Coupled Ginzburg-Landau Equations Research Articles

Bound states of solitary pulses in linearly coupled Ginzburg-Landau equations

We investigate the existence, formation and stability of multipulse bound states in a system of two Ginzburg-Landau equations coupled by linear terms. The system includes linear gain, diffusion, dispersion, and cubic nonlinearity in one component, and only linear losses in the other. This is a straightforward model of a doped dual-core nonlinear optical fiber in which only one core is pumped. The model supports exact stable solitary-pulse solutions. By means of systematic numerical simulations, we find that bound states of two, three, and more pulses with a uniquely determined separation between them exist. The three-pulse bound states are stable against symmetric perturbations, but prove to be unstable against asymmetric ones. Only the two-pulse states are found to be fully stable.

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.

Ordered and disordered defect chaos

Defect chaos is studied numerically in coupled Ginzburg–Landau equations for parametrically driven waves. The motion of the defects is traced in detail yielding their lifetimes, annihilation partners, and distances traveled. In a regime in which in the one-dimensional case the chaotic dynamics is due to double phase slips, the two-dimensional system exhibits a strongly ordered stripe pattern. When the parity-breaking instability to traveling waves is approached this order vanishes and the correlation function decays rapidly. In the ordered regime the defects have a typical lifetime, whereas in the disordered regime the lifetime distribution is exponential. The probability of large defect loops is substantially larger in the disordered regime.

Parametric forcing of waves with a nonmonotonic dispersion relation: Domain structures in ferrofluids

Surface waves on ferrofluids exposed to a dc magnetic field exhibit a nonmonotonic dispersion relation. The effect of a parametric driving on such waves is studied within suitable coupled Ginzburg-Landau equations. Due to the nonmonotonicity the neutral curve for the excitation of standing waves can have up to three minima. The stability of the waves with respect to long-wave perturbations is determined via a phase-diffusion equation. It shows that the band of stable wave numbers can split up into two or three subbands. The resulting competition between the wave numbers corresponding to the respective subbands leads quite naturally to patterns consisting of multiple domains of standing waves which differ in their wave number. The coarsening dynamics of such domain structures is addressed.

Justification of mean-field coupled modulation equations

We are interested in reflection symmetric (x↦–x) evolution problems on the infinite line. In the systems which we have in mind, a trivial ground state loses stability and bifurcates into a temporally oscillating, spatial periodic pattern. A famous example of such a system is the Taylor-Couette problem in the case of strongly counter-rotating cylinders. In this paper, we consider a system of coupled Kuramoto–Shivashinsky equations as a model problem for such a system. We are interested in solutions which are slow modulations in time and in space of the bifurcating pattern. Multiple scaling analysis is used in the existing literature to derive mean-field coupled Ginzburg–Landau equations as approximation equations for the problem. The aim of this paper is to give exact estimates between the solutions of the coupled Kuramoto–Shivashinsky equations and the associated approximations.

Phase Diffusion in Localized Spatio-Temporal Amplitude Chaos.

We present numerical simulations of coupled Ginzburg-Landau equations describing parametrically excited waves which reveal persistent dynamics due to the occurrence of phase slips in sequential pairs, with the second phase slip quickly following and negating the first. Of particular interest are solutions where these double phase slips occur irregularly in space and time within a spatially localized region. An effective phase diffusion equation utilizing the long-term phase conservation of the solution explains the localization of this new form of amplitude chaos.

Traveling wave–standing wave transition in the coupled complex Ginzburg–Landau equations

The coupled Ginzburg-Landau equations are studied numerically. The instability of a chaotic traveling wave state is characterized by means of a stability exponent. When the traveling wave state is unstable, several types of coexistent states of left and right traveling waves appear. Stationary and propagating soliton lattice states are numerically found as a stable coexistent state.

Ginzburg-Landau theory of the Abrikosov lattice in a d-wave superconductor

We consider the Ginzburg-Landau free energy for the case of coexisting s- and d-wave order parameters, coupled to a gauge field, for a system in which the minimum energy state in the absence of magnetic fields is uniform d-wave. The eigenvalue equation arising from the coupled Ginzburg-Landau equations, in the presence of a uniform magnetic field, is implicitly non-linear and has no obvious closed-form solution. We employ a simple variational two-component wave function to minimize the quadratic free energy, and use linear combinations of these wave functions to construct variational Abrikosov lattice solutions which minimize the total free energy. The resulting lattice has an oblique structure, similar to that observed by small angle neutron scattering.

Amplitude equations near pattern forming instabilities for strongly driven ferromagnets

A transversally driven isotropic ferromagnet being under the influence of a static external and an uniaxial internal anisotropy field is studied. We consider the dissipative Landau-Lifshitz equation as the fundamental equation of motion and present a complete local bifurcation analysis in 1+1 dimensions. We study the stability of the spatially homogeneous magnetization against inhomogeneous perturbations and describe the dynamics above threshold via amplitude equations. The dependence of their coefficients on the physical parameters of the system is determined explicitly. We find soft- and hard-mode instabilities, transitions between sub- and supercritical behaviour, various bifurcations of higher codimension, and present a series of explicit bifurcation diagrams. The analysis of the codimension-2 point where the soft- and hard-mode instabilities coincide leads to a system of two coupled Ginzburg-Landau equations. The results are quite general because no restrictions are imposed on the physical parameters of the system.

Amplitude equations near pattern forming instabilities for strongly driven ferromagnets

A transversally driven isotropic ferromagnet being under the influence of a static external and an uniaxial internal anisotropy field is studied. We consider the dissipative Landau-Lifshitz equation as the fundamental equation of motion and treat it in $1+1$~dimensions. The stability of the spatially homogeneous magnetizations against inhomogeneous perturbations is analyzed. Subsequently the dynamics above threshold is described via amplitude equations and the dependence of their coefficients on the physical parameters of the system is determined explicitly. We find soft- and hard-mode instabilities, transitions between sub- and supercritical behaviour, various bifurcations of higher codimension, and present a series of explicit bifurcation diagrams. The analysis of the codimension-2 point where the soft- and hard-mode instabilities coincide leads to a system of two coupled Ginzburg-Landau equations.

Rayleigh-Bénard problem with imposed weak through-flow: Two coupled Ginzburg-Landau equations.

The Rayleigh-B\'enard instability of a horizontal shear flow in a narrow channel is governed by two competing unstable modes: traveling transverse and stationary longitudinal convection rolls. The dynamics of the mode amplitudes obeys two coupled Ginzburg-Landau equations, which we derive in the present article. For mathematical simplicity we assume free-slip boundaries at the channel sidewalls. The analytical discussion and the numerical simulation of these amplitude equations show transitions between the two convective structures. Propagating fronts, which separate areas of transverse rolls and longitudinal ones, occur in the parameter range where both patterns are stable. Coexisting uniform states with nonzero amplitudes of both modes are unstable solutions of the equations. Results are in qualitative agreement with recent experiments.

Fronts between Hopf- and Turing-type domains in a two-component reaction-diffusion system

Propagating and standing fronts between Hopf- and Turing-type domains are observed experimentally on a one-dimensional array of resistively coupled nonlinear LC-oscillators describable by a two-component reaction-diffusion equation. Numerical and experimental results are compared in particular with respect to front velocities. In the neighbourhood of a codimension-two point two coupled Ginzburg-Landau equations, derived by multiple scale methods, are useful approximation.

Counterpropagating periodic pulses in coupled Ginzburg-Landau equations

A recently observed stable regime in the form of periodically colliding counterpropagating wave packets (pulses) in an annular convection channel at very small positive overcriticalities is described analytically in terms of coupled Ginzburg-Landau equations. First, the existence of this regime is demonstrated in the framework of the simplest system including only the group-velocity difference, weak gain, and nonlinear dissipative coupling between two modes. In this approximation, the shape of the counterpropagating waves remains indefinite. It is demonstrated that additional dispersive terms, regarded as a small perturbation, provide shaping of the wave packets and also give rise to the deviation of the phase velocity from that for purely linear waves.

THE INTERACTION OF STRUCTURES AND SPATIOTEMPORAL CHAOS IN MODELS OF COUPLED WAVES

Dynamic and stochastic effects of the interaction of individual localized structures and lattices of structures of a nonlinear field are considered within one-dimensional large-box Ginzburg-Landau and Swift-Hohenberg models. The criterion of soliton “survival” in competition with counterpropagating modes is derived analytically for a quasigradient case. The criterion of linear stability of a spatially homogeneous regime, that is similar to the Benjamin-Feir condition, is obtained for coupled Ginzburg-Landau equations. The relation between the correlation dimensions of the space series of counterpropagating modes and the dimension of the time series is investigated. It is shown that within a quasigradient model of two weakly coupled counterpropagating modes, the field can be represented as a superposition of two fixed stochastic spatial lattices which correspond to these modes and move through one another at constant velocity. The dimension of the time series at a given point in space is close to the sum of dimensions of the spatial distributions of counterpropagating modes. With the increase of coupling, the interaction of coupled modes “loosens” the equilibrium state corresponding to fixed lattices and the dimension of the time series grows.

Inelastic collisions of pulses in coupled Ginzburg-Landau equations

A collision of two solitary pulses (SPs) is considered in a system of two coupled bistable Ginzburg-Landau equations. The bistability means that each seperate equation admits two stable solutions, viz., the trivial one and an SP. The system is considered in a regime where the real parts of its coefficients are small as compared to the imaginary parts, the only coupling terms being non-dissipative cubic ones. It is demonstrated that the collision between the free SPs belonging to the two subsystems results in their fusion into a bound state (bisoliton), or leads to a full annihilation, provided the group velocity difference between the two subsystems is smaller than a certain threshold value. The results obtained lend a natural interpretation to recent experiments with SPs of subcritical traveling-wave convection.

The role of defects in the transition between different symmetries in convective patterns

The properties of the transition between two patterns with different symmetries are studied in an experiment on thermal convection in a fluid layer heated from below. The defect structure is also analyzed and it is found that the unstable phase is present in the defect cores. The experimental results are in agreement with a model of three coupled Ginzburg-Landau equations.

Elimination of hysteresis in a system of coupled Ginzburg-Landau equations.

We numerically study the effect of a stabilizing quintic term and a destabilizing cubic term in the coupled Ginzburg-Landau equations that describe the onset of traveling-wave convection in binary fluid mixtures. We present bifurcation diagrams which show that for large enough group velocity and coupling between the counter-propagating traveling waves, the expected hysteresis at onset is not present. We also show how a measure of the convected heat transport and its period vary with scaled Rayleigh number. While this model can explain certain features of recent experiments, a number of difficult issues remain.

Dynamics of traveling waves near the onset of convection in binary fluid mixtures

We report a study of oscillatory convection in ethanol/water mixtures in a rectangular container which is heated from below. For negative values of the separation ratio, Ψ, convection begins with a linear instability to a state of counterpropagating traveling waves. For sufficiently weak Ψ, the effects of propagation, nonlinear competition between waves traveling in opposite directions, and feedback due to endwall reflections lead to complex dynamical behavior just above the onset of convection. This regime is characterized by weak nonlinearity and small convective amplitudes. The observed states consist basically of a slow alternation between left- and right-going waves. However, the detailed character of these states is extremely sensitive to the geometry of the convection cell and the Rayleigh number. Many of these observations can be understood in terms of recent calculations using coupled Ginzburg-Landau equations which take into account the convective nature of the instability at low Rayleigh numbers.

Pattern selection and transition to turbulence in propagating waves

We present a study of oscillatory convection in two experimental systems: ethanol-water mixtures in a rectangular container heated from below and a thin layer of nematic liquid crystals under low frequency ac voltage. In both systems the first bifurcation is the transition to travelling waves (TW) with finite wave vector and frequency. We report experimental observations of a sequence of spatial structures and dynamical behaviour of nonlinear TW in a regime of a weak nonlinearity. Most of the rich variety of spatial and dynamical behaviour which we observe in one-dimensional finite geometries has been reproduced by numerical simulations based on a simple model of coupled Ginzburg-Landau equations which considers only the combination of translation and finite geometry. More complicated spatio-temporal behaviour of TW in cells with two-dimensional geometry which initiated by defect nucleation is attributed to the mechanism of modulational instability of TW.

External noise and the origin and dynamics of structure in convectively unstable systems

Some basic concepts and earlier work on external noise and the convectively unstable Ginzburg-Landau equation are reviewed, and some of the ideas presented in the earlier work are investigated further and extended. In particular, further consideration is given to convective chaos---chaos which only occurs in a moving frame of reference; and slugs--localized structures which are surrounded by a stable stationary state. Some new results on secondary convective instabilities and on periodic systems with a spatially varying instability are discussed. Work on the coupled Ginzburg-Landau equation is reviewed. Actual physical systems are discussed.