We consider the one-dimensional complex Ginzburg–Landau equation which is a generic modulation equation describing the nonlinear evolution of patterns in fluid dynamics. The existence of a Hopf bifurcation from the basic solution was proved by Park [Bifurcation and stability of the generalized complex Ginzburg–Landau equation, Pure Appl. Anal. 7(5) (2008) 1237–1253]. We prove in this paper that the solution bifurcates to traveling waves which have constant amplitudes. We also prove that there exist kink-profile traveling waves which have variable amplitudes. The structure of the traveling waves is examined and it is proved by means of the center manifold reduction method and some perturbation arguments, that the variable amplitude traveling waves are quasi-periodic and they connect two constant amplitude traveling waves.

Modulation instability of solutions to the complex Ginzburg–Landau equation

The modulation instability of continuous-wave (CW) solutions of the complex Ginzburg–Landau equation (CGLE), with arbitrary intensity-dependent nonlinearity, is studied. The variational approach and standard linear stability analysis are used to investigate the stability of CW and to obtain the criteria for modulation stability in the general form. Analytical stability criteria are established, enabling the construction of charts of stable fixed points of the cubic–quintic CGLE. We show that the evolution of modulationally stable and unstable CWs depends on the CGLE parameters. The analytical predictions for plane wave stability are confirmed by exhaustive numerical simulations.

An accurate Fourier splitting scheme for solving the cubic quintic complex Ginzburg–Landau equation

In this paper, we present a splitting scheme for the pseudo-spectral numerical method namely the Split-Step Fourier method (SSFM), in our approach we expand the exponential term in a manner that a succession of linear and nonlinear terms are distributed uniformly along one step size, the splitting will be performed symmetrically, this new scheme will be tested on one of the most used nonlinear partial deferential equation in optics, namely the cubic quintic complex Ginzburg–Landau (CQCGL) equation, in this work we demonstrate that the accuracy of the Split Step Fourier method scheme can be improved by expanding and distributing it in small parts within one step.

Symmetry breaking term effects on explosive localized solitons

We study the influence of an analog of self–steepening (SST), which is a term breaking the T →−T symmetry, on explosive localized solutions for the cubic–quintic complex Ginzburg–Landau equation in the anomalous dispersion regime. We find that while this explosive behavior occurs for a wide range of the parameter s, characterizing SST, the mean distance between explosions diverges close to a critical value sc. After this value the explosive solution becomes a fixed shape soliton that moves at constant speed. The transition between explosive and regular behavior is characterized by a transcritical bifurcation controlled by the SST parameter. We also proposed a mechanism which explains and predicts the mean distance between explosions as a function of s. We are glad to dedicate this article to Professor Helmut R. Brand on occasion of his 60th birthday.

The Kibble–Zurek mechanism in a subcritical bifurcation

We present a study of the freezing dynamics of topological defects in a subcritical system by testing the Kibble–Zurek (KZ) mechanism while crossing a tri-stable region in a one-dimensional quintic complex Ginzburg–Landau equation. The critical exponents of the KZ mechanism and the horizon (KZ-scaling regime) are predicted from the quasistatic study, and are in full accordance with the quenched study. The correlation length, in the KZ freezing regime, is corroborated from the number of topological defects and from the spatial correlation function of the order parameter. Furthermore, we characterize the dynamics to differentiate three out-of-equilibrium regimes: the adiabatic, the impulse and the free relaxation. We show that the impulse regime shares a common temporal domain with a fast exponential increase of the order parameter.

Dissipative Solitons in a Generalized Coupled Cubic–Quintic Ginzburg–Landau Equations

Three families of exact analytical solutions of a generalized nonlinear coupled cubic–quintic complex Ginzburg–Landau equations are obtained. These families of solitary waves which describe the evolution of progressive bright–bright, front–front, dark–dark, and other families of solitary waves are investigated. Using total energy of pulse modes, a loss–gain analysis due to nonlinearity in comparison to a linear gain is done. The stability of the solitary waves is examined using linear stability analysis and also a complete numerical stability of these families of solutions is provided. The results reveal that most of the solitary waves obtained here can propagate in a stable way under slight perturbation of white noise and the disturbance of parameters of the system.

Nonlinear structures of traveling waves in the cubic–quintic complex Ginzburg–Landau equation on a finite domain

We investigate the behavior of traveling waves in a spatial domain with the homogeneous boundary conditions by using the one-dimensional cubic–quintic complex Ginzburg–Landau equation. We focus our work on the absolute and convective instabilities and determine the dynamical regimes that are observed. As a consequence in the convectively unstable regime, the waves ultimately decay at any fixed position. Only when the threshold for the absolute instability is exceeded, the wave patterns may be maintained against the dissipation at the boundary. Consequently, coherent structures may be observed in the last case. We build a new state of phase diagram in the parameter plane spanned by the criticality parameter and the quintic nonlinear coefficient.

Exact solutions in nonlinearly coupled cubic–quintic complex Ginzburg–Landau equations

Abstract Exact analytical solutions for pulse propagation in a nonlinear coupled cubic–quintic complex Ginzburg–Landau equations are obtained. Three families of solitary waves which describe the evolutions of progressive bright–bright, front–front, dark–dark and other families of solitary waves are investigated. These exact solutions are analyzed both for competition of loss or gain due to nonlinearity and linearity of the system. The stability of the solitary waves is examined using analytical and numerical methods. The results reveal that the solitary waves obtained here can propagate in a stable way under slight perturbation of white noise and the disturbance of parameters of the system.

Exact front, soliton and hole solutions for a modified complex Ginzburg–Landau equation from the harmonic balance method

A novel approach of using harmonic balance (HB) method is presented to find front, soliton and hole solutions of a modified complex Ginzburg–Landau equation. Three families of exact solutions are obtained, one of which contains two parameters while the others one parameter. The HB method is an efficient technique in finding limit cycles of dynamical systems. In this paper, the method is extended to obtain homoclinic/heteroclinic orbits and then coherent structures. It provides a systematic approach as various methods may be needed to obtain these families of solutions. As limit cycles with arbitrary value of bifurcation parameter can be found through parametric continuation, this approach can be extended further to find analytic solution of complex quintic Ginzburg–Landau equation in terms of Fourier series.

Convection-induced stabilization of optical dissipative solitons

In spatially extended convective systems, the reflection symmetry breaking induced by drift effects leads to a striking nonlinear effect that drastically affects the formation and stability of dissipative solitons in optical parametric oscillators. The phenomenon of nonlinear-induced convection dynamics is revealed using a model of the complex quintic Ginzburg-Landau equation with nonlinear gradient terms in it. Mechanisms leading to stabilization of dissipative solitons by convection are singled out. The predictions are in very good agreement with numerical solutions found from the governing equations of the optical parametric oscillators.

Stability of traveling pulses of cubic–quintic complex Ginzburg–Landau equation including intrapulse Raman scattering

The complex cubic–quintic Ginzburg–Landau equation (CGLE) admits a special type of solutions called eruption solitons. Recently, the eruptions were shown to diminish or even disappear if a term of intrapulse Raman scattering (IRS) is added, in which case, self-similar traveling pulses exist. We perform a linear stability analysis of these pulses that shows that the unstable double eigenvalues of the erupting solutions split up under the effect of IRS and, following a different trajectory, they move on to the stable half-plane. The eigenfunctions characteristics explain some eruptions features. Nevertheless, for some CGLE parameters, the IRS cannot cancel the eruptions, since pulses do not propagate for the required IRS strength.

Interactions between optical bullets with different velocities in the three-dimensional cubic—quintic complex Ginzburg–Landau equation

By using the three-dimensional complex Ginzburg–Landau equation with cubic-quintic nonlinearity, this paper numerically investigates the interactions between optical bullets with different velocities in a dissipative system. The results reveal an abundance of interesting behaviours relating to the velocities of bullets: merging of the optical bullets into a single one at small velocities; periodic collisions at large velocities and disappearance of two bullets after several collisions in an intermediate region of velocity. Finally, it also reports that an extra bullet derives from the collision of optical bullets when optical bullets are at small velocities but with high energies.

Analytical proof of space-time chaos in Ginzburg-Landau equations

We prove that the attractor of the 1D quintic complex Ginzburg-Landau equation with a broken phase symmetry has strictly positive space-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillations in grids of weakly interacting solitons in a class of Ginzburg-Landau type equations. We provide an analytic proof for the existence of two-soliton configurations with chaotic temporal behavior, and construct solutions which are closed to a grid of such chaotic soliton pairs, with every pair in the grid well spatially separated from the neighboring ones for all time. The temporal evolution of the well-separated multi-soliton structures is described by a weakly coupled lattice dynamical system (LDS) for the coordinates and phases of the solitons. We develop a version of normal hyperbolicity theory for the weakly coupled LDSs with continuous time and establish for them the existence of space-time chaotic patterns similar to the Sinai-Bunimovich chaos in discrete-time LDSs. While the LDS part of the theory may be of independent interest, the main difficulty addressed in the paper concerns with lifting the space-time chaotic solutions of the LDS back to the initial PDE. The equations we consider here are space-time autonomous, i.e. we impose no spatial or temporal modulation which could prevent the individual solitons in the grid from drifting towards each other and destroying the well-separated grid structure in a finite time. We however manage to show that the set of space-time chaotic solutions for which the random soliton drift is arrested is large enough, so the corresponding space-time entropy is strictly positive.

Stable spatiotemporal dissipative soliton clusters in the complex Ginzburg–Landau equation

Stable spatiotemporal soliton clusters in the cubic–quintic complex Ginzburg–Landau equation are investigated theoretically. It is revealed that spatiotemporal soliton clusters carrying zero and nonzero topological charges can stably propagate and the clusters don't substantially rotate despite the value of topological charges due to the effect of friction force in such a model. It is found that if the separation of solitons is larger than a critical value, the cluster is maintained, otherwise solitons exhibit too strong an attraction for each other due to being in-phase, which leads to their instability. Prediction of the minimum separation of solitons for bound clusters is demonstrated by use of energy and momentum balance methods.

Transition from pulses to fronts in the cubic–quintic complex Ginzburg–Landau equation

The cubic-quintic complex Ginzburg-Landau is the amplitude equation for systems in the vicinity of an oscillatory sub-critical bifurcation (Andronov-Hopf), and it shows different localized structures. For pulse-type localized structures, we review an approximation scheme that enables us to compute some properties of the structures, like their existence range. From that scheme, we obtain conditions for the existence of pulses in the upper limit of a control parameter. When we study the width of pulses in that limit, the analytical expression shows that it is related to the transition between pulses and fronts. This fact is consistent with numerical simulations.

Elliptic General Analytic Solutions

To find analytically the traveling waves of partially integrable autonomous nonlinear partial differential equations, many methods have been proposed over the ages: “projective Riccati method,”“tanh‐method,”“exponential method,”“Jacobi expansion method,”“new … ,” etc. The common default to all these “truncation methods” is that they provide only some solutions, not all of them. By implementing three classical results of Briot, Bouquet, and Poincaré, we present an algorithm able to provide in closed form all those traveling waves that are elliptic or degenerate elliptic, i.e., rational in one exponential or rational. Our examples here include the Kuramoto–Sivashinsky equation and the cubic and quintic complex Ginzburg–Landau equations.

Wave dynamics in a modified quintic complex Ginzburg–Landau system

We consider physical systems described by the modified quintic complex Ginzburg–Landau equation and its derivative forms and examine numerically the dynamics of its shock type wave solution. Discussions on the behaviours of this shock wave are introduced and it is shown how the ratios of diverse velocities of this wave could be exploited to explain and collect information concerning the spatial patterns formation in the system.