Complex Swift-Hohenberg Equation Research Articles

Influence of higher-order nonlinear effects on optical solitons of the complex Swift-Hohenberg model in the mode-locked fiber laser

The complex Swift-Hohenberg equation (CSHE) has been widely studied in recent years. It is a more real model in the mode-locked fiber laser with the saturable absorber due to the addition of the diffraction term compared with the complex Ginzburg-Landau equation. In this work, the dynamics process of the traditional soliton and M−type soliton pulses in the mode-locked fiber laser is demonstrated based on the CSHE model with higher-order nonlinear effects. The results show that with the increase of the small signal gain coefficient, the number of soliton molecules adds gradually. Under the same transmission conditions, the transmission of M−type solitons in the laser are more stable than that of single solitons. By adding the self-steepening effect, it can be found that the time-domain shift due to higher-order dispersion effects is compensated. The self-frequency shift effect caused by the Raman scattering can produce not only time domain shift, but also frequency domain shift. Moreover, the addition of higher-order diffraction term can describe the spectral response of multiple peaks, and makes the pulse spectrum show the asymmetric propagation in the transmission process. Finally, the increase of the length of the single-mode fiber on the right side of the gain fiber in the optical circuit will not only shift the center position of the output pulse backward, but also make the pulse energy show a ladder type downward trend.

Localized patterns in a generalized Swift–Hohenberg equation with a quartic marginal stability curve

Abstract In some pattern-forming systems, for some parameter values, patterns form with two wavelengths, while for other parameter values, there is only one wavelength. The transition between these can be organized by a codimension-three point at which the marginal stability curve has a quartic minimum. We develop a model equation to explore this situation, based on the Swift–Hohenberg equation; the model contains, amongst other things, snaking branches of patterns of one wavelength localized in a background of patterns of another wavelength. In the small-amplitude limit, the amplitude equation for the model is a generalized Ginzburg–Landau equation with fourth-order spatial derivatives, which can take the form of a complex Swift–Hohenberg equation with real coefficients. Localized solutions in this amplitude equation help interpret the localized patterns in the model. This work extends recent efforts to investigate snaking behaviour in pattern-forming systems where two different stable non-trivial patterns exist at the same parameter values.

Complex Swift Hohenberg equation dissipative soliton fiber laser

Complex Swift Hohenberg equation (CSHE) has attracted intensive research interest over the years, as it enables realistic modeling of mode-locked lasers with saturable absorbers by adding a fourth-order term to the spectral response. Many researchers have reported a variety of numerical solutions of CSHE which reveal interesting pulse patterns and structures. In this work, we have demonstrated a CSHE dissipative soliton fiber laser experimentally using a unique spectral filter with a complicated transmission profile. The behavior and performance of the laser agree qualitatively with the numerical simulations based on CSHE. Our findings bring insight into dissipative soliton dynamics and make our mode-locked laser a powerful testbed for observing dissipative solitons of CSHE, which may open a new course in ultrafast fiber laser research.

Interaction of dissipative solitons stabilized by nonlinear gradient terms.

We study the interaction of stable dissipative solitons of the cubic complex Ginzburg-Landau equation which are stabilized only by nonlinear gradient terms. In this paper we focus for the interactions in particular on the influence of the nonlinear gradient term associated with the Raman effect. Depending on its magnitude, we find up to seven possible outcomes of theses collisions: Stationary bound states, oscillatory bound states, meandering oscillatory bound states, bound states with large-amplitude oscillations, partial annihilation, complete annihilation, and interpenetration. Detailed results and their analysis are presented for one value of the corresponding nonlinear gradient term, while the results for two other values are just mentioned briefly. We compare our results with those obtained for coupled cubic-quintic complex Ginzburg-Landau equations and with the cubic-quintic complex Swift-Hohenberg equation. It turns out that both meandering oscillatory bound states as well as bound states with large-amplitude oscillations appear to be specific for coupled cubic complex Ginzburg-Landau equations with a stabilizing cubic nonlinear gradient term. Remarkably, we find for the large-amplitude oscillations a linear relationship between oscillation amplitude and period.

PERIODIC AND QUASI-PERIODIC SOLUTIONS FOR THE COMPLEX SWIFT-HOHENBERG EQUATION

In this paper, we consider the complex Swift-Hohenberg(CSH) equation $ \frac{\partial u}{\partial t} = \lambda u-(\alpha+\mathrm{i}\beta)\left(1+\frac{\partial^2}{\partial x^2}\right)^2u-(\sigma+\mathrm{i}\rho)|u|^2u $ subject to periodic boundary conditions. Using an infinite dimensional KAM theorem, we prove that there exist a continuous branch of periodic solutions and a Cantorian branch of quasi-periodic solutions for the above equation.

Higher-order Spectral Filtering Effects on the Evolution of Stationary Dissipative Solitons

A semianalytical method to study the effects of the higher-order spectral filtering in the cubic-quintic complex Swift-Hohenberg equation (CSHE) through the dynamics of one soliton was applied. The approach is based on a reduction from an infinite-dimensional dynamical dissipative system to a finite-dimensional model. This formulation is helpful to study the ground state of the soliton dynamic since it depends on a trial function and a good set of parameters. With real coefficients, the CSHE exhibits stationary dissipative solitons in space with the equation parameters, and the higher-order spectral filtering has a real impact on the cartographies of stationary soliton domain. The detailed analysis reveals the effects of spectral filtering term on the stationary soliton parameters, and displays that it differently influences the cubic and quintic terms of the CSHE. The results highlight the major influence of the spectral filtering on the temporal width of the stationary soliton whereas it does not have a real impact on the amplitude and the spatial width.

Bifurcation and Stability Analysis of Pulsating Solitons

Pulsating soliton solutions bifurcation analysis of the two-dimensional (2D) Complex Swift-Hohenberg equation (CSHE) is presented. The approach is based on a reduction from an infinite-dimensional dynamical dissipative system to a finite-dimensional model. Thanks to the collective variable approach, we investigated the influence of the nonlinear gain and the saturation of the Kerr nonlinearity on the pulsations of the solitons. Research has shown that the transformation between pulsating soliton and fronts can be realized through a series of period-doubling bifurcations. The complete bifurcation diagrams of the total energy have been obtained for a definite range of the nonlinear gain and the saturation of the Kerr nonlinearity values. The detailed analysis reveals that when the saturation of the Kerr nonlinearity increases one-period pulsating solution bifurcates to double-period pulsations. While the increase of the nonlinear gain leads the double-period pulsations to return into one-period pulsation before transforming into a stationary pulsating solitons.

Swift–Hohenberg soliton explosions

New soliton explosions in the context of the complex Swift–Hohenberg equation (CSHE) have been found and analyzed in both temporal and spectral domains. In particular, very irregular explosions, with huge temporal extensions, were found. A region of existence of these explosive solitons in the plane (ν,ϵ) was also found. The impact of the filter profile and the higher-order effects (HOEs), namely, intrapulse Raman scattering and third-order dispersion, is analyzed. A comparison between the CSHE and the complex Ginzburg–Landau equation (CGLE) soliton explosions is performed. Simulations show that a proper combination of the filter profile and HOEs allows a fixed-shape pulse propagation for HOEs parameter values lower than in the CGLE case.

Universal description of pattern formation in optical oscillators under bichromatic injection

We study pattern formation in a complex Swift–Hohenberg equation with phase-sensitive (parametric) gain. Such an equation serves as a universal order parameter equation describing the onset of spontaneous oscillations in extended systems submitted to a bichromatic injection when the instability is toward long (transverse) wavelengths. Applications include two-level lasers and photorefractive oscillators. Under such an injection, the original continuous phase symmetry of the system is replaced by a discrete one and phase bistability emerges. This leads to the spontaneous formation of phase-locked spatial structures, such as phase domains and dark-ring (phase) cavity solitons. The stability of the homogeneous solutions is studied, and numerical simulations are made covering all the dynamical regimes of the model, which turn out to be very rich. Derivations of the rocked complex Swift–Hohenberg equation, using multiple scale techniques, are given for the two-level laser and the photorefractive oscillator.

Pulsating Solitons in the Two-Dimensional Complex Swift-Hohenberg Equation

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 displayed. The approach is based on the semi-analytical method of collective coordinate approach. This method is constructed on a reduction from an infinite-dimensional 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.

Chirped soliton solutions of the cubic-quintic Swift-Hohenberg equation with space-dependent coefficients

The optical pulses generation in mode-locked lasers are described, in the case of complicated spectral filtering, by the cubic–quintic complex Swift-Hohenberg equation (CQSHE). Using the ansatz method, we studied the existence of the bright and dark chirped soliton solutions of the space-dependent coefficients CQSHE. The physical parameters of the chirped soliton solutions have been obtained as functions of the variable coefficients of the model equation. The parametric conditions for the existence of the chirped soliton solutions are presented. By the numerical evolution of the bright and dark solitons, we studied the influence of high order effects, dispersion and nonlinearity’s, on the propagation of the soliton solutions.

Stability of Dissipative Optical Solitons in the 2D Complex Swift-Hohenberg Equation

This article deals with stationary localized solutions of the (2D) two-dimensional complex Swift-Hohenberg equation (CSHE). Our approach is based on the semi-analytical method of collective coordinate approach. According to the parameters of the equation and a suitable choice of ansatz, the stationary dissipative solitons of the 2D CSHE equation are mapped. This approach allows to describe the influence of the parameters of the equation on the various physical parameters of the pulse and their dynamics. Finally, the major impact of spectral filtering terms on the dynamic of the solitons is demonstrated.

High-energy plain and composite pulses in a laser modeled by the complex Swift–Hohenberg equation

In this work, new plain and composite high-energy solitons of the cubic–quintic Swift–Hohenberg equation were numerically found. Starting from a composite pulse found by Soto-Crespo and Akhmediev and changing some parameter values allowed us to find these high energy pulses. We also found the region in the parameter space in which these solutions exist. Some pulse characteristics, namely, temporal and spectral profiles and chirp, are presented. The study of the pulse energy shows its independence of the dispersion parameter but its dependence on the nonlinear gain. An extreme amplitude pulse has also been found.

Stability of the fixed points of the complex Swift-Hohenberg equation

We performed an investigation of the stability of fixed points in the complex Swift- Hohenberg equation using a variational formulation. The analysis is based on fixed points Euler-Lagrange equations and analytically showed that the Jacobian eigenvalues touched the imaginary axis and in general, Hopf bifurcation arises. The eigenvalues undergo a stability criterion in order to have Hopf's stability. Trial functions and linear loss dispersion parameter ε are responsible for the existence of stable pulse solutions in this system. We study behavior of the stable soliton-like solutions as we vary a bifurcation ε.

An Efficient Numerical Method for the Quintic Complex Swift-Hohenberg Equation

In this paper, we present an efficient time-splitting Fourier spectral method for the quintic complex Swift-Hohenberg equation. Using the Strang time-splitting tech- nique, we split the equation into linear part and nonlinear part. The linear part is solved with Fourier Pseudospectral method; the nonlinear part is solved analytically. We show that the method is easy to be applied and second-order in time and spectrally accurate in space. We apply the method to investigate soliton propagation, soliton interaction, and generation of stable moving pulses in one dimension and stable vortex solitons in two dimensions. AMS subject classifications: 65M70, 65Z05

Stability on Time-Dependent Domains

We explore the key differences in the stability picture between extended systems on time-fixed and time-dependent spatial domains. As a paradigm, we take the complex Swift–Hohenberg equation, which is the simplest nonlinear model with a finite critical wavenumber, and use it to study dynamic pattern formation and evolution on time-dependent spatial domains in translationally invariant systems, i.e., when dilution effects are absent. In particular, we discuss the effects of a time-dependent domain on the stability of spatially homogeneous and spatially periodic base states, and explore its effects on the Eckhaus instability of periodic states. New equations describing the nonlinear evolution of the pattern wavenumber on time-dependent domains are derived, and the results compared with those on fixed domains. Pattern coarsening on time-dependent domains is contrasted with that on fixed domains with the help of the Cahn–Hilliard equation extended here to time-dependent domains. Parallel results for the evolution of the Benjamin–Feir instability on time-dependent domains are also given.

Traveling waves and defects in the complex Swift-Hohenberg equation

The complex Swift-Hohenberg equation models pattern formation arising from an oscillatory instability with a finite wave number at onset and, as such, admits solutions in the form of traveling waves. The properties of these waves are systematically analyzed and the dynamics associated with sources and sinks of such waves investigated numerically. A number of distinct dynamical regimes is identified and analyzed using appropriate phase equations describing the evolution of long-wavelength instabilities of both the homogeneous oscillating state and constant amplitude traveling waves.

Dynamic bifurcation of the n-dimensional complex Swift-Hohenberg equation

This paper is concerned with the bifurcation of a complex Swift-Hohenberg equation. The attractor bifurcation of the complex Swift-Hohenberg equation on a onedimensional domain (0, L) is investigated. It is shown that the n-dimensional complex Swift-Hohenberg equation bifurcates from the trivial solution to an attractor under the Dirichlet boundary condition on a general domain and under a periodic boundary condition when the bifurcation parameter crosses some critical values. The stability property of the bifurcation attractor is analyzed.

Coarsening and frozen faceted structures in the supercritical complex Swift-Hohenberg equation

The supercritical complex Swift-Hohenberg equation models pattern formation in lasers, optical parametric oscillators and photorefractive oscillators. Simulations of this equation in one spatial dimension reveal that much of the observed dynamics can be understood in terms of the properties of exact solutions of phase-winding type. With real coefficients these states take the form of time-independent spatial oscillations with a constant phase difference between the real and imaginary parts of the order parameter and may be unstable to a longwave instability. Depending on parameters the evolution of this instability may or may not conserve phase. In the former case the system undergoes slow coarsening described by a Cahn-Hilliard equation; in the latter it undergoes repeated phase-slips leading either to a stable phase-winding state or to a faceted state consisting of an array of frozen defects connecting phase-winding states with equal and opposite phase. The transitions between these regimes are studied and their location in parameter space is determined.

Faceting and coarsening dynamics in the complex Swift-Hohenberg equation

The complex Swift-Hohenberg equation models pattern formation arising from an oscillatory instability with a finite wave number at onset and finds applications in lasers, optical parametric oscillators, and photorefractive oscillators. We show that with real coefficients this equation exhibits two classes of localized states: localized in amplitude only or localized in both amplitude and phase. The latter are associated with phase-winding states in which the real and imaginary parts of the order parameter oscillate periodically but with a constant phase difference between them. The localized states take the form of defects connecting phase-winding states with equal and opposite phase lag, and can be stable over a wide range of parameters. The formation of these defects leads to faceting of states with initially spatially uniform phase. Depending on parameters these facets may either coarsen indefinitely, as described by a Cahn-Hilliard equation, or the coarsening ceases leading to a frozen faceted structure.