Complex Swift-Hohenberg Equation Research Articles

Dynamic bifurcation of the complex Swift-Hohenberg equation

In this paper we are concerned with the dynamic bifurcation of the complex Swift-Hohenberg equation on a closed interval in $\mathbb R$. We consider the equations under the Dirichlet and the periodic boundary conditions. It is shown that the equation bifurcates from the trivial solution to an attractor when the control parameter crosses the critical value.

Numerical validation of the complex Swift-Hohenberg equation for lasers

Order parameter equations, such as the complex Swift-Hohenberg (CSH) equation, offer a simplified and universal description that hold close to an instability threshold. The universality of the description refers to the fact that the same kind of instability produces the same order parameter equation. In the case of lasers, the instability usually corresponds to the emitting threshold, and the CSH equation can be obtained from the Maxwell-Bloch (MB) equations for a class C laser with small detuning. In this paper we numerically check the validity of the CSH equation as an approximation of the MB equations, taking into account that its terms are of different asymptotic order, and that, despite of having been systematically overlooked in the literature, this fact is essential in order to correctly capture the weakly nonlinear dynamics of the MB. The approximate distance to threshold range for which the CSH equation holds is also estimated.

Numerical study of the vector complex Swift–Hohenberg equation

The vector complex Swift–Hohenberg equation is a natural extension of the scalar complex version that is widely used for the description of lasers and other non-linear optical systems. Numerical analysis of this equation reveals a great variety of patterns and structures such as traveling waves, spiral waves, defects, segregation and competition between stable solutions. Simple analytical arguments are given to explain these behaviours.

Scale disparities in the complex Swift-Hohenberg equation for lasers

The complex Swift-Hohenberg (CSH) equation is a generic order parameter equation that applies to many physical systems. In the case of class C lasers, it can be obtained from the Maxwell-Bloch equations using the assumptions of slow envelope and small detuning. We show that the resulting CSH equation inevitably contains different asymptotic order terms, associated with the dominance of the effect of dispersion over diffusion. These scale disparities are usually overlooked or simply not mentioned in the literature, assuming that a CSH equation with all terms of the same order still provides qualitative information. In this paper, the asymptotically nonuniform CSH equation is carefully deduced using a simpler scaling-free procedure, and a stability analysis of the simplest solutions together with some numerical simulations are presented, in which the mentioned scale disparities are clearly seen.

Motion of pulses and vortices in the cubic–quintic complex Ginzburg–Landau equation without viscosity

Motions of pulses and vortices are numerically studied with the cubic–quintic complex Ginzburg–Landau equation without viscous terms. There exist moving pulses and vortices with any velocities, because the equation is invariant for the Galilei transformation. We study mutual collisions of counter-propagating pulses and vortices, and motions of pulses and vortices in external potentials. Moving pulses and vortices pass through a potential wall like a tunnel effect. If some viscous terms are included, the model equation is equivalent to the quintic complex Swift–Hohenberg equation. We find a supercritical bifurcation from a stationary pulse to a moving pulse.

Modulation Instability Of Optical Waves In The Cubic-Quintic Complex Ginzburg-Landau Equation With Fourth-Order Dispersion And Gain Terms

The modulation instability of the one-dimensional cubic-quintic complex Ginzburg-Landau equation with fouth-order dispersion and gain terms, a. k. a., the quintic complex Swift-Hohenberg equation, is investgated. The effects of the fourth-order terms to the modulational instability is studied. We numerically investigate the dynamics of the modulational instability in the presence of the fourthorder dispersion and gain terms. - PACS numbers: 42.65.Tg, 42.81DP, 42.65Sf

Optical pulse propagation in fibers with random dispersion

The propagation of optical pulses in two types of fibers with randomly varying dispersion is investigated. The first type refers to a uniform fiber dispersion superimposed by random modulations with a zero mean. The second type is the dispersion-managed fiber line with fluctuating parameters of the dispersion map. Application of the mean field method leads to the nonlinear Schrödinger equation (NLSE) with a dissipation term, expressed by a fourth-order derivative of the wave envelope. The prediction of the mean field approach regarding the decay rate of a soliton is compared with that of the perturbation theory based on the inverse scattering transform (IST). A good agreement between these two approaches is found. Possible ways of compensation of the radiative decay of solitons using the linear and nonlinear amplification are explored. The corresponding mean field equation coincides with the complex Swift–Hohenberg equation. The condition for the autosolitonic regime in propagation of optical pulses along a fiber line with fluctuating dispersion is derived and the existence of autosoliton (dissipative soliton) is confirmed by direct numerical simulation of the stochastic NLSE. The dynamics of solitons in optical communication systems with random dispersion-management is further studied applying the variational principle to the mean field NLSE, which results in a system of ODEs for soliton parameters. Extensive numerical simulations of the stochastic NLSE, mean field equation and corresponding set of ODEs are performed to verify the predictions of the developed theory.

A complex Swift–Hohenberg equation coupled to the Goldstone mode in the nonlinear dynamics of flames

The nonlinear dynamics of a propagating flame front governed by a two-stage sequential chemical reaction is considered in the parameter range where the uniformly propagating front is unstable. We show that near the transition from the short wave to the long wave oscillatory instability the nonlinear dynamics is described by a Swift–Hohenberg equation with dominant dispersive term, coupled to an evolution equation for the zero mode associated with the translation symmetry of the propagating wave. The nonlinear dynamics described by this system of equations is studied both analytically and numerically. In the case of weak coupling between the two equations, we observe the spontaneous formation of spiral waves with rapidly moving cores, while strong coupling leads either to chaotic dynamics or to the formation of oscillons—spatially localized oscillating structures.

Periodic and optical soliton solutions of the quintic complex Swift–Hohenberg equation

Using a direct ansatz approach, we have found a number of periodic zero-velocity analytic solutions of the complex quintic Swift–Hohenberg equation (CSHE). These find application in assorted optical problems. Particular cases of periodic solutions, where the elliptic function modulus equals 1, are various localized solutions of the CSHE. Each of these solutions exists for a certain relation between the parameters of the equation. As a result, they are particular cases of the complete set of periodic and localised solutions which may exist for this equation. In fact, they are multi-parameter families of solutions and they can serve as a seeding set of solutions which could be useful in other optical studies. We have also derived energy and momentum balance equations for the solutions of CSHE and checked that our stationary solutions satisfy the energy balance equation.

Exact soliton solutions of the one-dimensional complex Swift–Hohenberg equation

Using Painlevé analysis, the Hirota multi-linear method and a direct ansatz technique, we study analytic solutions of the (1+1)-dimensional complex cubic and quintic Swift–Hohenberg equations. We consider both standard and generalized versions of these equations. We have found that a number of exact solutions exist to each of these equations, provided that the coefficients are constrained by certain relations. The set of solutions include particular types of solitary wave solutions, hole (dark soliton) solutions and periodic solutions in terms of elliptic Jacobi functions and the Weierstrass ℘ function. Although these solutions represent only a small subset of the large variety of possible solutions admitted by the complex cubic and quintic Swift–Hohenberg equations, those presented here are the first examples of exact analytic solutions found thus far.

Bénard–Marangoni convection of a binary mixture as an example of an oscillatory bifurcation under strong symmetry-breaking effects

Surface-tension-driven convection in a binary mixture is examined in the oscillatory regime, where the first instability at threshold leads to patterns built up by plane waves with a finite critical wave vector. Direct numerical treatment of the 3D hydrodynamic basic equations shows a variety of patterns close to onset. We found steady hexagons, traveling hexagons, traveling and standing waves as well as superpositions of these structures. A Karhunen–Loève analysis is used to study these patterns in more detail. We show further that qualitatively the same patterns can be obtained from a reduced and simplified model, the complex Swift–Hohenberg equation. We show by stability analysis that symmetry-breaking terms, here from the non-symmetric boundary conditions at the top and the bottom of the layer caused by the Marangoni effect, play a crucial role in pattern selection. We expect the obtained bifurcation sequences being generic for all instabilities with finite wavelength that occur via a Hopf bifurcation when strong symmetry-breaking effects are present.

Control-parameter-dependent Swift-Hohenberg equation as a model for bioconvection patterns.

We consider a complex Swift-Hohenberg equation with control-parameter-dependent coefficients and use it as a model to describe dynamical features seen in an experimental bacterial bioconvection pattern. In particular, we give numerical results showing the development of a phase-unstable pattern behind a moving front.

Diffusive stability of rolls in the two–dimensional real and complex swift–hohenberg equation

We show the nonlinear stability of small bifurcating stationary rolls u , , for the Swift–Hohenberg–equation on the domain R2. In Bloch wave representation the linearization around a marginal stable roll uε,x, has continuous spectrum up to 0 with a locally parabolic shape at the critical Bloch vector 0. Using an abstract renormalization theorem we show that small spatially localized integrable perturbations decay diffusively to zero. Moreover we estimate the size of the domain of attraction of a roll uε,x, in terms of its modulus and Fourier wavenumber. To explain the method we also treat the nonlinear stability of stationary rolls for the complex Swift–Hohenberg equation on R2

Swift-Hohenberg model of a laser with saturable absorber

The spatiotemporal dynamics of the bifurcating transverse modes of a laser with saturable absorber is demonstrated to be governed by a complex Swift-Hohenberg equation for the laser field coupled to an ordinary differential equation. Numerical investigations are reported to illustrate the nonlinear spatiotemporal evolution near and beyond threshold.

Herringbone pattern for an anisotropic complex Swift-Hohenberg equation

We show using numerical simulations that herringbone type traveling wave patterns arise as a dissipative structure in an anisotropic complex Swift-Hohenberg equation. In the herringbone pattern, zig and zag rolls alternate spatially along the x direction. The herringbone pattern becomes unstable as a control parameter is changed, then irregular patches of zig and zag structures appear.

Grain boundaries in two-dimensional traveling-wave patterns

Results of systematic numerical studies of grain boundaries (GBs) between traveling waves are reported. Starting from the two-dimensional (2D) complex Swift-Hohenberg (SH) equation, we demonstrate that it admits steadily moving GBs in the form of sinks and sources. The sinks are always stable, while the sources may be stable or unstable. Sometimes, chaotic patterns are also observed. Next, we reduce the 2D SH equation to two coupled 1D SH equations, assuming that the wave field is a superposition of two waves with fixed y-components of the corresponding wave vectors. Direct comparison demonstrates that typical wave patterns described by the 2D equation and by the coupled 1D equations are practically indistinguishable. Within the framework of the latter approximation, we compute dependence of velocities of the sinks and sources upon the orientation of the traveling waves. In particular, for nearly equal wave vectors on both sides of GB, the velocity proves to be close to the mean group velocity of the pattern. In the opposite limit of nearly antiparallel wave vectors, the sink's velocity is close to an earlier analytical prediction (two thirds of the mean group velocity). We have also considered further simplifications of the coupled 1D SH equations, reducing them to coupled Ginzburg-Landau equations, and then to a nonlinear phase-diffusion equation. The simplified equations still provide a satisfactory description of various dynamical regimes. Especially, the stability of the stationary source is discussed with the nonlinear phase-diffusion equation. Finally, we report preliminary results of simulations of patterns containing triple points, produced by collisions between nonparallel GBs. The triple point may be both stable and unstable.

Localized patterns for the quintic complex Swift-Hohenberg equation

We show using numerical simulations that a variety of localized patterns arise in a model equation: the quintic Swift-Hohenberg equation with complex coefficients. We demonstrate that various sizes of localized standing wave patterns are possible when the imaginary part of the complex coefficient is small. Localized traveling waves as well as localized standing waves with a fixed size are observed when the imaginary part is rather large. We also present stable localized patterns in two spatial dimensions and study their interaction.

Traveling-wave–standing-wave competition in a generalized complex Swift-Hohenberg equation

Traveling-wave–standing-wave competition in a generalized complex Swift-Hohenberg equation

Nonlinear Pattern Formation in Active Optical Systems: Shocks, Domains of Tilted Waves, and Cross-Roll Patterns

Vortices, shocks, domains of tilted waves, and cross-roll patterns are typical patterns of the complex Swift-Hohenberg equation, which describes spatiotemporal dynamics in nonlinear optical systems of large Fresnel number, such as lasers, optical parametric oscillators, and photorefractive oscillators. We show the occurrence of such ``essentially nonlinear'' patterns experimentally on a photorefractive oscillator and compare it with numerical solutions of the complex Swift-Hohenberg equation.

Generalized complex Swift-Hohenberg equation for optical parametric oscillators

A generalized complex Swift-Hohenberg equation including diffraction and nonlinear resonance terms is derived for spatially extended nondegenerate optical parametric oscillators (OPOs) with flat end mirrors. For vanishing pump detuning this equation becomes the complex Swift-Hohenberg (SH) equation valid also for lasers. Nevertheless the similarities between OPOs and lasers are limited, since the diffractive character of OPOs is lost when the diffraction coefficients of signal and idler fields are equal. This manifests, e.g., in the absence of advection by traveling waves (TWs), a clear difference with lasers. When pump detuning is nonzero a nonlinear resonance develops, as it occurs in degenerate OPOs. This nonlinear resonance is essential in order to properly describe the TWs that OPOs support, and describes the bistability between TWs. This leads to the appearance of localized structures, which we also report here.