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Journal of Applied Mathematics and Physics, 2018, 6, 2127-2141

http://www.scirp.org/journal/jamp

ISSN Online: 2327-4379

ISSN Print: 2327-4352

DOI:

10.4236/jamp.2018.610179 Oct. 29, 2018 2127 Journal of Applied Mathematics and Physics

Pulsating Solitons in the Two-Dimensional

Complex Swift-Hohenberg Equation

Aladji Kamagaté

1,2

, Alain-Brice Moubissi

Ecole Supérieure Africaine des Technologies d’Information et de Communication (ESATIC), Abidjan, Côte d’Ivoire

Agence Nationale de la Recherche, Paris, France

Université des Sciences et Techniques de Masuku, Franceville, Gabon

Abstract

In 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 dis-

played. The approach is based on the semi-

analytical method of collective

coordinate approach. This method is constructed on a reduction from an infi-

nite-dimensional dynamical dissipative system to a finite-dimensi

onal model.

The reduced model helps to obtain approximately the boundaries between

the stationary and pulsating solutions. We analyzed the dynamics and cha-

racteristics of the pulsating solitons. Then we obtained the bifurcation dia-

gram 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 satu-

ration of the Kerr nonlinearity increases, one period pulsating soliton solu-

tion bifurcates to double period pulsations.

Keywords

Pulsating Solution, Dissipative Soliton, Spatio-Temporal, Collective Coordinate

Approach, Ginzburg-Landau Equation, Complex Swift-Hohenberg Equation,

Spectral Filtering, Bifurcation

1. Introduction

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

Hohenberg [1] as a simple model for the Rayleigh-Bénard instability of roll

waves. This equation models pattern formation arising from an oscillatory insta-

bility [2] [3] with a finite wave number at onset and, as such, it admits solutions

How to cite this paper:

Kamagaté,

A. and

Moubissi

, A.-B. (2018) Pulsating Solitons

the Two

-Dimensional Complex Swift-Ho-

henberg Equation

Journal of Applied M

thematics and Physics

, 2127-2141.

https://doi.org/10.4236/jamp.2018.610179

Received:

September 11, 2018

Accepted:

October 26, 2018

Published:

October 29, 2018

8 by authors and

Scientific

Research Publishing Inc.

This work is licensed under the Creative

Commons Attribution International

License (CC BY

4.0).

http://creativecommons.org/licenses/by/4.0/

Open Access

A. Kamagaté, A.-B. Moubissi

DOI:

10.4236/jamp.2018.610179 2128 Journal of Applied Mathematics and Physics

in the form of traveling waves. The CSHE has been widely studied both analyti-

cally and numerically [4] [5]. It describes the dynamics of wide-aperture lasers of

class A and C close to the peak gain [6]. Besides lasers, the CSHE has been used

as a model for other nonlinear optical systems. For instance, photorefractive os-

cillator, Ker medium, semiconductor laser, and passively mode-locked lasers

that allow the generation of self-shaped ultra-short pulses in laser systems [7] are

well described by the Complex Swift-Hohenberg order parameter equation. The

CSHE is useful to describe solitons propagation in optical systems with linear

and nonlinear gain, and spectral filtering such as nonlinear polarization rotation

in fiber lasers or communication links including lumped fast absorbers [5]. It

seems clear that the generation of more complex impulse can be extremely de-

scribed by the higher order of the spectral filter.

Initially, one of the generic equations to analyze the dynamics of the dissipa-

tive soliton formation in laser systems with a fast saturable absorber is the com-

plex Cubic-quintic Ginzburg-Landau equation (CGLE) model [8] [9]. Neverthe-

less, the spectral filter of this model is restricted to the second-order term and

can only describe a spectral response with a single maximum, which is not the

case in many experiments. Indeed, the gain spectrum is usually wide and can

have multiple peaks. Thus, in order to make the model more realistic, we need to

add other terms of higher-order spectral filtering to the CGLE, leading to the

complex Swift-Hohenberg equation [9]. The formation of dissipative soliton in

these models has been widely studied in nonlinear dissipative optics. Their

properties and conditions of existence have been investigated extensively, from

fundamental point of view and due to the clear physical meaning in particular

applications [8] [10] [11].

Nonetheless, these studies use purely numerical approaches. Despite the fact

that some families of exact solution of the CSHE [12] can be obtained analyti-

cally, it is apparent that the CSHE can mainly be analyzed only using computer

simulations. Numerical analysis of the CSHE [6] reveals a great variety of pat-

terns and structures such as traveling waves, spiral waves, segregation and com-

petition between stable solutions. In [13] the authors check numerically the va-

lidity of the complex Swift-Hohenberg equation for the lasers. However, solving

numerically the complex Swift-Hohenberg equation for the two-dimensional so-

litons, for a given set of parameters and a given initial condition, is an extremely

lengthy procedure. To overcome this difficulty, semi-analytical methods based

on various physical backgrounds were developed [14]. For instance, in [15] the

authors applied the semi-analytical method to investigate soliton propagation

and generation of stable moving pulses in one dimension and stable vortex soli-

tons in two dimensions. These alternative theoretical semi-analytical tools can

perceive soliton solutions more efficiently in specific domains of the system pa-

rameters [9] [14] [16] [17].

Using the collective variable approach, we have expanded the regions of coex-

istence of 3D dissipative stationary and pulsating solitons in the complex Ginz-

A. Kamagaté, A.-B. Moubissi

DOI:

10.4236/jamp.2018.610179 2129 Journal of Applied Mathematics and Physics

burg-Landau equation with the cubic-quintic nonlinearity [18]. Recently, with

the same approach, we have demonstrated the stationary dissipative solutions of

the 2D complex Swift-Hohenberg equation. Particularly, we mapped the regions

of existence of stationary dissipative soliton in the

( )

νε

and

( )

βγ

planes

[5].

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 dis-

sipative 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 col-

lective 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.

2. Stability Studies by Collective Coordinates Theory

In this study, we address the complex Swift-Hohenberg equation in two dimen-

sions. It is helpful to describe soliton propagation in optical systems with linear

and nonlinear gain and spectral filtering. As well, the CSHE relates quantitative-

ly 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 polariza-

tion 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

[8] [19]:

z tt rr

tt ttt

iD i i i

ψ ψ ψ γψψ νψψ

δψ εψψ βψ µψψ γψ

− −− −

=+ ++ +

(1)

Without the additive term

2 ttt

γψ

this equation is the same as the CGLE one,

and here the coefficients

,, , ,,,D

µδ β νγγ

and

are real constants. The

right-hand-side of Equation (1) contains the dissipative terms:

represents

the higher-order spectral filter term.

δεβ

and

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,

( )

11D =+−

which is for the ano-

malous (normal) dispersion propagation regime and

which represents, if

negative, the saturation coefficient of the Kerr nonlinearity.

stands for Kerr

nonlinearity coefficient. In this present study, the dispersion is anomalous, and

is kept relatively small.

It is clear that the physical meaning of each term of the Equation (1) depends

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