Mean Field Effects for Counterpropagating Traveling Wave Solutions of Reaction-Diffusion Systems

In many problems, e.g., in combustion or solidification, one observes traveling waves that propagate with constant velocity and shape in the x direction, say, are independent of y and z and describe transitions between two equilibrium states, e.g., the burned and the unburned reactants. As parameters of the system are varied, these traveling waves can become unstable and give rise to waves having additional structure, such as traveling waves in the y and z directions, which can themselves be subject to instabilities as parameters are further varied. To investigate this scenario we consider a system of reaction-diffusion equations with a traveling wave solution as a basic state. We determine solutions bifurcating from the basic state that describe counterpropagating traveling waves in directions orthogonal to the direction of propagation of the basic state and determine their stability. Specifically, we derive long wave modulation equations for the amplitudes of the counterpropagating traveling waves that are coupled to an equation for a mean field, generated by the translation of the basic state in the direction of its propagation. The modulation equations are then employed to determine stability boundaries to long wave perturbations for both unidirectional and counterpropagating traveling waves. The stability analysis is delicate because the results depend on the order in which transverse and longitudinal perturbation wavenumbers are taken to zero. For the unidirectional wave we demonstrate that it is sufficient to consider the cases of (i) purely transverse perturbations, (ii) purely longitudinal perturbations, and (iii) longitudinal perturbations with a small transverse component. These yield Eckhaus type, zigzag type, and skew type instabilities, respectively. The latter arise as a specific result of interaction with the mean field. We also consider the degenerate case of very small group velocity, as well as other degenerate cases, which yield several additional instability boundaries. The stability analysis is then extended to the case of counterpropagating traveling waves

Effect of small-world topology on wave propagation on networks of excitable elements

We study excitation waves on a Newman-Watts small-world network model of coupled excitable elements. Depending on the global coupling strength, we find differing resilience to the added long-range links and different mechanisms of propagation failure. For high coupling strengths, we show agreement between the network and a reaction-diffusion model with additional mean-field term. Employing this approximation, we are able to estimate the critical density of long-range links for propagation failure.Comment: 19 pages, 8 figures and 5 pages supplementary materia

Excitation Waves on a Minimal Small-World Model

We examine traveling-wave solutions on a regular ring network with one additional long-range link that spans a distance d. The nodes obey the FitzHugh-Nagumo kinetics in the excitable regime. The additional shortcut induces a plethora of spatio-temporal behavior that is not present without it. We describe the underlying mechanisms for different types of patterns: propagation failure, period decreasing, bistability, shortcut blocking and period multiplication. For this purpose, we investigate the dependence on d, the network size, the coupling range in the original ring and the global coupling strength and present a phase diagram summarizing the different scenarios. Furthermore, we discuss the scaling behavior of the critical distance by analytical means and address the connection to spatially continuous excitable media.Comment: 14 pages, 11 figure

Transverse Patterns in Nonlinear Optical Resonators

The book is devoted to the formation and dynamics of localized structures (vortices, solitons) and extended patterns (stripes, hexagons, tilted waves) in nonlinear optical resonators such as lasers, optical parametric oscillators, and photorefractive oscillators. The theoretical analysis is performed by deriving order parameter equations, and also through numerical integration of microscopic models of the systems under investigation. Experimental observations, and possible technological implementations of transverse optical patterns are also discussed. A comparison with patterns found in other nonlinear systems, i.e. chemical, biological, and hydrodynamical systems, is given. This article contains the table of contents and the introductory chapter of the book.Comment: 37 pages, 14 figures. Table of contents and introductory chapter of the boo

Finite size effects near the onset of the oscillatory instability

A system of two complex Ginzburg - Landau equations is considered that applies at the onset of the oscillatory instability in spatial domains whose size is large (but finite) in one direction; the dependent variables are the slowly modulated complex amplitudes of two counterpropagating wavetrains. In order to obtain a well posed problem, four boundary conditions must be imposed at the boundaries. Two of them were already known, and the other two are first derived in this paper. In the generic case when the group velocity is of order unity, the resulting problem has terms that are not of the same order of magnitude. This fact allows us to consider two distinguished limits and to derive two associated (simpler) sub-models, that are briefly discussed. Our results predict quite a rich variety of complex dynamics that is due to both the modulational instability and finite size effects

Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials

We study a ring cavity filled with a slab of a right-handed material and a slab of a left-handed material. Both layers are assumed to be nonlinear Kerr media. First, we derive a model for the propagation of light in a left-handed material. By constructing a mean-field model, we show that the sign of diffraction can be made either positive or negative in this resonator, depending on the thicknesses of the layers. Subsequently, we demonstrate that the dynamical behavior of the modulation instability is strongly affected by the sign of the diffraction coefficient. Finally, we study the dissipative structures in this resonator and reveal the predominance of a two-dimensional up-switching process over the formation of spatially periodic structures, leading to the truncation of the homogeneous hysteresis cycle.Comment: 8 pages, 5 figure

Boundary effects on non-equilibrium localized structures in spatially extended systems

A study of the effects of system boundaries on bistable front propagation in nonequilibrium reaction-diffusion systems is presented. Two model partial differential equations displaying bistable fronts, with distinct experimental motivations and mathematical structure, are examined in detail utilizing simulations and perturbation techniques. We see that propagating fronts in both models bounce, trap, pin, or oscillate at the boundary, contingent on the imposed boundary condition, initial front speed and distance from the boundary. The similarities in front boundary interactions in these two models is traced to the fact that they display the same front instability (Ising-Bloch bifurcation) that controls the speed of propagation. A simplified dynamical picture based on ordinary differential equations that captures the essential features of front motion described by the original partial differential equations, is derived and analyzed for both models. In addition to addressing experimentally important boundary effects, we establish the universality of the Ising-Bloch bifurcation. Useful analytical insights into perturbative analysis of reaction diffusion systems are also presented

Non-Equilibrium Dynamics of Driven and Confined Colloidal Systems

[eng] In this thesis, I study the behavior of confined colloidal particles in aqueous suspension driven through an optical potential. For this purpose, I use micro-meter polystyrene particles, which I confine in the optical potential created with a system of optical tweezers. With the help of an Acousto Optical Deflector (AOD), which varies the laser position at a high frequency, I can create multiple quasi-simultaneous optical traps. This way, I can easily manipulate the particles and define the desired experimental conditions for the potential. I record videos of the particles' dynamics using optical microscopy. Thus, I obtain position information over time, which allows me to extract the necessary data to analyze the mechanisms that develop during forced transport. The results presented in this thesis expose the importance of Hydrodynamic Interactions (HI) when the transport of particles occurs due to a fluid drag. In addition, different situations are compared, including the change in the relative particle size concerning the separation between potential wells. In addition, I present a study on the emergence of solitons propagating in the opposite direction to the drag force. This situation, which appears when the experimental system is overcrowded, presents a mechanism where the transport dynamics accelerate, increasing the systems' efficiency.[spa] En esta tesis estudio el comportamiento de partículas coloidales en suspensión acuosa cuando son forzadas a moverse a través de un potencial óptico. Para ello, utilizo partículas micro- métricas de poliestireno, las cuales confino en el potencial óptico creado con un sistema de pinzas ópticas. Con la ayuda de un Deflector Acusto Óptico (AOD), el cual varía la posición del láser a una alta frecuencia, puedo crear múltiples trampas ópticas de manera casi simultánea. Esto me permite manipular las partículas con facilidad y definir las condiciones experimentales deseadas para el potencial. A través de microscopía óptica, obtengo imágenes en vídeo de la dinámica de las partículas. Así, obtengo la información de la posición a lo largo del tiempo, lo que me permite extraer los datos necesarios para analizar los mecanismos que se desarrollan durante el transporte forzado. Los resultados expuestos en esta tesis ponen de manifiesto la importancia de las Interacciones Hidrodinámicas (HI) en el transporte de partículas cuando son arrastradas por el fluido. Además, se comparan diferentes situaciones en las que se incluye el cambio en el tamaño relativo de las partículas respecto a la separación entre pozos de potencial. Además, presento un estudio sobre la aparición de solitones que se propagan en la dirección contraria en la que se ejerce la fuerza de arrastre. Esta situación, que aparece al sobrepoblar el sistema experimental, presenta un mecanismo en el que el transporte de materia se acelera, lo que incrementa la eficiencia