Pattern Formation And Solitons

The statistics of wrinkling flame front is invetigated by the quantum filed theory methods. We dwell on the WKB approximation in the functional integral which is analogous to the Wyld functional integral in turbulence. The main contribution to statistics is due to a coupled field-force configuration. This configuration is related to a kink between metastable exact pole solutions of the Syvashinsky equation. These kinks are responsible for both the formation of new cusps and the rapid power-law acceleration of the mean flame-front. The problem of asymptotic stability of the solutions is discussed.

I consider a bistable reaction-diffusion system on the interface of deep fluid interacting with Marangoni flow. The method of matched asymptotic expansions is used to resolve the singularity at a sharp interface between the alternative states, and to compute the self-induced flow velocity advecting the domain boundary. It is shown that Marangoni flow serves as an efficient mechanism preventing the spread of the state with a higher surface tension when it is dynamically favorable.

Numerical studies together with asymptotic and spectral analysis establish regimes where soliton pairs in degenerate optical parametric oscillators fuse, repel, or form bound states. A novel bound state stabilized by coupled internal oscillations is predicted.

We analyze photo-refractive incoherent soliton beams and their interactions in Kerr-like nonlinear media. The field in each of M incoherently interacting components is calculated using an integrable set of coupled nonlinear Schrodinger equations. In particular, we obtain a general N-soliton solution, describing propagation of multi-soliton complexes and their collisions. The analysis shows that the evolution of such higher-order soliton beams is determined by coherent and incoherent contributions from fundamental solitons. Common features and differences between these internal interactions are revealed and illustrated by numerical examples.

This paper summarizes the motivations and results obtained so far in the frame of a particular non-linearization of Classical Electrodynamics, which was called Extended Electrodynamics. The main purpose pursued with this non-linear extension of the classical Maxwell's equations is to have a reliable field-theoretical approach in describing (3+1) soliton-like electromagnetic formations, in particular, to build an extended and finite field model of free photons and photon complexes. The first chapter gives a corresponding analysis of Maxwell theory and introduces the new equations. The second chapter gives a full account of the results, including the photon-like solutions, in the vacuum case. A new concept, called scale factor, is defined and successfully used. Two ways for describing the intrinsic angular momentum are given. Interference of two photon-like solutions is also considered. The third chapter considers interaction with external fields (continuous media) on the base of establishing correspondence between the physical concept of nondissipation and the mathematical concept of integrability of Pfaff systems. A large family of solutions is found, allowing a (3+1) interpretation of all known (1+1) soliton solutions.

We analyze motion of a fluxon in a weakly damped ac-driven long Josephson junction with a periodically modulated maximum Josephson current density. We demonstrate both analytically and numerically that a pure {\it ac} bias current can drive the fluxon at a {\it resonant} mean velocity determined by the driving frequency and the spatial period of the modulation, provided that the drive amplitude exceeds a certain threshold value. In the range of strongly ``relativistic'' mean velocities, the agreement between results of a numerical solution of the effective (ODE) fluxon equation of motion and analytical results obtained by means of the harmonic-balance analysis is fairly good; morever, a preliminary PDE result tends to confirm the validity of the collective-coordinate (PDE-ODE) reduction. At nonrelativistic mean velocities, the basin of attraction, in position-velocity space, for phase-locked solutions becomes progressively smaller as the mean velocity is decreased.

The jam phases in a two-dimensional cellular automata model of traffic flow are investigated by computer simulations. Two different types of the jam phases are found. The spatially diagonal long-range correlation obeys the power law at the low-density jam configurations. The diagonal correlation exponentially decays at the high-density jam. The exponent of the short-range correlation in the diagonal direction is introduced to define the transition between these two phases. We also discuss the stability of the jams.

We consider the following hypothesis: some of KdV equation shock-like waves are invariant with respect to the combination of the Galilean symmetry and KdV equation higher symmetries. Also we demonstrate our approach on the example of Burgers equation.

We present a new set of kinematic equations for front motion in bistable media. The equations extend earlier kinematic approaches by coupling the front curvature with the order parameter associated with a parity breaking front bifurcation. In addition to naturally describing the core region of rotating spiral waves the equations can be be used to study the nucleation of spiral-wave pairs along uniformly propagating fronts. The analysis of spiral-wave nucleation reduces to the simpler problem of droplet, or domain, nucleation in one space dimension.

An interphase boundary may be immobilized due to nonlinear diffractional interactions in a feedback optical device. This effect reminds of the Turing mechanism, with the optical field playing the role of a diffusive inhibitor. Two examples of pattern formation are considered in detail: arrays of kinks in 1d, and solitary spots in 2d. In both cases, a large number of equilibrium solutions is possible due to the oscillatory character of diffractional interaction.