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arXiv1996

Local and Global Existence of Multiple Waves Near Formal Approximations

Assuming that a formal approximation of multiple waves has been obtained by matched asymptotic methods, we derive a {\em Spatial Shadowing lemma} to construct exact solutions near the formal approximation. In Part I, we consider a general singularly perturbed parabolic system. $$ εu_t + (-ε^2)^m D^{2m}_x u = f(u,εu_x,\cdots,(εD_x)^{2m-1} u,x,ε). $$ We show that if the formal approximation is precise, there is always an exact solution nearby for at least a short time. Examples include Cahn-Hilliard equation and viscous profile of conservation laws. In Part II, we show under some more assumptions, the process in Part I can be repeated to obtain global solutions if the formal approximation is a global one. Examples include reaction-diffusion equations and phase field equations.

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arXiv1996

Numerical Proof of Self-Similarity in Burgers' Turbulence

We study the statistical properties of solutions to Burgers' equation, $v_t + vv_x = νv_{xx}$, for large times, when the initial velocity and its potential are stationary Gaussian processes. The initial power spectral density at small wave numbers follows a steep power-law $E_0(k) \sim |k|^n$ where the exponent $n$ is greater than two. We compare results of numerical simulations with dimensional predictions, and with asymptotic analytical theory. The theory predicts self-similarity of statistical characteristics of the turbulence, and also leads to a logarithmic correction to the law of energy decay in comparison with dimensional analysis. We confirm numerically the existence of self-similarity for the power spectral density, and the existence of a logarithmic correction to the dimensional predictions.

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arXiv1996

Shadowing matching errors for wave-front-like solutions

Consider a singularly perturbed system $$εu_t=ε^2 u_{xx} + f(u,x,ε),\quad u\in {\Bbb R}^n,x\in{\Bbb R},t\geq 0. $$ Assume that the system has a sequence of regular and internal layers occurring alternatively along the $x$-direction. These ``multiple wave'' solutions can formally be constructed by matched asymptotic expansions. To obtain a genuine solution, we derive a {\em Spatial Shadowing Lemma} which assures the existence of an exact solution that is near the formal asymptotic series provided (1) the residual errors are small in all the layers, and (2) the matching errors are small along the lateral boundaries of the adjacent layers. The method should work on some other systems like $εu_t=-(-ε^2 D_{xx})^m u+ \dots.$

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arXiv1996

The Newell-Whitehead-Segel Equation for Traveling Waves

An equation to describe nearly one-dimensional traveling-waves patterns is put forward. This is a dispersive generalization of the classical Newell-Whitehead-Segel (NWS) equation. Transverse stability of plane waves is considered within the framework of this equation. It is shown that the dispersion terms drastically alter the stability. A necessary stability condition is obtained in the form of a transverse Benjamin-Feir criterion. If this condition is met, a quarter of the plane-wave existence band (in terms of the squared wave number) is unstable, while three quarters are transversely stable. Next, linear defects in the form of grain boundaries (GB's) are studied. An effective Burgers equation is derived from the dispersive NWS equation, in the framework of which a GB is tantamount to a shock wave. It is shown that the GB's are generic solutions. Asymmetric GB's are moving at a constant velocity, which is found. The integrability of the Burgers equation allows one as well to analyze transient processes and interactions between parallel GB's. The shock-wave solutions obtained in this work may also find applications in nonlinear fiber optics.

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arXiv1996

The Exactly Solvable Simplest Model for Queue Dynamics

We present an exactly solvable model for queue dynamics. Our model is very simple but provides the essential property for such dynamics. As an example, the model has the traveling cluster solution as well as the homogeneous flow solution. The model is the limiting case of Optimal Velocity (OV) model, which is proposed for the car following model to induce traffic jam spontaneously.

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arXiv1996

Delay of Vehicle Motion in Traffic Dynamics

We demonstrate that in Optimal Velocity Model (OVM) delay times of vehicles coming from the dynamical equation of motion of OVM almost explain the order of delay times observed in actual traffic flows without introducing explicit delay times. Delay times in various cases are estimated: the case of a leader vehicle and its follower, a queue of vehicles controlled by traffic lights and many-vehicle case of highway traffic flow. The remarkable result is that in most of the situation for which we can make a reasonable definition of a delay time, the obtained delay time is of order 1 second.

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arXiv1996

Large-Scale Flow and Spiral Core Instability in Rayleigh-Benard Convection

The spiral core instability, observed in large aspect ratio Rayleigh-Benard convection, is studied numerically in the framework of the Swift-Hohenberg equation coupled to a large-scale flow. It is shown that the instability leads to non-trivial core dynamics and is driven by the self-generated vorticity. Moreover, the recently reported transition from spirals to hexagons near the core is shown to occur only in the presence of a non-variational nonlinearity, and is triggered by the spiral core instability. Qualitative agreement between the simulations and the experiments is demonstrated.

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arXiv1996

Bifurcating bright and dark solitary waves of the nearly nonlinear cubic-quintic Schrodinger equation

The existence of bright and dark multi-bump solitary waves for Ginzburg-Landau type perturbations of the cubic-quintic Schrodinger equation is considered. The waves in question are not perturbations of known analytic solitary waves, but instead arise as a bifurcation from a heteroclinic cycle in a three dimensional ODE phase space. Using geometric singular perturbation techniques, regions in parameter space for which 1-bump bright and dark solitary waves will bifurcate are identified. The existence of N-bump dark solitary waves is shown via an application of the Exchange Lemma with Exponentially Small Error. N-bump bright solitary waves are shown to exist as a consequence of the previous work of S. Maier-Paape and this author.

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arXiv1996

A Mechanism for Spatio-Temporal Disorder in Bistable Reaction-Diffusion Systems

In bistable systems, the stability of front structures often influences the dynamics of extended patterns. We show how the combined effect of an instability to curvature modulations and proximity to a pitchfork front bifurcation leads to spontaneous nucleation of spiral waves along the front. This effect is demonstrated by direct simulations of a FitzHugh-Nagumo (FHN) model and by simulations of order parameter equations for the front velocity and curvature. Spontaneous spiral-wave nucleation often results in a state of spatio-temporal disorder involving repeated events of spiral wave nucleation, domain spliting and spiral wave annihilation.

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arXiv1997

Synchronization of globally-coupled phase oscillators: singularities and scaling for general couplings

The onset of collective behavior in a population of globally coupled oscillators with randomly distributed frequencies is studied for phase dynamical models with arbitrary coupling; the effect of a stochastic temporal variation in the frequencies is also included. The Fokker-Planck equation for the coupled Langevin system is reduced to a kinetic equation for the oscillator distribution function. Instabilities of the phase-incoherent state are studied by center manifold reduction to the amplitude dynamics of the unstable modes. Depending on the coupling, the coefficients in the normal form can be singular in the limit of weak instability when the diffusive effect of the noise is neglected. A detailed analysis of these singularities to all orders in the normal form expansion is presented. Physically, the singularities are interpreted as predicting an altered scaling of the entrained component near the onset of synchronization.

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arXiv1997

Transition from Spatiotemporal Chaos to Ideal Straight Rolls in Rayleigh-Benard Convection

For Rayleigh-Benard convection in a square cell with a fluid of Prandtl number one, we report experimental results on the transition between a stationary pattern of ideal straight rolls (ISR) and the spatiotemporal chaotic state of spiral defect chaos (SDC). In contrast to experiments in circular geometries, we found ISR states below a particular value of the control parameter and SDC states above this value. By characterizing the pattern with a global measure, the pattern entropy, we found that the transition from SDC to ISR showed similarities to phase transitions in equilibrium finite-size systems.

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arXiv1997

Liquefaction of a Horizontally Vibrated Granular Bed: Friction, Dilation and Segregation

We present experimental observations of the onset of flow (liquefaction) for horizontally vibrated granular materials. As the acceleration increases above certain value, the top layer of granular material liquefies, while the remainder of the layer moves with the shaker in solid body motion. With increasing acceleration, more of the layer becomes fluidized. The initial bifurcation is backward, and the amount of hysteresis depends mainly on frictional properties of the granular media. A small amount of fluidization by gas flow lifts the hysteresis. Modest differences in the frictional properties of otherwise identical particles leads to rapid segregation.

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arXiv1997

Phase Segregation Dynamics In Particle Systems with Long Range Interactions II: Interface motion

We study properties of the solutions of a family of second order integro-differential equations, which describe the large scale dynamics of a class of microscopic phase segregation models with particle conserving dynamics. We first establish existence and uniqueness as well as some properties of the instantonic solutions. Then we concentrate on formal asymptotic (sharp interface) limits. We argue that the obtained interface evolution laws (a Stefan-like problem and the Mullins-Sekerka solidification model) coincide with the ones which can be obtained in the analogous limits from the Cahn-Hilliard equation, the fourth order PDE which is the standard macroscopic model for phase segregation with one conservation law.

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arXiv1997

A Lubrication Model of Coating Flows over a Curved Substrate in Space

Consider the three-dimensional flow of a viscous Newtonian fluid upon an abitrarily curved substrate when the fluid film is thin as occurs in many draining, coating and biological flows. We derive a model of the dynamics of the film, the model being expressed in terms of the film thickness and the curvature tensor of the substrate. The model accurately includes the effects of the curvature of the substrate, via a physical multiple-scale approach, and gravity and inertia, via more rigorous centre manifold techniques. Numerical simulations exhibit some generic features of the dynamics of such thin fluid films on substrates with complex curvature.

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arXiv1997

On Envelope Dynamics In 2D Faraday Waves

A weakly nonlinear model for two-dimensional Faraday waves over infinite depth is derived and studied. Sideband instability of monochromatic standing waves as well as non-monochromatic solutions are studied analytically. Persistent irregular regimes are found numerically.

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arXiv1997

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.

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arXiv1997

Weakly nonlinear analysis in spatially extended systems as a formal perturbation scheme

The well known concept, to reduce the spatio-temporal dynamics beyond instabilities of trivial states to amplitude modulated patterns, is reviewed from the point of view of a formal perturbation expansion for general dissipative partial differential equations. For codimension one instabilities closed analytical formulas for all coefficients of the resulting amplitude equation are given, with no further restriction on the basic equations of motion. Both the autonomous and the explicitly time-dependent case are discussed. For the latter, the problem of strong resonances is addressed separately. The formal character of the expansion allows for an analysis of higher-codimension instabilities like the Turing-Hopf instability and for the discussion of principal limits of the amplitude approach in the present form.

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