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I n this paper, we performed an investigation of the dissipative solitons of the two-dimensional (2D) Complex Swift-Hohenberg equation (CSHE). Stationary to pulsating soliton bifurcation analysis of the 2D CSHE is displayed. The approach is based on the semi-analytical method of collective coordi nate approach. This method is constructed on a reduction from an infinite-dimensiona l dynamical dissipative system to a finite-dimensional model. The reduced model helps to obtain approximately the boundaries between the stationary and pulsating solutions. We analyzed the dynamics and characteristics of the pulsating solitons. Then we obtained the bifurcation diagram for a definite range of the saturation of the Kerr nonlinearity values. This diagram reveals the effect of the saturation of the Kerr nonlinearity on the period pulsations. The results show that when the parameter of the saturation of the Kerr nonlinearity increases , one period pulsating soliton solution bifurcates to double period pulsations.

The complex Swift-Hohenberg equation (CSHE) was first suggested by Swift and Hohenberg [

Initially, one of the generic equations to analyze the dynamics of the dissipative soliton formation in laser systems with a fast saturable absorber is the complex Cubic-quintic Ginzburg-Landau equation (CGLE) model [

Nonetheless, these studies use purely numerical approaches. Despite the fact that some families of exact solution of the CSHE [

Using the collective variable approach, we have expanded the regions of coexistence of 3D dissipative stationary and pulsating solitons in the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity [

Here, our main purpose is to investigate the pulsating solution of the 2D CSHE using a variational formulation. On the fact that the dynamics of the dissipative solutions are much more complex, and the numerical simulations are extremely tedious tasks, the variational approach is useful to study the ground state since it depends on a trial function and a good set of parameters.

The rest of the paper is organized as follows. We remind in section 2 the collective variable approach and our procedure of determination of the stability domains of the pulsating solutions. Section 3 is devoted to the findings of the 2D pulsating CSHE solutions. We illustrate the bifurcation behavior and show that they can be stable over a wide range of parameter values. Finally, we summarize with our conclusions in section 4.

In this study, we address the complex Swift-Hohenberg equation in two dimensions. It is helpful to describe soliton propagation in optical systems with linear and nonlinear gain and spectral filtering. As well, the CSHE relates quantitatively as qualitatively many nonlinear effects, which occur during the propagation. This equation is also useful for communication links with lumped fast saturable absorbers or fiber lasers with additive-pulse mode-locking or nonlinear polarization rotation. The CSHE higher order of the spectral filter is extremely essential to analyze the generation of more complex impulse, which makes it preferable in certain situations to the CGLE. The CSHE can be read in this normalized form [

ψ z − i D ψ t t / 2 − i ψ r r / 2 − i γ | ψ | 2 ψ − i ν | ψ | 4 ψ = δ ψ + ε | ψ | 2 ψ + β ψ t t + μ | ψ | 4 ψ + γ 2 ψ t t t (1)

Without the additive term γ 2 ψ t t t this equation is the same as the CGLE one, and here the coefficients μ , δ , β , D , ν , γ , γ 2 and ε are real constants. The right-hand-side of Equation (1) contains the dissipative terms: γ 2 represents the higher-order spectral filter term. δ , ε , β and μ are the coefficients for linear loss (if negative), nonlinear gain (if positive), spectral filtering (if positive) and saturation of the nonlinear gain (if negative), respectively. The left-hand side holds the conservative terms: namely, D = + 1 ( − 1 ) which is for the anomalous (normal) dispersion propagation regime and ν which represents, if negative, the saturation coefficient of the Kerr nonlinearity. γ stands for Kerr nonlinearity coefficient. In this present study, the dispersion is anomalous, and ν is kept relatively small.

It is clear that the physical meaning of each term of the Equation (1) depends on the real problem which must be examined. In optics, when applied to the propagation of the pulses in a laser system, as is the case in our study, ψ = ψ ( r , t , z ) represents the normalized optical envelope and is a complex function of three real variables. The optical envelope describes not only the pulse as a collective entity localized in time and space but also all other localized or non-localized excitations, such as noise, which are always present in the real system.

The retarded time in the frame moving with the pulse is given by t, and z is the propagation distance or the cavity round-trip number. Finally r ( r = x 2 + y 2 ) represents the transverse coordinate, taking account of the spatial diffraction effects.

The dynamics of light pulses described by Equation (1) can be stationary [

We have proved in our previous work [

In reference to our work [

ψ ( r , t , z ) = f ( x 1 , x 2 , ⋯ , x n , t ) + q ( z , t ) (2)

where f the ansatz function is a function of the collective variables ( x n ) and is chosen to draw, at best, the configuration of the optical pulse ψ . The choice of the ansatz function that introduces the collective variables in the theory is crucial for obtaining solutions with the desired properties and important for the success of the technique. The component q is a residual field that represents all other excitations in the system (noise, radiation, dressing field, etc.).

Subsequently, by neglecting the residual field, one can consider the fact that the pulse propagation can be completely characterized by the ansatz function. This approximation is called the bare approximation [

f = A exp ( − t 2 w t 2 − r 2 w r 2 + i 2 c t t 2 + i 2 c r r 2 + i p ) (3)

here the collective variables A , w t , w r , c t , c r and p represent respectively the amplitude, the temporal and spatial widths of the pulse, the chirp along t axis, the spatial chirp and p the global phase. The collective variables evolve along the propagation direction z and the dynamic of the dissipative soliton. In these dynamic and evolutions, the chirps are highly important.

Using the bare approximation to the 2D CSHE (see all the details in [

A ˙ = A δ + 3 4 A 3 ε − 2 w t 2 A β + 5 9 A 5 μ − A c t D − 2 A c r + 3 ( 2 c t 2 − w t 2 c t 4 + 3 w t 4 ) A γ 2 ,

w ˙ t = 2 w t c t D − 1 4 w t A 2 ε − 2 9 A 4 w t μ + ( 1 − w t 4 c t 2 ) 2 β w t + ( w t 8 c t 4 − 1 ) 12 w t 3 γ 2 ,

w ˙ r = 4 w r c r − 1 4 w r A 2 ε − 2 9 A 4 w r μ ,

c ˙ t = 2 ( 1 w t 4 − c t 2 ) D − 8 w t 2 c t β − 1 2 w t 2 A 2 γ − 4 9 w t 2 A 4 ν + 48 c t ( 1 w t 4 + c t 2 ) γ 2 ,

c ˙ r = − 4 c r 2 − 1 2 w r 2 A 2 γ + 4 w r 4 − 4 9 w r 2 A 4 ν ,

p ˙ = 2 β c t + 3 4 A 2 γ − D w t 2 − 2 w r 2 + 5 9 A 4 ν − 12 c t ( 1 w t 2 − w t 2 c t 2 ) γ 2 (4)

We observe that the CSHE, Equation (1) is reduced to an ordinary differential equation given by the soliton parameters A , w t , w r , c t , c r and p. It is clear that these equations give no explicit information with regard to the different solutions of the Equation (1) and their stability. They simply reveal in detail the influence of each parameter of equation CSHE (1) under soliton parameters. In addition, they give us the first idea on the dynamic of the light pulse.

The trial function is suitable for its simplicity, and it makes the procedure of derivation of the variational equations relatively easy. This approach provides the basic parameters of the fixed points (FPs), and a mapping of different types of solutions. The FPs of the system are found by imposing the left-hand side of Equation (4) to be zero ( X ˙ = 0 with X = A , w t , w r , c t , c r , p ) [

If the real part of at least one of the eigenvalues ( λ j ) is positive, the corresponding FP is unstable. The unstable fixed points correspond to the pulsating and non-stationary solutions of the 2D CSHE Equation (1).

If the real parts of all the eigenvalues of the matrix M i j are negative, the corresponding FPs are stable. The stable fixed points correspond to stationary solutions of the 2D CSHE Equation (1). Using the initial condition,

ψ ( r , t , 0 ) = 2.86 exp ( − t 2 0.7 − r 2 1.36 ) (5)

and investigating the parameter regions situated in the neighbourhood of the parameters D = γ = 1 , β = − 0.3 , δ = − 0.5 , μ = − 0.1 and γ 2 = 0.05 , we have highlighted the stationary and pulsating solutions in the ( ν , ε ) plane. For a given set of ν and ε the use of the Newton-Raphson allows finding the corresponding fixed point before determining its stability. The mapping

2D complex Swift-Hohenberg equation. We studied intensively the distinguishing feature of the stationary solution of the 2D CSHE in [

The nonlinear dissipative systems are more dynamics than complicated because they include energy exchange with external sources. Besides, they admit pulsating solitons, in contrast to Hamiltonian systems. In [

In order to investigate pulsating soliton in the 2D CSHE, we fix the parameters of the equation as D = γ = 1 , β = − 0.3 , δ = − 0.5 , μ = − 0.1 and γ 2 = 0.05 . Afterwards, we choose one point between the blue region and the red line (

changes periodically rather than the widths. They have the same period, but we can see that the temporal width quickly evolved with respect to the spatial width. The two dynamics are also in opposition of phase. It clearly shows a difference characteristic and oscillation between the temporal and the transverse widths of the 2D complex Swift-Hohenberg equation.

When the value of the saturation of the Kerr nonlinearity changes from ν = − 0.247 to ν = − 0.240 for the same value of nonlinear gain ε = 50 , the period of pulsating soliton doubles, that is, the amplitude repeats itself after two pulsations, as shown in

However, we notice that the number of pulsations decreases when the value of the saturation of the Kerr nonlinearity goes from ν = − 0.247 to ν = − 0.240 . In addition, the amplitude of the oscillations increases.

The analysis of these two figures (

The parameter ν influences both qualitatively and quantitatively the pulsations of the soliton. As well, the saturation of the Kerr nonlinearity plays a key role by changing a single periodic pulsating soliton to a double period pulsating soliton.

nonlinearity from ν = − 0.240 to ν = − 0.196 .

The total energy plays a whole role in the study of the complex dissipative systems. Here it gives us the main information about the soliton dynamics. It is not conserved but evolves in accordance with the so-called balance equation. When a pulsating solution is reached, the total energy is an oscillating function of the propagation distance. Thus, all the solitons parameters (amplitude, widths, chirp ...) stay pulsating throughout propagation, as has been seen in

We plotted the dynamics of the energy for ε = 50 and ν = − 0.240 (

pulse wave remain pulsating, but the pulse energy is much higher (

also becomes more important, a characteristic sign of a larger impulse.

During this present study, it distinctly appeared that the saturation of the Kerr nonlinearity ν has a real impact on the dynamics of the pulsating solitons. Changing this parameter and keeping the other parameters constant, the stationary soliton becomes pulsating with one period, and then the pulsating soliton bifurcates to double period pulsations. To illustrate this phenomenon, we plot in the

For a given set of the saturation of the Kerr nonlinearity, the curve (

We have carried out the dynamical behavior of pulsating solitons in the two-dimensional Complex Swift-Hohenberg equation. Thanks to collective variable approach, the regions of coexistence of pulsating dissipative solitons are obtained. This semi-analytical method is a useful tool to predict pulsating solutions when a suitable trial function is chosen in physical systems of high dimension. The detailed analysis shows that the nonlinear gain and the saturation of the Kerr nonlinearity influence both qualitatively and quantitatively the pulsations of the pulsating solitons. The complete bifurcation diagram has been obtained for a definite range of the saturation of the Kerr nonlinearity values. The diagram reveals that when the saturation of the Kerr nonlinearity increases, one period pulsating solution bifurcates to double period pulsations. This study reveals the rich dynamics of the pulsating solutions in the 2D CSHE and could be completed by the analysis of their properties. It can also help to describe specific aspects that occur in wide-aperture laser cavity.

The authors declare no conflicts of interest regarding the publication of this paper.

Kamagaté, A. and Moubissi, A.-B. (2018) Pulsating Solitons in the Two-Dimensional Complex Swift-Hohenberg Equation. Journal of Applied Mathematics and Physics, 6, 2127-2141. https://doi.org/10.4236/jamp.2018.610179