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preprint1994arXiv

Front Structures in a Real Ginzburg-Landau Equation Coupled to a Mean Field

Localized traveling wave trains or pulses have been observed in various experiments in binary mixture convection. For strongly negative separation ratio, these pulse structures can be described as two interacting fronts of opposite orientation. An analytical study of the front solutions in a real Ginzburg-Landau equation coupled to a mean field is presented here as a first approach to the pulse solution. The additional mean field becomes important when the mass diffusion in the mixture is small as is the case in liquids. Within this framework it can lead to a hysteretic transition between slow and fast fronts when the Rayleigh number is changed.

preprint1999arXiv

Lax Pair Formulation and Multi-soliton Solution of the Integrable Vector Nonlinear Schrodinger Equation

The integrable vector nonlinear Schrodinger (vector NLS) equation is investigated by using Zakharov-Shabat (ZS) scheme. We get a Lax pair formulation of the vector NLS model. Multi-soliton solution of the equation is also derived by using inverse scattering method of ZS scheme. We also find that there is an elastic and inelastic collision of the bright and dark multi-solitons of the system.

preprint1996arXiv

Fluid Models for Kinetic Effects on Coherent Nonlinear Alfven Waves. II. Numerical Solutions

The influence of various kinetic effects (e.g. Landau damping, diffusive and collisional dissipation, and finite Larmor radius terms) on the nonlinear evolution of finite amplitude Alfvenic wave trains in a finite-beta environment is systematically investigated using a novel, kinetic nonlinear Schrodinger (KNLS) equation. The dynamics of Alfven waves is sensitive to the sense of polarization as well as the angle of propagation with respect to the ambient magnetic field. Numerical solution for the case with Landau damping reveals the formation of dissipative structures, which are quasi-stationary, S-polarized directional (and rotational) discontinuities which self-organize from parallel propagating, linearly polarized waves. Parallel propagating circularly polarized packets evolve to a few circularly polarized Alfven harmonics on large scales. Stationary arc-polarized rotational discontinuities form from obliquely propagating waves. Collisional dissipation, even if weak, introduces enhanced wave damping when beta is very close to unity. Cyclotron motion effects on resonant particle interactions introduce cyclotron resonance into the nonlinear Alfven wave dynamics.

preprint1997arXiv

The dynamics of vortex lines in the three-dimensional complex Ginzburg-Landau equation: instability, stretching, entanglement, and helices

The dynamics of curved vortex filaments is studied analytically and numerically in the framework of a three-dimensional complex Ginzburg-Landau equation (CGLE). It is shown that a straight vortex line is unstable with respect to spontaneous stretching and bending in a substantial range of parameters of the CGLE, resulting in formation of persistent entangled vortex configurations. The boundary of the three-dimensional instability in parameter space is determined. Near the stability boundary, the supercritical saturation of the instability is found, resulting in the formation of stable helicoidal vortices.

preprint1995arXiv

A Kink Soliton Characterizing Traffic Congestion

We study traffic congestion by analyzing a one dimensional traffic flow model. Developing an asymptotic method to investigate the long time behavior near a critical point, we derive the modified Korteweg-de Vries (mKdV) equation as the lowest order model. There are an infinite number of kink solitons to the mKdV equation, while it has been found by numerical simulations that the kink pattern arising in traffic congestion is uniquely determined irrespective of initial conditions. In order to resolve this selection problem, we consider higher order corrections to the mKdV equation and find that there is a kink soliton which can deform continuously with the perturbation represented by the addition of these corrections. With numerical confirmation, we show that this continuously deformable kink soliton characterizes traffic congestion. We also discuss the relationship between traffic congestion and slugging phenomenon in granular flow.

preprint1997arXiv

Stretching of vortex lines and generation of vorticity in the three-dimensional complex Ginzburg-Landau Equation

The dynamics of curved vortex filaments is studied analytically and numerically in the framework of a three-dimensional complex Ginzburg-Landau equation (CGLE). It is proved that a straight vortex line is unstable with respect to spontaneous stretching and bending in a certain range of parameters of the CGLE, resulting in formation of persistent entangled vortex configurations. The analysis shows that the standard approach relating the velocity of the filament with the local curvature, is insufficient to describe the instability and self-generation of vorticity.

preprint1998arXiv

Controlled Dynamics of Interfaces in a Vibrated Granular Layer

We present experimental study of a topological excitation, {\it interface}, in a vertically vibrated layer of granular material. We show that these interfaces, separating regions of granular material oscillation with opposite phases, can be shifted and controlled by a very small amount of an additional subharmonic signal, mixed with the harmonic driving signal. The speed and the direction of interface motion depends sensitively on the phase and the amplitude of the subharmonic driving.

preprint1998arXiv

Solitonic Gluons

We describe a physical mechanism for creating multi-soliton bound states where solitons are glued together by attraction between the beams they guide, `solitonic gluons'. We verify the concept of the solitonic gluons experimentally, observing a suppression of the repulsion between dark solitons due to an attractive force between bright guided beams.

preprint1995arXiv

MULTIPLE FRONT PROPAGATION IN A POTENTIAL NON-GRADIENT SYSTEM

A classification of dynamical systems in terms of their variational properties is reviewed. Within this classification, front propagation is discussed in a non-gradient relaxational potential flow. The model is motivated by transient pattern phenomena in nematics. A front propagating into an unstable homogenous state leaves behind an unstable periodic pattern, which decays via a second front and a second periodic state. An interface between unstable periodic states is shown to be a source of propagating fronts in opposite directions.

preprint1997arXiv

The Role of the Korteweg-de Vries Hierarchy in the N-Soliton Dynamics of the Shallow Water Wave Equation

We apply a multiple-time version of the reductive perturbation method to study long waves as governed by the shallow water wave model equation. As a consequence of the requirement of a secularity-free perturbation theory, we show that the well known N-soliton dynamics of the shallow water wave equation, in the particular case of $α=2 β$, can be reduced to the N-soliton solution that satisfies simultaneously all equations of the Korteweg-de Vries hierarchy.

preprint1996arXiv

Extended Symbolic Dynamics in Bistable CML: Existence and Stability of Fronts

We consider a diffusive Coupled Map Lattice (CML) for which the local map is piece-wise affine and has two stable fixed points. By introducing a spatio-temporal coding, we prove the one-to-one correspondence between the set of global orbits and the set of admissible codes. This relationship is applied to the study of the (uniform) fronts' dynamics. It is shown that, for any given velocity in $[-1,1]$, there is a parameter set for which the fronts with that velocity exist and their shape is unique. The dependence of the velocity of the fronts on the local map's discontinuity is proved to be a Devil's staircase. Moreover, the linear stability of the global orbits which do not reach the discontinuity follows directly from our simple map. For the fronts, this statement is improved and as a consequence, the velocity of all the propagating interfaces is computed for any parameter. The fronts are shown to be also nonlinearly stable under some restrictions on the parameters. Actually, these restrictions follow from the co-existence of uniform fronts and non-uniformly travelling fronts for strong coupling. Finally, these results are extended to some $C^{\infty}$ local maps.

preprint1996arXiv

Twisted and Nontwisted Bifurcations Induced by Diffusion

We discuss a diffusively perturbed predator-prey system. Freedman and Wolkowicz showed that the corresponding ODE can have a periodic solution that bifurcates from a homoclinic loop. When the diffusion coefficients are large, this solution represents a stable, spatially homogeneous time-periodic solution of the PDE. We show that when the diffusion coefficients become small, the spatially homogeneous periodic solution becomes unstable and bifurcates into spatially nonhomogeneous periodic solutions. The nature of the bifurcation is determined by the twistedness of an equilibrium/homoclinic bifurcation that occurs as the diffusion coefficients decrease. In the nontwisted case two spatially nonhomogeneous simple periodic solutions of equal period are generated, while in the twisted case a unique spatially nonhomogeneous double periodic solution is generated through period-doubling. Key Words: Reaction-diffusion equations; predator-prey systems; homoclinic bifurcations; periodic solutions.

preprint1997arXiv

Localized and Cellular Patterns in a Vibrated Granular Layer

We propose a phenomenological model for pattern formation in a vertically vibrated layer of granular material. This model exhibits a variety of stable cellular patterns including standing rolls and squares as well as localized objects (oscillons and worms), similar to recent experimental observations(Umbanhowar et al., 1996). The model is an amplitude equation for the parametrical instability coupled to the mass conservation law. The structure and dynamics of the solutions resemble closely the properties of localized and cellular patterns observed in the experiments.

preprint1998arXiv

Oscillon-type Structures and Their Interaction in a Swift-Hohenberg Model

Motivated by the observation of localized circular excitations (`oscillons') in vertically vibrated granular layers (P.B. Umbanhowar, F. Melo and H.L. Swinney, Nature 382 (1996) 793), we numerically investigate an extension of a Swift-Hohenberg model that exhibits a subcritical transition to square patterns. For sufficient subcriticality, stable oscillon structures are found. The localization mechanism is quite general and is due to non-adiabatic effects. Bound structures of oscillons of equal and opposite polarity are found with bound states of like polarity being less robust. Much of the phenomena are consistent with the experimental observations and suggest that oscillons are not specific to patterns in granular media or to parametrically driven systems. Experimental tests are suggested that would determine whether this minimal framework is sufficient to describe the phenomena.

preprint1997arXiv

Spiral Motion in a Noisy Complex Ginzburg-Landau Equation

The response of spiral waves to external perturbations in a stable regime of the two-dimensional complex Ginzburg-Landau equation (CGLE) is investigated. It is shown that the spiral core has a finite mobility and performs Brownian motion when driven by white noise. Combined with simulation results, this suggests that defect-free and quasi-frozen states in the noiseless CGLE are unstable against free vortex excitation at any non-zero noise strength.

preprint1997arXiv

Propagation and Structure of Planar Streamer Fronts

Streamers often constitute the first stage of dielectric breakdown in strong electric fields: a nonlinear ionization wave transforms a non-ionized medium into a weakly ionized nonequilibrium plasma. New understanding of this old phenomenon can be gained through modern concepts of (interfacial) pattern formation. As a first step towards an effective interface description, we determine the front width, solve the selection problem for planar fronts and calculate their properties. Our results are in good agreement with many features of recent three-dimensional numerical simulations. In the present long paper, you find the physics of the model and the interfacial approach further explained. As a first ingredient of this approach, we here analyze planar fronts, their profile and velocity. We encounter a selection problem, recall some knowledge about such problems and apply it to planar streamer fronts. We make analytical predictions on the selected front profile and velocity and confirm them numerically. (abbreviated abstract)

preprint1995arXiv

INVERSE SCATTERING TRANSFORM ANALYSIS OF STOKES-ANTI-STOKES STIMULATED RAMAN SCATTERING

Zakharov-Shabat--Ablowitz-Kaup-Newel-Segur representation for Stokes-anti-Stokes stimulated Raman scattering is proposed. Periodical waves, solitons and self-similarity solutions are derived. Transient and bright threshold solitons are discussed.

preprint1996arXiv

Scenarios of domain pattern formation in a reaction-diffusion system

We performed an extensive numerical study of a two-dimensional reaction-diffusion system of the activator-inhibitor type in which domain patterns can form. We showed that both multidomain and labyrinthine patterns may form spontaneously as a result of Turing instability. In the stable homogeneous system with the fast inhibitor one can excite both localized and extended patterns by applying a localized stimulus. Depending on the parameters and the excitation level of the system stripes, spots, wriggled stripes, or labyrinthine patterns form. The labyrinthine patterns may be both connected and disconnected. In the the stable homogeneous system with the slow inhibitor one can excite self-replicating spots, breathing patterns, autowaves and turbulence. The parameter regions in which different types of patterns are realized are explained on the basis of the asymptotic theory of instabilities for patterns with sharp interfaces developed by us in Phys. Rev. E. 53, 3101 (1996). The dynamics of the patterns observed in our simulations is very similar to that of the patterns forming in the ferrocyanide-iodate-sulfite reaction.

preprint1997arXiv

Phenomenological Theory for Spatiotemporal Chaos in Rayleigh-Benard Convection

We present a phenomenological theory for spatiotemporal chaos (STC) in Rayleigh-Benard convection, based on the generalized Swift-Hohenberg model. We apply a random phase approximation to STC and conjecture a scaling form for the structure factor $S(k)$ with respect to the correlation length $ξ_2$. We hence obtain analytical results for the time-averaged convective current $J$ and the time-averaged vorticity current $Ω$. We also define power-law behaviors such as $J \sim ε^μ$, $Ω\sim ε^λ$ and $ξ_2 \sim ε^{-ν}$, where $ε$ is the control parameter. We find from our theory that $μ= 1$, $ν\ge 1/2$ and $λ= 2 μ+ ν$ for phase turbulence and that $μ= 1$, $ν\ge 1/2$ and $λ= 2 μ+ 2 ν$ for spiral-defect chaos. These predictions, together with the scaling conjecture for $S(k)$, are confirmed by our numerical results. Finally we suggest that Porod's law, $S(k) \sim 1/ξ_2 k^3$ for large $k$, might be valid in STC.

preprint1996arXiv

Vortex Dynamics in Dissipative Systems

We derive the exact equation of motion for a vortex in two- and three- dimensional non-relativistic systems governed by the Ginzburg-Landau equation with complex coefficients. The velocity is given in terms of local gradients of the magnitude and phase of the complex field and is exact also for arbitrarily small inter-vortex distances. The results for vortices in a superfluid or a superconductor are recovered.

preprint1995arXiv

Aggregation Patterns in Stressed Bacteria

We study the formation of spot patterns seen in a variety of bacterial species when the bacteria are subjected to oxidative stress due to hazardous byproducts of respiration. Our approach consists of coupling the cell density field to a chemoattractant concentration as well as to nutrient and waste fields. The latter serves as a triggering field for emission of chemoattractant. Important elements in the proposed model include the propagation of a front of motile bacteria radially outward form an initial site, a Turing instability of the uniformly dense state and a reduction of motility for cells sufficiently far behind the front. The wide variety of patterns seen in the experiments is explained as being due the variation of the details of the initiation of the chemoattractant emission as well as the transition to a non-motile phase.

preprint1997arXiv

Nonlinear Dynamics of a Single Ferrofluid-Peak in an Oscillating Magnetic Field

If a magnetic field normal to the surface of a magnetic fluid is increased beyond a critical value a spontaneous deformation of the surface arises (normal field instability). The instability is subcritical and leads to peaks of a characteristic shape. We investigate the neighborhood of this instability experimentally under the influence of a temporal modulation of the magnetic field. We use a small vessel, where only one peak arises. The modulation can either be stabilizing or destabilizing, depending on the frequency and amplitude. We observe a cascade of odd-numbered response-periods up to period 11, and also a domain of even-numbered periods. We propose a minimal model involving a cutoff-condition which captures the essence of the experimental observations. PACS: 47.20.-k, 47.20.Ky, 75.50.Mm Keywords: magnetic fluid; nonlinear oscillator; subharmonic response; surface instability;

preprint1997arXiv

Propagation Failure in Excitable Media

We study a mechanism of pulse propagation failure in excitable media where stable traveling pulse solutions appear via a subcritical pitchfork bifurcation. The bifurcation plays a key role in that mechanism. Small perturbations, externally applied or from internal instabilities, may cause pulse propagation failure (wave breakup) provided the system is close enough to the bifurcation point. We derive relations showing how the pitchfork bifurcation is unfolded by weak curvature or advective field perturbations and use them to demonstrate wave breakup. We suggest that the recent observations of wave breakup in the Belousov-Zhabotinsky reaction induced either by an electric field or a transverse instability are manifestations of this mechanism.

preprint1998arXiv

Theory of Spike Spiral Waves in a Reaction-Diffusion System

We discovered a new type of spiral wave solutions in reaction-diffusion systems --- spike spiral wave, which significantly differs from spiral waves observed in FitzHugh-Nagumo-type models. We present an asymptotic theory of these waves in Gray-Scott model. We derive the kinematic relations describing the shape of this spiral and find the dependence of its main parameters on the control parameters. The theory does not rely on the specific features of Gray-Scott model and thus is expected to be applicable to a broad range of reaction-diffusion systems.

133 works