Pattern Formation And Solitons

Nonlinear classical Hamiltonian lattices exhibit generic solutions in the form of discrete breathers. These solutions are time-periodic and (typically exponentially) localized in space. The lattices exhibit discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. Necessary ingredients for their occurence are the existence of upper bounds on the phonon spectrum (of small fluctuations around the groundstate) of the system as well as the nonlinearity in the differential equations. We will present existence proofs, formulate necessary existence conditions, and discuss structural stability of discrete breathers. The following results will be also discussed: the creation of breathers through tangent bifurcation of band edge plane waves; dynamical stability; details of the spatial decay; numerical methods of obtaining breathers; interaction of breathers with phonons and electrons; movability; influence of the lattice dimension on discrete breather properties; quantum lattices - quantum breathers. Finally we will formulate a new conceptual aproach capable of predicting whether discrete breather exist for a given system or not, without actually solving for the breather. We discuss potential applications in lattice dynamics of solids (especially molecular crystals), selective bond excitations in large molecules, dynamical properties of coupled arrays of Josephson junctions, and localization of electromagnetic waves in photonic crystals with nonlinear response.

We discuss the process by which energy, initially evenly distributed in a nonlinear lattice, can localize itself into large amplitude excitations. We show that, the standard modulational instability mechanism, which can initiate the process by the formation of small amplitude breathers, is completed efficiently, in the presence of discreteness, by energy exchange mechanisms between the nonlinear excitations which favor systematically the growth of the larger excitations. The process is however self regulated because the large amplitude excitations are finally trapped by the Peierls-Nabarro potential.

A new class of ``toy models'' for subaqueous bedform formation are proposed and examined. These models all show a similar mechanism of wavelength selection via bedform unification, and they may have applications to bedform stratigraphy. The models are also useful for exploring general issues of pattern formation and complexity in stochastically driven far from equilibrium systems.

We study the dynamics of the discrete nonlinear Schr{ö}dinger lattice initialized such that a very long transitory period of time in which standard Boltzmann statistics is insufficient is reached. Our study of the nonlinear system locked in this {\em non-Gibbsian} state focuses on the dynamics of discrete breathers (also called intrinsic localized modes). It is found that part of the energy spontaneously condenses into several discrete breathers. Although these discrete breathers are extremely long lived, their total number is found to decrease as the evolution progresses. Even though the total number of discrete breathers decreases we report the surprising observation that the energy content in the discrete breather population increases. We interpret these observations in the perspective of discrete breather creation and annihilation and find that the death of a discrete breather cause effective energy transfer to a spatially nearby discrete breather. It is found that the concepts of a multi-frequency discrete breather and of internal modes is crucial for this process. Finally, we find that the existence of a discrete breather tends to soften the lattice in its immediate neighborhood, resulting in high amplitude thermal fluctuation close to an existing discrete breather. This in turn nucleates discrete breather creation close to a already existing discrete breather.

We construct lattice Hamiltonians with homogeneous interaction potentials which allow for explicit breather solutions. Especially we obtain exponentially localized solutions for d -dimensional lattices with d=2,3 .

We propose a simple discrete stochastic model for calcium dynamics in living cells. Specifically, the calcium concentration distribution is assumed to give rise to a set of probabilities for the opening/closing of channels which release calcium thereby changing those probabilities. We study this model in one dimension, analytically in the mean-field limit of large number of channels per site N, and numerically for small N. As the number of channels per site is increased, the transition from a non-propagating region of activity to a propagating one changes in nature from one described by directed percolation to that of deterministic depinning in a spatially discrete system. Also, for a small number of channels a propagating calcium wave can leave behind a novel fluctuation-driven state, in a parameter range where the limiting deterministic model exhibits only single pulse propagation.

We reveal that even weak inherent discreteness of a nonlinear model can lead to instabilities of the localized modes it supports. We present the first example of an oscillatory instability of dark solitons, and analyse how it may occur for dark solitons of the discrete nonlinear Schrodinger and generalized Ablowitz-Ladik equations.

I review recent work on the ``phase diagram'' of the one-dimensional complex Ginzburg-Landau equation for system sizes at which chaos is extensive. Particular attention is paid to a detailed description of the spatiotemporally disordered regimes encountered. The nature of the transition lines separating these phases is discussed, and preliminary results are presented which aim at evaluating the phase diagram in the infinite-size, infinite-time, thermodynamic limit.

Dispersion curves to a oscillatory reaction-diffusion system with the self-consistent flow have obtained by means of numerical calculations. The flow modulates the shape of dispersion curves and characteristics of traveling waves. The point of inflection which separates the dispersion curves into two branches corresponding to trigger and phase waves, moves according to the value of the advection constant. The dynamics of phase wave in reaction-diffusion-advection equations has been studied by limit cycle perturbations. The dispersion relation obtained from the phase equation shows that the competition between diffusion and advection constants modulates the oscillation frequency from the bulk oscillation in the long-wave dynamics. Such a competition implies that phase waves with the flow have a wider variety of dynamics than waves without the flow.

Part of the field dependent dissipation in ferrofluids occurs due to the rotational motion of the ferromagnetic grains relative to the viscous flow of the carrier fluid. The classical theoretical description due to Shliomis uses a mesoscopic treatment of the particle motion to derive a relaxation equation for the non-equilibrium part of the magnetization. Complementary, the hydrodynamic approach of Liu involves only macroscopic quantities and results in dissipative Maxwell equations for the magnetic fields in the ferrofluid. Different stress tensors and constitutive equations lead to deviating theoretical predictions in those situations, where the magnetic relaxation processes cannot be considered instantaneous on the hydrodynamic time scale. We quantify these differences for two situations of experimental relevance namely a resting fluid in an oscillating oblique field and the damping of parametrically excited surface waves. The possibilities of an experimental differentiation between the two theoretical approaches is discussed.