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A semi-analytical approach for the pulsating solutions of the 3D complex Cubic-quintic Ginzburg-Landau Equation (CGLE) is presented in this article. A collective variable approach is used to obtain a system of variational equations which give the evolution of the light pulses parameters as a function of the propagation distance. The collective coordinate approach is incomparably faster than the direct numerical simulation of the propagation equation. This allows us to obtain, efficiently, a global mapping of the 3D pulsating soliton. In addition it allows describing the influence of the parameters of the equation on the various physical parameters of the pulse and their dynamics.

Dissipative systems in nonlinear optics have a dynamic involving many different phenomena such as nonlinear gain, the saturable losses, the dispersion and others effects. The interaction between these different physical manifestations leads to a rich variety of structures [

For dissipative systems in nonlinear optics, stable solitons can arise in one, two, and three dimensions [

With a good choice of the parameters of the system, stable solitons over very large distances of propagation are perfectly obtained. However, a variation of the parameters of the system changes the types of solutions via bifurcations. Dissipative soliton stability depends crucially on the energy balance and exists as long as there is a continuous energy supply to the system. Its shape, amplitude, and velocity are all fixed and defined by the parameters of the system [

Properties and conditions of the existence of solitons have been extensively studied in (1D) one-dimension and (2D) two-dimension. The (3D) three-dimensional case is still largely in its infancy and requires an extremely lengthy and costly procedure. Indeed solving numerically a (3D) equation for a given set of parameters and a given initial condition can take up to several days in a standard PC. In this context, it is important to develop theoretical tools that can perceive soliton solutions more efficiently and envisage their domains of existence.

It has been recently demonstrated in previous works that the collective variable approach is a useful tool which can reduce significantly the computation time. This approach is useful for predicting approximately the domains of existence of stable light bullets in the parameter space of the (3D) complex cubic-quintic Ginzburg- Landau equation [

Here, previous studies are focusing on this type of (3D) optical pulse in order to obtain the properties and conditions of their existence. According to the specific choice of the trial function, different internal dynamics of the pulsating dissipative soliton are shown.

In this study, the propagation of pulse in a system described by an extended complex Cubic-quintic Ginzburg- Landau equation model is considered. This equation (CGLE) is one of the universal equations used to describe dissipative systems. Many nonequilibrium phenomena, such as the generation of spatio-temporal dissipative structure in lasers [

Equation (1) is written in normalized form. The optical envelope ψ is a complex function of four real variables

The left-hand-side contains the conservative terms, namely

Now, to best of our Knowledge, there is no analytical solution for the (3D) complex Cubic-quintic Ginzburg- Landau equation. Indeed, research conducted to date use namely direct numerical simulations of the CGLE. This procedure is extremely and costly [

The lack of general analytical solutions for the complex Cubic-quintic Ginzburg-Landau equation CGLE leads us to the necessity of using simple approaches to explore the existence of certain class of solutions. A few approximate, semi-analytical methods based on various physical backgrounds were developed and applied to study nonlinear pulse propagation. Several of them make use of a trial function and its associated finite-dimen- sional dynamical system.

Our study is therefore to develop theoretical tools that can perceive soliton solutions more efficiently and envisage their domains of existence. Using collective variable approach allow to rich variety of solutions that include stationary and pulsating dissipative solitons.

The dynamics of ultra-short light pulses in a fiber system can be described by the Equation (1). But the field _{i} with

where f, the ansatz function or trial function is a function of the collective variables. And

The choice of the trial function is important for the success of the technique. It is impossible to get any idea of the exact profile without solving numerically the CGLE (1). Hence, the choice of a particular trial function has a certain degree of arbitrariness, stressing furthermore the approximate nature of the collective variable approach. After choosing the ansatz function one can pursue the process of characterization of the pulse by neglecting the residual field. This approximation is called the bare approximation [

In this study the propagation of solitons in a system described by an extended (3D) complex Cubic-quintic Ginzburg-Landau equation model is considered. This model includes cubic and quintic nonlinearities of dispersive and dissipative types. Transverse operators are added to take account of the spatial diffraction in the wave paraxial approximation.

To obtain a better understanding of dynamic processes which influence the behavior of pulse during its propagation, it is considered that the real field

Thus, it is possible to reduce the equation of impulsive field to an ordinary differential equation (ODE) describing the evolution of the parameters of the soliton during propagation. The advantage is that, the ordinary differential equation can be solved numerically with relative ease.

However, the precise shape of the trial function is crucial to have solutions with the desired properties. For the complex systems, this choice is preceded by careful analysis and based on a comparison of some analytical approximate solutions before an exact numerical solving of the CGLE can be made for comparison.

In previous work [

On the basis of preliminary results of previous works [

A, w_{t}, w_{x}, w_{y}, c_{t}, c_{x}, c_{y}, and p represent the collective variables. With t, x and y, the temporal and transverses variables along x and y axis respectively. A stands for soliton amplitude, w_{t} the temporal width along t, w_{x} the transverse width along x axis and w_{y} the transverse width along y axis. c_{t} represents the temporal chirp parameter, c_{x} the transverse chirp along x, c_{y} the transverse chirp along y and p is the global phase that evolves along with propagation. When a stationary regime is reached, the phase becomes a linear function of the propagation distance z.

The choice of the trial function is done according to the master Equation (1) and the type of solutions pursued. Consequently, the precise shape of the trial function that introduces the collective variable in the analysis of the dynamics of soliton becomes rather crucial. After choosing the trial function one can pursue the process of characterization of the pulse in two completely different ways depending on the level of accuracy desired. First, one can make use of the exact pulse field to obtain the pulse parameters. The second approach of characterization would be to carry out a variational analysis neglecting the residual field. The approximation of neglecting the residual field is called the bare approximation [

After choosing the trial function one can carry out variational analysis neglecting the residual field (the bare approximation). Applying the bare approximation to the 3D CGLE, that is, substituting the field

The collective variables evolve according to the following set of eight coupled ordinary differential equation are easily obtained:

Obviously the ordinary differential equations in Equation (4) depend on the choice the Equation (3).

These equations give no explicit information with regard to the different solutions of the Equation (1) and their stability. They give us the first idea on the dynamic of the light pulse. The variational equations allow seeing the influence of each Equation (1) parameters on the various physical parameters of the soliton. They are usually functions of time that evolve subject to the constraints of the system and finally converge to fixed point or a limit cycle.

A meticulous analysis of the variational equations show that the evolution of the amplitude (A) is dominated by the linear loss (δ), the nonlinear gain (ε) and its saturation (μ), as well as that the terms of spectral filtering (β) and dispersion term (D). This confirms quite well that the perfect balance between losses and gains is required to maintain the shape and stability of the soliton. The temporal (w_{t}) and spatial widths (w_{x}, w_{y}) also depend on the nonlinear gain (ε) and its saturation (μ). As expected, the terms of spectral filtering (β) and dispersion term (D) affect the temporal width. As well the spatial parameters c_{x}, c_{y} and temporal parameters c_{t} are influenced in the same way by the Kerr term saturation of the optical nonlinearity (ν), but the temporal term is also affected by the terms of spectral filtering (β) and dispersion term (D). Finally, not any parameters of the soliton are influenced by (p), the global phase.

One advantage of the collective variable approach is that the total energy can also expressed as function of the trial function parameters. Here it is interesting to gain insight from this simple and useful quantity, which is defined as

This above equation point out that the total energy is strongly ruled by the amplitude (A) the temporal (w_{t}) and spatial widths (w_{x}, w_{y}).

Hence, the significance of our collective variable approach helps to analyze the variational equations and the influence of various parameters. Following this in-depth analysis, the first major step in our study is to provide a mapping of the regions of existence of stable solutions in the parameter space of the (3D) CGLE.

From the analytical results of the variational Equation (4), the fixed points are carefully analyzed and their stability studied. The fixed points (FPs) of the system are found by imposing the left-hand side of Equation (4) to be zero. The threshold of existence of FPs can be estimated by the relation

A major goal of our study is to provide a quick approximate mapping of the regions of existence of stable and unstable solutions in the parameter space of the Equation (1). Stationary and pulsating solutions exist in regions defined by the space of the parameters of the equation, but it is extremely difficult to map these regions by varying all these parameters. To remedy this, it is convenient to set all parameters of the equation except two chosen as variable parameters. Thus the different solutions obtained are defined in this plan. Chosen variable parameters are the dispersion D and the cubic nonlinear gain ε. So for each set of parameters, the existence of the fixed point and its stability is studied.

By investigating the parameter regions situated in the neighborhood of the parameters μ = −0.1, δ = −0.4, β = 0.1, ν = −0.08 and γ = 1 and according to our previous study [

For a given set of CGLE parameters (μ, δ, β, ν, et γ), with an initial pulse

For each value pair (D, ε), the fourth-order Runge-Kutta algorithm provides the fixed point whose stability is analyzed.

The mapping

Regions of existence of stable light bullets following the same technique that we used for dissipative solitons in the (3 + 1)D dimensional case [

Inside this cartography, two different regions according to the values of the dispersion and the nonlinear gain are observed. The coloured area contains the stable fixed points, which are the basin of shows. And near this points, all the others points converge. The stable fixed points regions represents the domain of stationary solitons of the Equation (1) found from collective variable approach. In this domain, all the solitons parameters (amplitude, width, chirp…) stay stationary throughout propagation. Above the stationary domain, instable fixed points which can be dived in two categories: the limit-cycle attractor and the instable solutions could be observed. Our main interest is to study the dynamic of the pulse in the limit-cycle attractor area (dotted line domain). Indeed close to boundaries of the existence domains of stationary solitons solutions, there is more often an intermediate region in which pulsating solutions can be found. The existence of pulsating solutions is indeed a general feature of most nonlinear dissipative systems. This behavior of pulsating soliton can be attributed of a limit-cycle attractor; it then possesses inherent stability the same way as stationary stable solutions do. Above the pulsating domain, instable solitons which are little physical interest in this current study are noticed. The cartography (

the soliton, is a pulsating one, the total energy is an oscillating function of z. In

In

The influence of the saturation coefficient of the Kerr nonlinearity ν on the oscillations is investigated. To this end, parameters a μ = −0.1, ε = 0.6, D = 6, δ = −0.4, β = 0.1, γ = 1 are set with different values of ν (−0.07, −0.08, −0.09). The dynamics of the energy with respect to the propagation distance z are followed.

In this work, based on collective variable approach, we have expanded the studies of 3D dissipative pulsating solitons in the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity. In particular, the regions of coexistence of stationary and pulsating dissipative soliton are obtained. The collective variable approach is very efficient for approximating stable pulsating solutions when a suitable trial function is chosen. A nearly harmonic evolution of the widths along x and y axis is shown and the x and y oscillations are out of phase with the same magnitude. The dynamics of soliton can be controlled by the choice of the system parameters. So according to the values of the nonlinear gain, the system has to undergo a bifurcation from the stationary solution, to obtain pulsations with two oscillation periods. A pulsating soliton whose spectrum contains two main frequencies (two oscillation periods instead of one), associated to intensities of the same magnitude could also be predicted. The latter effect may be used, in principle, to grow photonic channels and multichannel arrays in bulk optical media. The collective variable technique is incomparably quicker than direct numerical computations. Of course, it should be used at the final stage of studies to confirm, complement, or invalidate the collective variable approach predictions.

Olivier Asseu,Ambroise Diby,Pamela Yoboué,Aladji Kamagaté, (2016) Spatio-Temporal Pulsating Dissipative Solitons through Collective Variable Methods. Journal of Applied Mathematics and Physics,04,1032-1041. doi: 10.4236/jamp.2016.46108