Pattern Formation And Solitons

Noise power spectra in spatially extended dynamical systems are investigated, using as a model the Complex Ginzburg-Landau equation with a stochastic term. Analytical and numerical investigations show that the temporal noise spectra are of 1/f^a form, where a=2-D/2 with D the spatial dimension of the system. This suggests that nonequilibrium order-disorder phase transitions may play a role for the universally observed 1/f noise.

Using the method of separation of variables, we developed a vector nonparaxial theory from the nonlinear wave equations in strong field approximation of a Kerr media. Investigating three waves on different frequencies, which satisfied the condition 2w3=w1+w2, we found that exact localized vortex parametric soliton solutions existed for the set of four nonlinear 3D+1vector wave equations. The solutions are obtained for a fixed angle between the main frequencies and the signal waves A=60 deg. This method is applicable for angular functions, which satisfy additional conditions. It is shown that exact parametric vortex solitary waves exist for solution with eigenrotation momentum L=1. The method is generalized for arbitrary number of signal waves appearing symmetrically in pairs in respect to w3. The existence of such type of vortex solitons, in the case of degenerate four-wave mixing is discussed. As all of the waves are localized and have momentum L=1, they are called optical mesons.

The motion of a thin layer of granular material on a plate undergoing sinusoidal vibrations is considered. We develop equations of motion for the local thickness and the horizontal velocity of the layer. The driving comes from the violent impact of the grains on the plate. A linear stability theory reveals that the waves are excited non-resonantly, in contrast to the usual Faraday waves in liquids. Together with the experimentally observed continuum scaling, the model suggests a close connection between the neutral curve and the dispersion relation of the waves, which agrees quite well with experiments. For strong hysteresis we find localized oscillon solutions.

It is shown that the renormalization group (RG) method for global analysis can be formulated in the context of the classical theory of envelopes: Several examples from partial differential equations are analyzed. The amplitude equations which are usually derived by the reductive perturbation theory are shown to be naturally derived as the equations describing the envelopes of the local solutions obtained in the perturbation theory.

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.

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.

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.

We present a new class of exact solutions for the so-called {\it Laplacian Growth Equation} describing the zero-surface-tension limit of a variety of 2D pattern formation problems. Contrary to common belief, we prove that these solutions are free of finite-time singularities (cusps) for quite general initial conditions and may well describe real fingering instabilities. At long times the interface consists of N separated moving Saffman-Taylor fingers, with ``stagnation points'' in between, in agreement with numerous observations. This evolution resembles the N-soliton solution of classical integrable PDE's.

A simple model is presented for the formation of rolling grain ripples on a flat sand bed by the oscillatory flow generated by a surface wave. An equation of motion is derived for the individual ripples, seen as "particles", on the otherwise flat bed. The model account for the initial apperance of the ripples, the subsequent coarsening of the ripples and the final equilibrium state. The model is related to physical parameters of the problem, and an analytical approximation for the equilibrium spacing of the ripples is developed. It is found that the spacing between the ripples scale with the square-root of the non-dimensional shear stress (the Shields parameter) on a flat bed. The results of the model are compared with measurements, and reasonable agreement between the model and the measurements is demonstrated.

Multiplicity of phase states within frequency locked bands in periodically forced oscillatory systems may give rise to front structures separating states with different phases. A new front instability is found within bands where ω forcing / ω system =2n ( n>1 ). Stationary fronts shifting the oscillation phase by π lose stability below a critical forcing strength and decompose into n traveling fronts each shifting the phase by π/n . The instability designates a transition from stationary two-phase patterns to traveling n -phase patterns.