Pattern Formation And Solitons

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.

A proper formulation in the perturbative renormalization group method is presented to deduce amplitude equations. The formulation makes it possible not only avoiding a serious difficulty in the previous reduction to amplitude equations by eliminating all of the secular terms but also consistent derivation of higher-order correction to amplitude equations.

We have studied shrinkage crack patterns which form when a thin layer of an alumina/water slurry dries. Both isotropic and directional drying were studied. The dynamics of the pattern formation process and the geometric properties of the isotropic crack patterns are similar to what is expected from recent models, assuming weak disorder. There is some evidence for a gradual increase in disorder as the drying layer become thinner, but no sudden transition, in contrast to what has been seen in previous experiments. The morphology of the crack patterns is influenced by drying gradients and front propagation effects, with sharp gradients having a strong orienting and ordering effect.

We studied the stability of linear vortex filaments in 3-dimensional (3D) excitable media, using both analytical and numerical methods. We found an intrinsic 3D instability of vortex filaments that is diffusion-induced, and is due to the slower diffusion of the inhibitor. This instability can result either in a single helical filament or in chaotic scroll breakup, depending on the specific kinetic model. When the 2-dimensional dynamics were in the chaotic regime, filament instability occurred via on-off intermittency, a failure of chaos synchronization in the third dimension.

Spiral waves are a ubiquitous feature of the nonequilibrium dynamics of a great variety of excitable systems. In the limit of a large separation in timescale between fast excitation and slow recovery, one can reduce the spiral problem to one involving the motion of a free surface separating the excited and quiescent phases. In this work, we study the free surface problem in the limit of small diffusivity for the slow field variable. Specifically, we show that a previously found spiral solution in the diffusionless limit can be extended to finite diffusivity, without significant alteration. This extension involves the creation of a variety of boundary layers which cure all the undesirable singularities of the aforementioned solution. The implications of our results for the study of spiral stability are briefly discussed.

For the time-dependent Ginzburg-Landau equation on the real line, we construct solutions which converge, as x→±∞ , to periodic stationary states with different wave-numbers η ± . These solutions are stable with respect to small perturbations, and approach as t→+∞ a universal diffusive profile depending only on the values of η ± . This extends a previous result of Bricmont and Kupiainen by removing the assumption that η ± should be close to zero. The existence of the diffusive profile is obtained as an application of the theory of monotone operators, and the long-time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term.

We find a new type of optical vector soliton that originates from trapping of a dipole mode by a soliton-induced waveguide. These solitons, which appear as a consequence of the vector nature of the two component system, are more stable than the previously found optical vortex-mode solitons and represent a new type of extremely robust nonlinear vector structure.

We study parametric resonance of interacting waves having the same wave vector and frequency. In addition to the well-known period-doubling instability we show that under certain conditions the instability is caused by a Hopf bifurcation leading to quasiperiodic traveling waves. It occurs, for example, if the group velocities of both waves have different signs and the damping is weak. The dynamics above the threshold is briefly discussed. Examples concerning ferromagnetic spin waves and surface waves of ferro fluids are discussed.

We present the first large scale numerical simulation of three-dimensional Rayleigh-Bénard convection near onset, under free-free boundary conditions for a fluid of Prandtl number σ=0.5 . We find that a spatiotemporally chaotic state emerges immediately above onset, which we investigate as a function of the reduced control parameter ϵ . We conclude that the transition from conduction to spatiotemporal chaos is second order and of ``mean field'' character. We also present a simple theory for the time-averaged convective current. Finally, we show that the time-averaged structure factor satisfies a scaling behavior with respect to the correlation length near onset.

The vortex-like solutions are studied in the framework of the gauge model of disclinations in elastic continuum. A complete set of model equations with disclination driven dislocations taken into account is considered. Within the linear approximation an exact solution for a low-angle wedge disclination is found to be independent from the coupling constants of the theory. As a result, no additional dimensional characteristics (like the core radius of the defect) are involved. The situation changes drastically for 2\pi vortices where two characteristic lengths, l_\phi and l_W, become of importance. The asymptotical behaviour of the solutions for both singular and nonsingular 2\pi vortices is studied. Forces between pairs of vortices are calculated.