title:
Existence and Stability of Propagating Fronts for an Autocatalytic
  Reaction-Diffusion System

abstract:
  We study a one-dimensional reaction-diffusion system which describes an
isothermal autocatalytic chemical reaction involving both a quadratic (A + B ->
2B) and a cubic (A + 2B -> 3B) autocatalysis. The parameters of this system are
the ratio D = D_B/D_A of the diffusion constants of the reactant A and the
autocatalyst B, and the relative activity k of the cubic reaction. First, for
all values of D > 0 and k >= 0, we prove the existence of a family of
propagating fronts (or travelling waves) describing the advance of the
reaction. In particular, in the quadratic case k=0, we recover the results of
Billingham and Needham [BN]. Then, if D is close to 1 and k is sufficiently
small, we prove using energy functionals that these propagating fronts are
stable against small perturbations in exponentially weighted Sobolev spaces.
This extends to our system part of the stability results which are known for
the scalar Fisher equation.


title:
Stretching of vortex lines and generation of vorticity in the
  three-dimensional complex Ginzburg-Landau Equation

abstract:
  The dynamics of curved vortex filaments is studied analytically and
numerically in the framework of a three-dimensional complex Ginzburg-Landau
equation (CGLE). It is proved that a straight vortex line is unstable with
respect to spontaneous stretching and bending in a certain range of parameters
of the CGLE, resulting in formation of persistent entangled vortex
configurations. The analysis shows that the standard approach relating the
velocity of the filament with the local curvature, is insufficient to describe
the instability and self-generation of vorticity.


title:
Stability of a Nonequilibrium Interface in a Driven Phase Segregating
  System

abstract:
  We investigate the dynamics of a nonequilibrium interface between coexisting
phases in a system described by a Cahn-Hilliard equation with an additional
driving term. By means of a matched asymptotic expansion we derive equations
for the interface motion. A linear stability analysis of these equations
results in a condition for the stability of a flat interface. We find that the
stability properties of a flat interface depend on the structure of the driving
term in the original equation.


title: Amplitude Equation for Lattice Maps, a Renormalization Group Approach

abstract:
  We consider the development of instabilities of homogeneous stationary
solutions of discrete time lattice maps. Under some generic hypothesis we
derive an amplitude equation which is the space-time continuous Ginzburg-Landau
equation. Using dynamical renormalization group methods we control the accuracy
of this approximation in a large ball of its basin of attraction.


title: Amplitude Equations for Electrostatic Waves: multiple species

abstract:
  The amplitude equation for an unstable electrostatic wave is analyzed using
an expansion in the mode amplitude $A(t)$. In the limit of weak instability,
i.e. $\gamma\to 0^+$ where $\gamma$ is the linear growth rate, the nonlinear
coefficients are singular and their singularities predict the dependence of
$A(t)$ on $\gamma$. Generically the scaling $|A(t)|=\gamma^{5/2}r(\gamma t)$ as
$\gamma\to 0^+$ is required to cancel the coefficient singularities to all
orders. This result predicts the electric field scaling $|E_k|\sim\gamma^{5/2}$
will hold universally for these instabilities (including beam-plasma and
two-stream configurations) throughout the dynamical evolution and in the
time-asymptotic state. In exceptional cases, such as infinitely massive ions,
the coefficients are less singular and the more familiar trapping scaling
$|E_k|\sim\gamma^2$ is recovered.

categories: patt-sol cond-mat nlin.PS physics.plasm-ph


title:
Phase Space Derivation of a Variational Principle for One Dimensional
  Hamiltonian Systems

abstract:
  We consider the bifurcation problem u'' + \lambda u = N(u) with two point
boundary conditions where N(u) is a general nonlinear term which may also
depend on the eigenvalue \lambda. A new derivation of a variational principle
for the lowest eigenvalue \lambda is given. This derivation makes use only of
simple algebraic inequalities and leads directly to a more explicit expression
for the eigenvalue than what had been given previously.


title:
Asymptotic description of transients and synchronized states of globally
  coupled oscillators

abstract:
  A two-time scale asymptotic method has been introduced to analyze the
multimodal mean-field Kuramoto-Sakaguchi model of oscillator synchronization in
the high-frequency limit. The method allows to uncouple the probability density
in different components corresponding to the different peaks of the oscillator
frequency distribution. Each component evolves toward a stationary state in a
comoving frame and the overall order parameter can be reconstructed by
combining them. Synchronized phases are a combination of traveling waves and
incoherent solutions depending on parameter values. Our results agree very well
with direct numerical simulations of the nonlinear Fokker-Planck equation for
the probability density. Numerical results have been obtained by finite
differences and a spectral method in the particular case of bimodal (symmetric
and asymmetric) frequency distribution with or without external field. We also
recover in a very easy and intuitive way the only other known analytical
results: those corresponding to reflection-symmetric bimodal frequency
distributions near bifurcation points.

categories: patt-sol adap-org cond-mat.dis-nn nlin.AO nlin.PS


title: Fingering Instability in a Water-Sand Mixture

abstract:
  The temporal evolution of a water-sand interface driven by gravity is
experimentally investigated. By means of a Fourier analysis of the evolving
interface the growth rates are determined for the different modes appearing in
the developing front. To model the observed behavior we apply the idea of the
Rayleigh-Taylor instability for two stratified fluids. Carrying out a linear
stability analysis we calculate the growth rates from the corresponding
dispersion relations for finite and infinite cell sizes. Based on the
theoretical results the viscosity of the suspension is estimated to be
approximately 100 times higher than that of pure water, in agreement with other
experimental findings.


title:
Time-periodic phases in populations of nonlinearly coupled oscillators
  with bimodal frequency distributions

abstract:
  The mean field Kuramoto model describing the synchronization of a population
of phase oscillators with a bimodal frequency distribution is analyzed (by the
method of multiple scales) near regions in its phase diagram corresponding to
synchronization to phases with a time periodic order parameter. The richest
behavior is found near the tricritical point were the incoherent, stationarily
synchronized, ``traveling wave'' and ``standing wave'' phases coexist. The
behavior near the tricritical point can be extrapolated to the rest of the
phase diagram. Direct Brownian simulation of the model confirms our findings.


title:
Weakly nonlinear analysis in spatially extended systems as a formal
  perturbation scheme

abstract:
  The well known concept, to reduce the spatio-temporal dynamics beyond
instabilities of trivial states to amplitude modulated patterns, is reviewed
from the point of view of a formal perturbation expansion for general
dissipative partial differential equations. For codimension one instabilities
closed analytical formulas for all coefficients of the resulting amplitude
equation are given, with no further restriction on the basic equations of
motion. Both the autonomous and the explicitly time-dependent case are
discussed. For the latter, the problem of strong resonances is addressed
separately. The formal character of the expansion allows for an analysis of
higher-codimension instabilities like the Turing-Hopf instability and for the
discussion of principal limits of the amplitude approach in the present form.


title:
Phenomenological Theory for Spatiotemporal Chaos in Rayleigh-Benard
  Convection

abstract:
  We present a phenomenological theory for spatiotemporal chaos (STC) in
Rayleigh-Benard convection, based on the generalized Swift-Hohenberg model. We
apply a random phase approximation to STC and conjecture a scaling form for the
structure factor $S(k)$ with respect to the correlation length $\xi_2$. We
hence obtain analytical results for the time-averaged convective current $J$
and the time-averaged vorticity current $\Omega$. We also define power-law
behaviors such as $J \sim \epsilon^\mu$, $\Omega \sim \epsilon^\lambda$ and
$\xi_2 \sim \epsilon^{-\nu}$, where $\epsilon$ is the control parameter. We
find from our theory that $\mu = 1$, $\nu \ge 1/2$ and $\lambda = 2 \mu + \nu$
for phase turbulence and that $\mu = 1$, $\nu \ge 1/2$ and $\lambda = 2 \mu + 2
\nu$ for spiral-defect chaos. These predictions, together with the scaling
conjecture for $S(k)$, are confirmed by our numerical results. Finally we
suggest that Porod's law, $S(k) \sim 1/\xi_2 k^3$ for large $k$, might be valid
in STC.

categories: patt-sol chao-dyn cond-mat.stat-mech nlin.CD nlin.PS physics.flu-dyn


title:
Optical characterization of chevron texture formation in nematic
  electroconvection

abstract:
  We characterize the director structures in electroconvection patterns of
nematic liquid crystals by means of polarising microscopy. This study was
stimulated by a theory put forward by Rossberg et al. (patt-sol/9701005) who
propose a mechanism for chevron texture formation. We characterize the relation
between the wave vector modulation and in-plane twist modulations of the
director field. In addition to the standard optical setup, a circular analyzer
is used. The results provide new insight into the interplay of director and
wave vector field leading to the chevron texture and fluctuating conductive
rolls. Our experimental results confirm the theoretical predictions.


title: Nonequilibrium phase separation in traffic flows

abstract:
  Traffic jam in an optimal velocity model with the backward reference function
is analyzed. An analytic scaling solution is presented near the critical point
of the phase separation. The validity of the solution has been confirmed from
the comparison with the simulation of the model.


title:
Breaking the symmetry in bimodal frequency distributions of globally
  coupled oscillators

abstract:
  The mean-field Kuramoto model for synchronization of phase oscillators with
an asymmetric bimodal frequency distribution is analyzed. Breaking the
reflection symmetry facilitates oscillator synchronization to rotating wave
phases. Numerical simulations support the results based of bifurcation theory
and high-frequency calculations. In the latter case, the order parameter is a
linear superposition of parameters corresponding to rotating and
counterrotating phases.


title:
Universal algebraic relaxation of fronts propagating into an unstable
  state

abstract:
  We analyze ``pulled'' or ``linearly marginally stable'' fronts propagating
into unstable states. While ``pushed'' fronts into meta- and unstable states
relax exponentially, pulled fronts relax algebraically, and simultaneously the
standard derivation of effective interface equations breaks down. We calculate
all universal relaxation terms of uniformly translating pulled fronts. The
leading $1/t$ and $1/t^{3/2}$ corrections to the velocity are determined by the
dispersion relation of the linearized equation only. Our analysis sheds new
light on the propagation mechanism of pulled fronts.


title: Surface tension and dynamics of fingering patterns

abstract:
  We study the minimal class of exact solutions of the Saffman-Taylor problem
with zero surface tension, which contains the physical fixed points of the
regularized (non-zero surface tension) problem. New fixed points are found and
the basin of attraction of the Saffman-Taylor finger is determined within that
class. Specific features of the physics of finger competition are identified
and quantitatively defined, which are absent in the zero surface tension case.
This has dramatic consequences for the long-time asymptotics, revealing a
fundamental role of surface tension in the dynamics of the problem. A
multifinger extension of microscopic solvability theory is proposed to
elucidate the interplay between finger widths, screening and surface tension.


title:
Strongly nonlinear convection in binary fluids: Minimal model for
  extended states using symmetry decomposed modes

abstract:
  Spatially extended stationary and traveling states in the strongly nonlinear
regime of convection in layers of binary fluid mixtures heated from below are
described by a few-mode-model. It is derived from the proper hydrodynamic
balance equations including experimentally relevant boundary conditions with a
non-standard Galerkin approximation that uses numerically obtained, symmetry
decomposed modes. Properties of the model are elucidated and compared with full
numerical solutions of the field equations.


title:
Order Parameter Equations for Front Transitions: Nonuniformly Curved
  Fronts

abstract:
  Kinematic equations for the motion of slowly propagating, weakly curved
fronts in bistable media are derived. The equations generalize earlier
derivations where algebraic relations between the normal front velocity and its
curvature are assumed. Such relations do not capture the dynamics near
nonequilibrium Ising-Bloch (NIB) bifurcations, where transitions between
counterpropagating Bloch fronts may spontaneously occur. The kinematic
equations consist of coupled integro-differential equations for the front
curvature and the front velocity, the order parameter associated with the NIB
bifurcation. They capture the NIB bifurcation, the instabilities of Ising and
Bloch fronts to transverse perturbations, the core structure of a spiral wave,
and the dynamic process of spiral wave nucleation.


title: Noise sustained pattern growth: Bulk versus boundary effects

abstract:
  The effect of thermally generated bulk stochastic forces on the statistical
growth dynamics of forwards bifurcating propagating macroscopic patterns is
compared with the influence of fluctuations at the boundary of a semiinfinite
system, $0<x$. To that end the linear complex Ginzburg-Landau amplitude
equation with additive stochastic forcing is solved by a spatial Laplace
transformation in the presence of arbitrary boundary conditions for the
fluctuations of the pattern amplitude at $x=0$. A situation where the latter
are advected with an imposed through-flow from an outside upstream part towards
the inlet boundary at $x=0$ is investigated in more detail. The spatiotemporal
growth behavior in the convectively unstable regime is compared with recent
work by Swift, Babcock, and Hohenberg [Physica A 204, 625 (1994)] where a
special boundary condition is imposed.


title: Nature of Roll to Spiral-Defect-Chaos Transition

abstract:
  We investigate the nature of the parallel-roll to spiral-defect-chaos (SDC)
transition in Rayleigh-Benard convection, based on the generalized
Swift-Hohenberg model. We carry out extensive, systematic numerical studies by,
on one branch, increasing the control parameter gradually from the parallel
roll regime to the SDC regime and, on the other branch, decreasing it in the
opposite manner. We find that the data of several time-averaged global
quantities all form hysteretic loops from the two branches. We also discuss
several possible scenarios for the transition and analyze our data for SDC
accordingly. We conclude that the roll-to-SDC transition is first-order in
character and that the correlation length diverges at the conduction to
convection onset. We further postulate that this transition can be understood
somewhat similar to the hexagon-to-roll transition in non-Boussinesq fluids.
Finally we comment on the differences between our conclusion and those in two
experiments.


title:
Spatio-temporal dynamics of coupled array of Murali-Lakshmanan-Chua
  circuits

abstract:
  The circuit recently proposed by Murali, Lakshmanan and Chua (MLC) is one of
the simplest non-autonomous nonlinear electronic circuits which shows a variety
of dynamical phenomena including various bifurcations, chaos and so on. In this
paper we study the spatio-temporal dynamics in one and two dimensional arrays
of coupled MLC circuits both in the absence as well as in the presence of
external periodic force. In the absence of any external force, the propagation
phenomena of travelling wave front and its failure have been observed from
numerical simulations....


title:
Amplitude equations for coupled electrostatic waves in the limit of weak
  instability

abstract:
  We consider the simplest instabilities involving multiple unstable
electrostatic plasma waves corresponding to four-dimensional systems of mode
amplitude equations. In each case the coupled amplitude equations are derived
up to third order terms. The nonlinear coefficients are singular in the limit
in which the linear growth rates vanish together. These singularities are
analyzed using techniques developed in previous studies of a single unstable
wave. In addition to the singularities familiar from the one mode problem,
there are new singularities in coefficients coupling the modes. The new
singularities are most severe when the two waves have the same linear phase
velocity and satisfy the spatial resonance condition $k_2=2k_1$. As a result
the short wave mode saturates at a dramatically smaller amplitude than that
predicted for the weak growth rate regime on the basis of single mode theory.
In contrast the long wave mode retains the single mode scaling. If these
resonance conditions are not satisfied both modes retain their single mode
scaling and saturate at comparable amplitudes.


title: Complexation of Wavenumbers in Solitons

abstract:
  It is shown that, by letting wavenumbers and frequencies complex in Hirota's
bilinear method, new classes of exact solutions of soliton equations can be
obtained systematically. They include not only singular or N-homoclinic
solutions but also N-wavepacket solutions.


title: Localized moving breathers in a 2-D hexagonal lattice

abstract:
  We show for the first time that highly localized in-plane breathers can
propagate in specific directions with minimal lateral spreading in a model 2-D
hexagonal non-linear lattice. The lattice is subject to an on-site potential in
addition to longitudinal nonlinear inter-particle interactions. This study
investigates the prediction that stable breather-like solitons could be formed
as a result of energetic scattering events in a given layered crystal and would
propagate in atomic-chain directions in certain atomic planes. This prediction
arose from a long-term study of previously unexplained dark lines in natural
crystals of muscovite mica.


title:
Uses of Envelopes for Global and Asymptotic Analysis; geometrical
  meaning of renormalization group equation

abstract:
  We give a comprehensive review of the renormalization group method for global
and asymptotic analysis, putting an emphasis on the relevance to the classical
theory of envelopes and the existence of invariant manifolds of the dynamics
under consideration. We clarify that an essential point of the method is to
convert the problem from solving differential equations to obtaining suitable
initial (or boundary) conditions.
  We mention that the notion of envelopes is also useful for constructing
global and asymptotic behavior of wave functions of quantum systems such as the
ones with the quartic potential or double-well potential.


title: Kinematic Equations for Front Motion and Spiral-Wave Nucleation

abstract:
  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.


title: Spiral Motion in a Noisy Complex Ginzburg-Landau Equation

abstract:
  The response of spiral waves to external perturbations in a stable regime of
the two-dimensional complex Ginzburg-Landau equation (CGLE) is investigated. It
is shown that the spiral core has a finite mobility and performs Brownian
motion when driven by white noise. Combined with simulation results, this
suggests that defect-free and quasi-frozen states in the noiseless CGLE are
unstable against free vortex excitation at any non-zero noise strength.


title: Extended Electrodynamics I. Basic Notions, Principles and Equations

abstract:
  This paper aims to present an elaborate view on the motivation and
realization of the idea to extend Maxwell's electrodynamics to Extended
Electrodynamics in a reasonable and appropriate way in order to make it
possible to describe electromagnetic (3+1)-soliton-like objects in vacuum and
in the presence of continuous media (external fields), exchanging
energy-momentum with the electromagnetic field.


title:
Extended Electrodynamics: II. Properties and invariant characteristics
  of the non-linear vacuum solutions

abstract:
  This paper aims to consider the general properties of the non-linear
solutions to the vacuum equations of Extended Electrodynamics. The *-invariance
and the conformal invariance of the equations are mentioned. It is also proved
that all non-linear solutions have zero invariants: F_{\mu\nu}F^{\mu\nu} =
(*F)_{\mu\nu}F^{\mu\nu} = 0. The three invariant characteristics of the
non-linear solutions: amplitude, phase and scale factor are introdiced and
discussed.


title: Propagation Failure in Excitable Media

abstract:
  We study a mechanism of pulse propagation failure in excitable media where
stable traveling pulse solutions appear via a subcritical pitchfork
bifurcation. The bifurcation plays a key role in that mechanism. Small
perturbations, externally applied or from internal instabilities, may cause
pulse propagation failure (wave breakup) provided the system is close enough to
the bifurcation point. We derive relations showing how the pitchfork
bifurcation is unfolded by weak curvature or advective field perturbations and
use them to demonstrate wave breakup. We suggest that the recent observations
of wave breakup in the Belousov-Zhabotinsky reaction induced either by an
electric field or a transverse instability are manifestations of this
mechanism.


title:
Extended Electrodynamics: III. Free Photons and (3+1)-Soliton- like
  Vacuum Solutions

abstract:
  This paper aims to give explicitly all non-linear vacuum solutions to our
non-linear field equations, and to define in a coordinate free manner the
important subclass of non-linear solutions, which we call almost photon-like.
By means of a correct introduction of the local and integral intrinsic angular
momentums of these solutions, we saparate the photon-like solutions through the
requirement their integral intrinsic angular momentums to be equal to the
Planck's constant. Finally, we consider such solutions, moving radially to or
from a given center, using standard spherical coordinates.


title: Nonlinear Competition Between Small and Large Hexagonal Patterns

abstract:
  Recent experiments by Kudrolli, Pier and Gollub on surface waves,
parametrically excited by two-frequency forcing, show a transition from a small
hexagonal standing wave pattern to a triangular ``superlattice'' pattern. We
show that generically the hexagons and the superlattice wave patterns bifurcate
simultaneously from the flat surface state as the forcing amplitude is
increased, and that the experimentally-observed transition can be described by
considering a low-dimensional bifurcation problem. A number of predictions come
out of this general analysis.


title: Annular electroconvection with shear

abstract:
  We report experiments on convection driven by a radial electrical force in
suspended annular smectic A liquid crystal films. In the absence of an
externally imposed azimuthal shear, a stationary one-dimensional (1D) pattern
consisting of symmetric vortex pairs is formed via a supercritical transition
at the onset of convection. Shearing reduces the symmetries of the base state
and produces a traveling 1D pattern whose basic periodic unit is a pair of
asymmetric vortices. For a sufficiently large shear, the primary bifurcation
changes from supercritical to subcritical. We describe measurements of the
resulting hysteresis as a function of the shear at radius ratio $\eta \sim
0.8$. This simple pattern forming system has an unusual combination of
symmetries and control parameters and should be amenable to quantitative
theoretical analysis.

categories: patt-sol cond-mat.soft nlin.PS physics.flu-dyn


title: Mutator Dynamics on a Smooth Evolutionary Landscape

abstract:
  We investigate a model of evolutionary dynamics on a smooth landscape which
features a ``mutator'' allele whose effect is to increase the mutation rate. We
show that the expected proportion of mutators far from equilibrium, when the
fitness is steadily increasing in time, is governed solely by the transition
rates into and out of the mutator state. This results is a much faster rate of
fitness increase than would be the case without the mutator allele. Near the
fitness equilibrium, however, the mutators are severely suppressed, due to the
detrimental effects of a large mutation rate near the fitness maximum. We
discuss the results of a recent experiment on natural selection of E. coli in
the light of our model.


title: Introduction to Extended Electrodynamics

abstract:
  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.


title: Propagation failure of traveling waves in a discrete bistable medium

abstract:
  Propagation failure (pinning) of traveling waves is studied in a discrete
scalar reaction-diffusion equation with a piecewise linear, bistable reaction
function. The critical points of the pinning transition, and the wavefront
profile at the onset of propagation are calculated exactly. The scaling of the
wave speed near the transition, and the leading corrections to the front shape
are also determined. We find that the speed vanishes logarithmically close to
the critical point, thus the model belongs to a different universality class
than the standard Nagumo model, defined with a smooth, polynomial reaction
function.


title:
Adiabatic reduction near a bifurcation in stochastically modulated
  systems

abstract:
  We re-examine the procedure of adiabatic elimination of fast relaxing
variables near a bifurcation point when some of the parameters of the system
are stochastically modulated. Approximate stationary solutions of the
Fokker-Planck equation are obtained near threshold for the pitchfork and
transcritical bifurcations. Stochastic resonance between fast variables and
random modulation may shift the effective bifurcation point by an amount
proportional to the intensity of the fluctuations. We also find that
fluctuations of the fast variables above threshold are not always Gaussian and
centered around the (deterministic) center manifold as was previously believed.
Numerical solutions obtained for a few illustrative examples support these
conclusions.


title: Phase chaos in the anisotropic complex Ginzburg-Landau Equation

abstract:
  Of the various interesting solutions found in the two-dimensional complex
Ginzburg-Landau equation for anisotropic systems, the phase-chaotic states show
particularly novel features. They exist in a broader parameter range than in
the isotropic case, and often even broader than in one dimension. They
typically represent the global attractor of the system. There exist two
variants of phase chaos: a quasi-one dimensional and a two-dimensional
solution. The transition to defect chaos is of intermittent type.


title: Amplitude equations and pattern selection in Faraday waves

abstract:
  A nonlinear theory of pattern selection in parametric surface waves (Faraday
waves) is presented that is not restricted to small viscous dissipation. By
using a multiple scale asymptotic expansion near threshold, a standing wave
amplitude equation is derived from the governing equations. The amplitude
equation is of gradient form, and the coefficients of the associated Lyapunov
function are computed for regular patterns of various symmetries as a function
of a viscous damping parameter gamma. For gamma approximately 1, the selected
wave pattern comprises a single standing wave (stripe pattern). For gamma much
less than 1, patterns of square symmetry are obtained in the capillary regime
(large frequencies). At lower frequencies (the mixed gravity-capillary regime),
a sequence of six-fold (hexagonal), eight-fold,... patterns are predicted. For
even lower frequencies (gravity waves) a stripe pattern is again selected. Our
predictions of the stability regions of the various patterns are in
quantitative agreement with recent experiments conducted in large aspect ratio
systems.


title:
The dynamics of vortex lines in the three-dimensional complex
  Ginzburg-Landau equation: instability, stretching, entanglement, and helices

abstract:
  The dynamics of curved vortex filaments is studied analytically and
numerically in the framework of a three-dimensional complex Ginzburg-Landau
equation (CGLE). It is shown that a straight vortex line is unstable with
respect to spontaneous stretching and bending in a substantial range of
parameters of the CGLE, resulting in formation of persistent entangled vortex
configurations. The boundary of the three-dimensional instability in parameter
space is determined. Near the stability boundary, the supercritical saturation
of the instability is found, resulting in the formation of stable helicoidal
vortices.


title:
Wavy film flows down an inclined plane. Part I: Perturbation theory and
  general evolution equation for the film thickness

abstract:
  Wavy film flow of incompressible Newtonian fluid down an inclined plane is
considered. The question is posed as to the parametric conditions under which
the description of evolution can be approximately reduced for all time to a
single evolution equation for the film thickness. An unconventional
perturbation approach yields the most general evolution equation and least
restrictive conditions on its validity. The advantages of this equation for
analytical and numerical studies of 3D waves in inclined films are pointed out.


title: New stability results for long-wavelength convection patterns

abstract:
  We consider the transition from a spatially uniform state to a steady,
spatially-periodic pattern in a partial differential equation describing
long-wavelength convection. This both extends existing work on the study of
rolls, squares and hexagons and demonstrates how recent generic results for the
stability of spatially-periodic patterns may be applied in practice. We find
that squares, even if stable to roll perturbations, are often unstable when a
wider class of perturbations is considered. We also find scenarios where
transitions from hexagons to rectangles can occur. In some cases we find that,
near onset, more exotic spatially-periodic planforms are preferred over the
usual rolls, squares and hexagons.


title: Validation and Calibration of Models for Reaction-Diffusion Systems

abstract:
  Space and time scales are not independent in diffusion. In fact, numerical
simulations show that different patterns are obtained when space and time steps
($\Delta x$ and $\Delta t$) are varied independently. On the other hand,
anisotropy effects due to the symmetries of the discretization lattice prevent
the quantitative calibration of models. We introduce a new class of explicit
difference methods for numerical integration of diffusion and
reaction-diffusion equations, where the dependence on space and time scales
occurs naturally. Numerical solutions approach the exact solution of the
continuous diffusion equation for finite $\Delta x$ and $\Delta t$, if the
parameter $\gamma_N=D \Delta t/(\Delta x)^2$ assumes a fixed constant value,
where $N$ is an odd positive integer parametrizing the alghorithm. The error
between the solutions of the discrete and the continuous equations goes to zero
as $(\Delta x)^{2(N+2)}$ and the values of $\gamma_N$ are dimension
independent. With these new integration methods, anisotropy effects resulting
from the finite differences are minimized, defining a standard for validation
and calibration of numerical solutions of diffusion and reaction-diffusion
equations. Comparison between numerical and analytical solutions of
reaction-diffusion equations give global discretization errors of the order of
$10^{-6}$ in the sup norm. Circular patterns of travelling waves have a maximum
relative random deviation from the spherical symmetry of the order of 0.2%, and
the standard deviation of the fluctuations around the mean circular wave front
is of the order of $10^{-3}$.


title:
Thermal Fluctuations, Subcritical Bifurcation, and Nucleation of
  Localized States in Electroconvection

abstract:
  Measurements of the mean-square amplitude of thermally-induced fluctuations
of a thin layer of the nematic liquid crystal I52 subjected to a voltage V < Vc
are reported. They yield the limit of stability Vc of the spatially uniform
conduction state to infinitesimal perturbations. Localized long-lived
convecting structures known as worms form spontaneously from the fluctuations
for V well below Vc. This, as well as measurements of the lifetime of the
conduction state for V < Vc, suggests a thermally-activated mechanism of worm
formation; but a theory for this non-potential system is lacking.


title:
Three basic issues concerning interface dynamics in nonequilibrium
  pattern formation

abstract:
  These are lecture notes of a course given at the 9th International Summer
School on Fundamental Problems in Statistical Mechanics, held in Altenberg,
Germany, in August 1997. In these notes, we discuss at an elementary level
three themes concerning interface dynamics that play a role in pattern forming
systems: (i) We briefly review three examples of systems in which the normal
growth velocity is proportional to the gradient of a bulk field which itself
obeys a Laplace or diffusion type of equation (solidification, viscous fingers
and streamers), and then discuss why the Mullins-Sekerka instability is common
to all such gradient systems. (ii) Secondly, we discuss how underlying an
effective interface description of systems with smooth fronts or transition
zones, is the assumption that the relaxation time of the appropriate order
parameter field(s) in the front region is much smaller than the time scale of
the evolution of interfacial patterns. Using standard arguments we illustrate
that this is generally so for fronts that separate two (meta)stable phases: in
such cases, the relaxation is typically exponential, and the relaxation time in
the usual models goes to zero in the limit in which the front width vanishes.
(iii) We finally summarize recent results that show that so-called ``pulled''
or ``linear marginal stability'' fronts which propagate into unstable states
have a very slow universal power law relaxation. This slow relaxation makes the
usual ``moving boundary'' or ``effective interface'' approximation for problems
with thin fronts, like streamers, impossible.


title: Diffusive Mixing of Stable States in the Ginzburg-Landau Equation

abstract:
  For the time-dependent Ginzburg-Landau equation on the real line, we
construct solutions which converge, as $x \to \pm\infty$, to periodic
stationary states with different wave-numbers $\eta_\pm$. These solutions are
stable with respect to small perturbations, and approach as $t \to +\infty$ a
universal diffusive profile depending only on the values of $\eta_\pm$. This
extends a previous result of Bricmont and Kupiainen by removing the assumption
that $\eta_\pm$ 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.


title: A Continuum Description of Vibrated Sand

abstract:
  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.


title: Front Propagation: Precursors, Cutoffs and Structural Stability

abstract:
  We discuss the problem of fronts propagating into metastable and unstable
states. We examine the time development of the leading edge, discovering a
precursor which in the metastable case propagates out ahead of the front at a
velocity more than double that of the front and establishes the characteristic
exponential behavior of the steady-state leading edge. We also study the
dependence of the velocity on the imposition of a cutoff in the reaction term.
These studies shed new light on the problem of velocity selection in the case
of propagation into an unstable state. We also examine how discreteness in a
particle simulation acts as an effective cutoff in this case.


title:
Square patterns in Rayleigh-Benard convection with rotation about a
  vertical axis

abstract:
  We present experimental results for Rayleigh-Benard convection with rotation
about a vertical axis at dimensionless rotation rates in the range 0 to 250 and
upto 20% above the onset. Critical Rayleigh numbers and wavenumbers agree with
predictions of linear stability analysis. For rotation rates greater than 70
and close to onset, the patterns are cellular with local four-fold coordination
and differ from the theoretically expected Kuppers-Lortz unstable state. Stable
as well as intermittent defect-free square lattices exist over certain
parameter ranges. Over other ranges defects dynamically disrupt the lattice but
cellular flow and local four-fold coordination is maintained.


title:
Localization of Waves without Bistability: Worms in Nematic
  Electroconvection

abstract:
  A general localization mechanism for waves in dissipative systems is
identified that does not require the bistability of the basic state and the
nonlinear plane-wave state. The mechanism explains the two-dimensional
localized wave structures (`worms') that recently have been observed in
experiments on electroconvection in nematic liquid crystals where the
transition to extended waves is supercritical. The mechanism accounts for the
propagation direction of the worms and certain aspects of their interaction.
The dynamics of the localized waves can be steady or irregular.


title: Hexagons and Interfaces in a Vibrated Granular Layer

abstract:
  The order parameter model based on parametric Ginzburg-Landau equation is
used to describe high acceleration patterns in vibrated layer of granular
material. At large amplitude of driving both hexagons and interfaces emerge.
Transverse instability leading to formation of ``decorated'' interfaces and
labyrinthine patterns, is found. Additional sub-harmonic forcing leads to
controlled interface motion.


title: Continuum-type stability balloon in oscillated granular layers

abstract:
  The stability of convection rolls in a fluid heated from below is limited by
secondary instabilities, including the skew-varicose and crossroll
instabilities. We observe a stability boundary defined by the same
instabilities in stripe patterns in a vertically oscillated granular layer.
Molecular dynamics simulations show that the mechanism of the skew-varicose
instability in granular patterns is similar to that in convection. These
results suggest that pattern formation in granular media can be described by
continuum models analogous to those used in fluid systems.


title:
The Accurate Modelling of Thin 3D Fluid Flows with Inertia on Curved
  Substrates

abstract:
  Models and simulations of the flow of thin films of fluids have many
important applications in industrial and natural processes. We consider the
motion of a thin layer of an incompressible, Newtonian fluid over an arbitrary
solid, stationary curved substrate. Earlier work by Roy, et al has modeled slow
fluid flows in which inertia is negligible. However, in many applications a
model expressed in terms of both the fluid layer thickness and the average
lateral velocity is needed to resolve faster dynamics. Here we use centre
manifold techniques to derive such a non-dimensional, approximate model of the
three dimensional flow.


title: A Phase Front Instability in Periodically Forced Oscillatory Systems

abstract:
  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
$\omega_{forcing}/\omega_{system}=2n$ ($n>1$). Stationary fronts shifting the
oscillation phase by $\pi$ lose stability below a critical forcing strength and
decompose into $n$ traveling fronts each shifting the phase by $\pi/n$. The
instability designates a transition from stationary two-phase patterns to
traveling $n$-phase patterns.


title: The Universal Gaussian in Soliton Tails

abstract:
  We show that in a large class of equations, solitons formed from generic
initial conditions do not have infinitely long exponential tails, but are
truncated by a region of Gaussian decay. This phenomenon makes it possible to
treat solitons as localized, individual objects. For the case of the KdV
equation, we show how the Gaussian decay emerges in the inverse scattering
formalism.


title: Spike autosolitons in the Gray-Scott model

abstract:
  We performed a comprehensive study of the spike autosolitons: self-sustained
solitary inhomogeneous states, in the classical reaction-diffusion system ---
the Gray-Scott model. We developed singular perturbation techniques based on
the strong separation of the length scales to construct asymptotically the
solutions in the form of a one-dimensional static autosolitons,
higher-dimensional radially-symmetric static autosolitons, and two types of
traveling autosolitons. We studied the stability of the static autosolitons in
one and three dimensions and analyzed the properties of the static and the
traveling autosolitons.


title: Path instabilities of air bubbles rising in clean water

abstract:
  Experiments are conducted to study the path and shape of single air bubbles
(diameter range 0.10- 0.20cm) rising freely in clean water. The experimental
results demonstrate that the bubble shape has a bistable state, i. e. the
bubble chooses to be in spherical or ellipsoidal shape depending on its
generation mechanism. The path of a spherical/ellipsoidal bubble is found to
change from a straight path to a zigzag/spiral path via a
supercritical/subcritical bifurcation when the Reynolds number of the bubble
exceeds a threshold.


title:
New Approaches to Soliton Quantization and Existence for Particle
  Physics

abstract:
  This paper provides mathematical details related to another new paper which
suggests: (1) new approaches to the analysis of soliton stability; (2) families
of Lagrangian field theories where solitons might possibly exist even without
topological charge; (3) alternative approaches to quantizing solitons, with
testable nuclear implications. This paper evaluates the possibility of strong
energy-minimizing states in four families of systems, two promising and two not
promising. In these examples, it presents new methods for second-order
stability analysis, and analyzes persistent multifurcation. Section 6 presents
three alternative formalisms for quantizing solitons (topological or
nontopological), all of which have major implications for the foundations of
quantum theory: (1) the standard formalism, based on functional integration,
reinterpreted as an imaginary Markhov Random Field (iMRF) across time and
space, with parallels to fuzzy logic; (2) two radically conservative
formalisms, consistent with the core of Einstein's vision, based on a true MRF
model. Bell's Theorem, bosonization, time-symmetry and macroscopic asymmetry
are discussed, along with some testable alternative possibilities and heresies,
like nonDoppler redshift.


title: The Dynamics of Granular Segregation Patterns in a Long Drum Mixer

abstract:
  We have studied the early time evolution of granular segregation patterns in
a horizontal rotating cylinder partially filled with a sand/salt mixture. The
growth of concentration fluctuations starting from premixed initial conditions
was analyzed using Fourier techniques. In one mixture, we observed generally
merging dynamics in the segregated bands up to the onset of saturation. At the
threshold of saturation, we found a spectrum of wavelengths with a broad peak.
The peak position was nearly independent of the tube rotation rate. The overall
growth rate of Fourier modes had a maximum at a particular value of the angular
rotation frequency. In a slightly different mixture, we observed transient
traveling waves when the larger grains were in the majority. We measured the
wave speed as a function of several parameters using presegregated initial
conditions to launch waves of various wavelengths.

categories: patt-sol chao-dyn cond-mat.soft nlin.CD nlin.PS


title:
Oscillon-type Structures and Their Interaction in a Swift-Hohenberg
  Model

abstract:
  Motivated by the observation of localized circular excitations (`oscillons')
in vertically vibrated granular layers (P.B. Umbanhowar, F. Melo and H.L.
Swinney, Nature 382 (1996) 793), we numerically investigate an extension of a
Swift-Hohenberg model that exhibits a subcritical transition to square
patterns. For sufficient subcriticality, stable oscillon structures are found.
The localization mechanism is quite general and is due to non-adiabatic
effects. Bound structures of oscillons of equal and opposite polarity are found
with bound states of like polarity being less robust. Much of the phenomena are
consistent with the experimental observations and suggest that oscillons are
not specific to patterns in granular media or to parametrically driven systems.
Experimental tests are suggested that would determine whether this minimal
framework is sufficient to describe the phenomena.


title: Influence of the Soret effect on convection of binary fluids

abstract:
  Convection in horizontal layers of binary fluids heated from below and in
particular the influence of the Soret effect on the bifurcation properties of
extended stationary and traveling patterns that occur for negative Soret
coupling is investigated theoretically. The fixed points corresponding to these
two convection structures are determined for realistic boundary conditions with
a many mode Galerkin scheme for temperature and concentration and an accurate
one mode truncation of the velocity field. This solution procedure yields the
stable and unstable solutions for all stationary and traveling patterns so that
complete phase diagrams for the different convection types in typical binary
liquid mixtures can easily be computed. Also the transition from weakly to
strongly nonlinear states can be analyzed in detail. An investigation of the
concentration current and of the relevance of its constituents shows the way
for a simplification of the mode representation of temperature and
concentration field as well as for an analytically manageable few mode
description.


title: Analysis of Optimal Velocity Model with Explicit Delay

abstract:
  We analyze Optimal Velocity Model (OVM) with explicit delay. The properties
of congestion and the delay time of car motion are investigated by analytical
and numerical methods. It is shown that the small explicit delay time has
almost no effects. In the case of the large explicit delay time, a new phase of
congestion pattern of OVM seems to appear.


title: Model for convection in binary liquids

abstract:
  A minimal, analytically manageable Galerkin type model for convection in
binary mixtures subject to realistic boundary conditions is presented. The
model elucidates and reproduces the typical bifurcation topology of extended
stationary and oscillatory convective states seen for negative Soret coupling:
backwards stationary and Hopf bifurcations, saddle node bifurcations to stable
strongly nonlinear stationary and traveling wave (TW) states, and merging of
the TW solution branch with stationary states. Also unstable standing wave
solutions are obtained. A systematic analysis of the concentration balance for
liquid mixture parameters has lead to a representation of the concentration
field in terms of two linear and two nonlinear modes. This truncation captures
the important large--scale effects in the laterally averaged concentration
field resulting from advective and diffusive mixing. Also the fact that with
increasing flow intensity along the TW solution branch the frequency decreases
monotonically in the same way as the mixing increases --- the variance of the
concentration distribution decreases --- is ensured and reproduced well.
Universal scaling relations between flow intensity, frequency, and variance of
the concentration distribution (degree of mixing) in a TW are predicted by the
model and have been confirmed by numerical solutions of the full equations. The
validity of the model is checked by comparison with numerical solutions of the
full field equations.


title:
Pattern formation of reaction-diffusion system having self-determined
  flow in the amoeboid organism of Physarum plasmodium

abstract:
  The amoeboid organism, the plasmodium of Physarum polycephalum, behaves on
the basis of spatio-temporal pattern formation by local
contraction-oscillators. This biological system can be regarded as a
reaction-diffusion system which has spatial interaction by active flow of
protoplasmic sol in the cell. Paying attention to the physiological evidence
that the flow is determined by contraction pattern in the plasmodium, a
reaction-diffusion system having self-determined flow arises. Such a coupling
of reaction-diffusion-advection is a characteristic of the biological system,
and is expected to relate with control mechanism of amoeboid behaviours. Hence,
we have studied effects of the self-determined flow on pattern formation of
simple reaction-diffusion systems. By weakly nonlinear analysis near a trivial
solution, the envelope dynamics follows the complex Ginzburg-Landau type
equation just after bifurcation occurs at finite wave number. The flow term
affects the nonlinear term of the equation through the critical wave number
squared. Contrary to this, wave number isn't explicitly effective with lack of
flow or constant flow. Thus, spatial size of pattern is especially important
for regulating pattern formation in the plasmodium. On the other hand, the flow
term is negligible in the vicinity of bifurcation at infinitely small wave
number, and therefore the pattern formation by simple reaction-diffusion will
also hold. A physiological role of pattern formation as above is discussed.


title: The Stability of Standing Waves

abstract:
  We determine the modulational stability of standing waves with small group
velocity in quasi-onedimensional systems slightly above the threshold of a
supercritical Hopf bifurcation. The stability limits are given by two different
long-wavelength destabilisation mechanisms and generically also a
short-wavelength destabilisation. The Eckhaus parabola is shifted off-center
and can be convex from below or above. For nonzero group velocity the Newell
criterion, which near the cross-over from standing to traveling waves becomes a
rather weak condition, does not determine the destabilisation of all standing
waves in one dimension. The cross-over to the non-local equations that are
asymptotically valid near threshold is discussed in detail. Close to the
transition from standing to traveling waves complex dynamics can arise due to
the competition of counter-propagating waves and the wavenumber selection by
sources. Our results yield necessary conditions for the stability of traveling
rectangles in quasi-twodimensional systems with axial anisotropy and form a
starting point for understanding the spatio-temporal chaos of traveling oblique
rolls observed in electroconvection of nematic liquid crystals.


title: Fronts and interfaces in bistable extended mappings

abstract:
  We study the interfaces' time evolution in one-dimensional bistable extended
dynamical systems with discrete time. The dynamics is governed by the
competition between a local piece-wise affine bistable mapping and any
couplings given by the convolution with a function of bounded variation. We
prove the existence of travelling wave interfaces, namely fronts, and the
uniqueness of the corresponding selected velocity and shape. This selected
velocity is shown to be the propagating velocity for any interface, to depend
continuously on the couplings and to increase with the symmetry parameter of
the local nonlinearity. We apply the results to several examples including
discrete and continuous couplings, and the planar fronts' dynamics in
multi-dimensional Coupled Map Lattices. We eventually emphasize on the
extension to other kinds of fronts and to a more general class of bistable
extended mappings for which the couplings are allowed to be nonlinear and the
local map to be smooth.


title: Phase instabilities in hexagonal patterns

abstract:
  The general form of the amplitude equations for a hexagonal pattern including
spatial terms is discussed. At the lowest order we obtain the phase equation
for such patterns. The general expression of the diffusion coefficients is
given and the contributions of the new spatial terms are analysed in this
paper. From these coefficients the phase stability regions in a hexagonal
pattern are determined. In the case of Benard-Marangoni instability our results
agree qualitatively with numerical simulations performed recently.


title: Boundary Limitation of Wavenumbers in Taylor-Vortex Flow

abstract:
  We report experimental results for a boundary-mediated wavenumber-adjustment
mechanism and for a boundary-limited wavenumber-band of Taylor-vortex flow
(TVF). The system consists of fluid contained between two concentric cylinders
with the inner one rotating at an angular frequency $\Omega$. As observed
previously, the Eckhaus instability (a bulk instability) is observed and limits
the stable wavenumber band when the system is terminated axially by two rigid,
non-rotating plates. The band width is then of order $\epsilon^{1/2}$ at small
$\epsilon$ ($\epsilon \equiv \Omega/\Omega_c - 1$) and agrees well with
calculations based on the equations of motion over a wide $\epsilon$-range.
When the cylinder axis is vertical and the upper liquid surface is free (i.e.
an air-liquid interface), vortices can be generated or expelled at the free
surface because there the phase of the structure is only weakly pinned. The
band of wavenumbers over which Taylor-vortex flow exists is then more narrow
than the stable band limited by the Eckhaus instability. At small $\epsilon$
the boundary-mediated band-width is linear in $\epsilon$. These results are
qualitatively consistent with theoretical predictions, but to our knowledge a
quantitative calculation for TVF with a free surface does not exist.


title:
Rayleigh-B\'{e}nard convection in a homeotropically aligned nematic
  liquid crystal

abstract:
  We report experimental results for convection near onset in a thin layer of a
homeotropically aligned nematic liquid crystal heated from below as a function
of the temperature difference $\Delta T$ and the applied vertical magnetic
field $H$ and compare them with theoretical calculations. The experiments cover
the field range $8 \alt h \equiv H/ H_{F} \alt 80$ ($H_F =$ is the
Fr\'eedericksz field). For $h$ less than a codimension-two field $h_{ct} \simeq
46$ the bifurcation is subcritical and oscillatory, with travelling- and
standing-wave transients. Beyond $h_{ct}$ the bifurcation is stationary and
subcritical until a tricritical field $h_t= 57.2$ is reached, beyond which it
is supercritical. The bifurcation sequence as a function of $h$ found in the
experiment confirms the qualitative aspects of the theoretical predictions.
However, the value of $h_{ct}$ is about 10% higher than the predicted value and
the results for $k_c$ are systematically below the theory by about 2% at small
$h$ and by as much as 7% near $h_{ct}$. At $h_{ct}$, $k_c$ is continuous within
the experimental resolution whereas the theory indicates a 7% discontinuity.
The theoretical tricritical field $h_t^{th} = 51$ is somewhat below the
experimental one. The fully developed flow above $R_c$ for $h < h_{ct}$ is
chaotic. For $h_{ct} < h < h_t$ the subcritical stationary bifurcation also
leads to a chaotic state. The chaotic states persist upon reducing the Rayleigh
number below $R_c$, i.e. the bifurcation is hysteretic. Above the tricritical
field $h_t$, we find a bifurcation to a time independent pattern which within
our resolution is non-hysteretic.


title: Magnetic Faraday-Instability

abstract:
  In a magnetic fluid parametrically driven surface waves can be excited by an
external oscillating magnetic field. A static magnetic field changes the
restoring forces and damping coefficients of the various surface waves. This
property enables the excitation of both subharmonic and harmonic responses of
the standing waves.


title:
Parametrically excited surface waves in magnetic fluids: observation of
  domain structures

abstract:
  Observations of parametrically excited surface waves in a magnetic fluid are
presented. Under the influence of a magnetic field these waves have a
non--monotonic dispersion relation, which leads to a richer behavior than in
ordinary liquids. We report observation of three novel effects, namely:
  i) domain structures,
  ii) oscillating defects and
  iii) relaxational phase oscillations.


title: Velocity correlations in granular materials

abstract:
  A system of inelastic hard disks in a thin pipe capped by hot walls is
studied with the aim of investigating velocity correlations between particles.
Two effects lead to such correlations: inelastic collisions help to build
localized correlations, while momentum conservation and diffusion produce long
ranged correlations. In the quasi-elastic limit, the velocity correlation is
weak, but it is still important since it is of the same order as the deviation
from uniformity. For system with stronger inelasticity, the pipe contains a
clump of particles in highly correlated motion. A theory with empirical
parameters is developed. This theory is composed of equations similar to the
usual hydrodynamic laws of conservation of particles, energy, and momentum.
Numerical results show that the theory describes the dynamics satisfactorily in
the quasi-elastic limit, however only qualitatively for stronger inelasticity.


title:
Secondary instabilities form a codimension-2 point accompanied by a
  homoclinic bifurcation

abstract:
  A study of secondary instabilities in ac-driven electroconvection of a
planarly aligned nematic liquid crystal is presented. At low frequencies one
has a transition from normal rolls to a zig-zag pattern and at high frequencies
a direct transition from normal to abnormal rolls. The crossover defines a
codimension-2 point. This point is also the origin of a line of homoclinic
bifurcations, which enable the transition from zig-zags to abnormal rolls at
lower frequencies.


title:
Electrically driven convection in a thin annular film undergoing
  circular Couette flow

abstract:
  We investigate the linear stability of a thin, suspended, annular film of
conducting fluid with a voltage difference applied between its inner and outer
edges. For a sufficiently large voltage, such a film is unstable to
radially-driven electroconvection due to charges which develop on its free
surfaces. The film can also be subjected to a Couette shear by rotating its
inner edge. This combination is experimentally realized using films of smectic
A liquid crystals. In the absence of shear, the convective flow consists of a
stationary, azimuthally one-dimensional pattern of symmetric, counter-rotating
vortex pairs. When Couette flow is applied, an azimuthally traveling pattern
results. When viewed in a co-rotating frame, the traveling pattern consists of
pairs of asymmetric vortices. We calculate the neutral stability boundary for
arbitrary radius ratio $\alpha$ and Reynolds number ${{\cal R} e}$ of the shear
flow, and obtain the critical control parameter ${\cal R}_c (\alpha, {{\cal R}
e})$ and the critical azimuthal mode number ${m_c (\alpha, {{\cal R} e})}$. The
Couette flow suppresses the onset of electroconvection, so that ${\cal R}_c
(\alpha, {{\cal R} e}) > {\cal R}_c (\alpha,0)$. The calculated suppression is
compared with experiments performed at $\alpha = 0.56 $ and $0 \leq {{\cal R}
e} \leq 0.22 $.

categories: patt-sol chao-dyn cond-mat.soft nlin.CD nlin.PS physics.flu-dyn


title: Turing Instability in a Boundary-fed System

abstract:
  The formation of localized structures in the chlorine dioxide-idodine-malonic
acid (CDIMA) reaction-diffusion system is investigated numerically using a
realistic model of this system. We analyze the one-dimensional patterns formed
along the gradients imposed by boundary feeds, and study their linear stability
to symmetry-breaking perturbations (Turing instability) in the plane transverse
to these gradients. We establish that an often-invoked simple local linear
analysis which neglects longitudinal diffusion is inappropriate for predicting
the linear stability of these patterns. Using a fully nonuniform analysis, we
investigate the structure of the patterns formed along the gradients and their
stability to transverse Turing pattern formation as a function of the values of
two control parameters: the malonic acid feed concentration and the size of the
reactor in the dimension along the gradients. The results from this
investigation are compared with existing experiments.


title:
Numerical Bifurcation Diagram for the Two-Dimensional Boundary-fed CDIMA
  System

abstract:
  We present numerical solution of the chlorine dioxide-iodine-malonic acid
reaction-diffusion system in two dimensions in a boundary-fed system using a
realistic model. The bifurcation diagram for the transition from non-symmetry
breaking structures along boundary feed gradients to transverse symmetry
breaking patterns in a single layer is numerically determined. We find this
transition to be discontinuous. We make connection with earlier results and
discuss prospects for future work.


title: Benard-Marangoni Convection in Two Layered Liquids

abstract:
  We describe experiments on B{\'e}nard-Marangoni convection in horizontal
layers of two immiscible liquids. Unlike previous experiments, which used gases
as the upper fluid, we find a square planform close to onset which undergoes a
secondary bifurcation to rolls at higher temperature differences. The scale of
the convection pattern is that of the thinner lower fluid layer for which
buoyancy and surface tension forces are comparable. The wavenumber of the
pattern near onset agrees with the prediction of the linear stability analysis
for the full two-layer problem. The square planform is in qualitative agreement
with recent one- and two-layer nonlinear theories, which fail however to
predict the transition to rolls.

categories: patt-sol chao-dyn nlin.CD nlin.PS physics.flu-dyn


title:
Simple and Superlattice Turing Patterns in Reaction-Diffusion Systems:
  Bifurcation, Bistability, and Parameter Collapse

abstract:
  This paper investigates the competition between both simple (e.g. stripes,
hexagons) and ``superlattice'' (super squares, super hexagons) Turing patterns
in two-component reaction-diffusion systems. ``Superlattice'' patterns are
formed from eight or twelve Fourier modes, and feature structure at two
different length scales. Using perturbation theory, we derive simple analytical
expressions for the bifurcation equation coefficients on both rhombic and
hexagonal lattices. These expressions show that, no matter how complicated the
reaction kinectics, the nonlinear reaction terms reduce to just four effective
terms within the bifurcation equation coefficients. Moreover, at the hexagonal
degeneracy -- when the quadratic term in the hexagonal bifurcation equation
disappears -- the number of effective system parameters drops to two, allowing
a complete characterization of the possible bifurcation results at this
degeneracy. The general results are then applied to specific model equations,
to investigate the stability of different patterns within those models.


title:
Dynamical Scaling Behavior of the Swift-Hohenberg Equation Following a
  Quench to the Modulated State

abstract:
  We study the kinetics of phase transitions in a Rayleigh-Benard system after
onset of convection using 2D Swift-Hohenberg equation. An initially uniform
state evolves to one whose ground state is spatially periodic. We confirmed
previous results which showed that dynamical scaling occurs at medium quench
($\epsilon = 0.25$) with scaling exponents 1/5 and 1/4 under zero noise and
finite noise respectively. We find logarithmic scaling behavior for a deep
quench ($\epsilon = 0.75$) at zero noise. A simple method is devised to measure
the proxy of domain wall length. We find that the energy and domain wall length
exhibit scaling behavior with the same exponent. For $\epsilon = 0.25$, the
scaling exponents are 1/4 and 0.3 at zero and finite noise respectively.


title: On the degenerated soft-mode instability

abstract:
  We consider instabilities of a single mode with finite wavenumber in
inversion symmetric spatially one dimensional systems, where the character of
the bifurcation changes from sub- to supercritical behaviour. Starting from a
general equation of motion the full amplitude equation is derived
systematically and formulas for the dependence of the coefficients on the
system parameters are obtained. We emphasise the importance of nonlinear
derivative terms in the amplitude equation for the behaviour in the vicinity of
the bifurcation point. Especially the numerical values of the corresponding
coefficients determine the region of coexistence between the stable trivial
solution and stable spatially periodic patterns. Our approach clearly shows
that similar considerations fail for the case of oscillatory instabilities.


title: Dynamics and Selection of Giant Spirals in Rayleigh-Benard Convection

abstract:
  For Rayleigh-Benard convection of a fluid with Prandtl number \sigma \approx
1, we report experimental and theoretical results on a pattern selection
mechanism for cell-filling, giant, rotating spirals. We show that the pattern
selection in a certain limit can be explained quantitatively by a
phase-diffusion mechanism. This mechanism for pattern selection is very
different from that for spirals in excitable media.


title: Theory of Spike Spiral Waves in a Reaction-Diffusion System

abstract:
  We discovered a new type of spiral wave solutions in reaction-diffusion
systems --- spike spiral wave, which significantly differs from spiral waves
observed in FitzHugh-Nagumo-type models. We present an asymptotic theory of
these waves in Gray-Scott model. We derive the kinematic relations describing
the shape of this spiral and find the dependence of its main parameters on the
control parameters. The theory does not rely on the specific features of
Gray-Scott model and thus is expected to be applicable to a broad range of
reaction-diffusion systems.


title: Solitonic Gluons

abstract:
  We describe a physical mechanism for creating multi-soliton bound states
where solitons are glued together by attraction between the beams they guide,
`solitonic gluons'. We verify the concept of the solitonic gluons
experimentally, observing a suppression of the repulsion between dark solitons
due to an attractive force between bright guided beams.


title: Nonlinear theory of soliton-induced waveguides

abstract:
  We develop a nonlinear theory of soliton-induced waveguides that describes a
finite-amplitude probe beam guided by a spatial dark soliton in a saturable
nonlinear medium. We suggest an effective way to control the interaction of
these soliton-induced waveguides and also show that, in sharp contrast with
scalar dark solitons, the dark-soliton waveguides can attract each other and
even form stationary bound states.


title:
Dynamics of kinks in the Ginzburg-Landau equation: Approach to a
  metastable shape and collapse of embedded pairs of kinks

abstract:
  We consider initial data for the real Ginzburg-Landau equation having two
widely separated zeros. We require these initial conditions to be locally close
to a stationary solution (the ``kink'' solution) except for a perturbation
supported in a small interval between the two kinks. We show that such a
perturbation vanishes on a time scale much shorter than the time scale for the
motion of the kinks. The consequences of this bound, in the context of earlier
studies of the dynamics of kinks in the Ginzburg-Landau equation, [ER], are as
follows: we consider initial conditions $v_0$ whose restriction to a bounded
interval $I$ have several zeros, not too regularly spaced, and other zeros of
$v_0$ are very far from $I$. We show that all these zeros eventually disappear
by colliding with each other. This relaxation process is very slow: it takes a
time of order exponential of the length of $I$.


title: Power-Law Behaviors in Nonlinearly Coupled Granular Chain under Gravity

abstract:
  We find power-law behaviors of grain velocity in both propagation and
backscattering in a gravitationally compacted granular chain with nonlinear
contact force. We focus on the leading peak of the velocity signal which
decreases in a power-law $d^{-\alpha}$, where $d$ is the location of the peak,
as the signal goes down. The ratio of backscattered to incident leading
velocity also follows a power-law $d_i^{-\beta}$, where $d_i$ is the depth of
impurity. The up-going backscattered signal is nearly solitary. Therefore, the
overall change of the leading velocity peak is given by the power-law
$d_i^{-(\alpha+\beta)}$. We find $\alpha=0.250$ and $\beta=0.167$ for the
Hertzian contact force.


title: Two boundary model for freezing front propagation in biological tissue

abstract:
  The response of the living tissue to the effects of strong heating or cooling
can cause the blood flow rate to vary by an order of magnitude. A mathematical
model for the freezing of living tissue is formulated which takes into account
the nonlocal temperature dependence of the blood flow rate when the temperature
distribution in the tissue is substantially nonuniform, as in cryosurgery.

categories: patt-sol nlin.PS physics.bio-ph physics.med-ph q-bio


title:
Second-order perturbation theory for the parametrically driven NLS
  solitons

abstract: Withdrawn by the author. Further research is needed.


title: Perturbation theory for gap solitons

abstract: The paper was withdrawn by the author. Further research is needed.


title: Emergence of Order in Textured Patterns

abstract:
  A characterization of textured patterns, referred to as the disorder function
\bar\delta(\beta), is used to study properties of patterns generated in the
Swift-Hohenberg equation (SHE). It is shown to be an intensive,
configuration-independent measure. The evolution of random initial states under
the SHE exhibits two stages of relaxation. The initial phase, where local
striped domains emerge from a noisy background, is quantified by a power law
decay \bar\delta(\beta) \sim t^{-{1/2} \beta}. Beyond a sharp transition a
slower power law decay of \bar\delta(\beta), which corresponds to the
coarsening of striped domains, is observed. The transition between the phases
advances as the system is driven further from the onset of patterns, and
suitable scaling of time and \bar\delta(\beta) leads to the collapse of
distinct curves. The decay of $\bar\delta(\beta)$ during the initial phase
remains unchanged when nonvariational terms are added to the underlying
equations, suggesting the possibility of observing it in experimental systems.
In contrast, the rate of relaxation during domain coarsening increases with the
coefficient of the nonvariational term.


title: Tunable Pinning of Burst-Waves in Extended Systems with Discrete Sources

abstract:
  We study the dynamics of waves in a system of diffusively coupled discrete
nonlinear sources. We show that the system exhibits burst waves which are
periodic in a traveling-wave reference frame. We demonstrate that the burst
waves are pinned if the diffusive coupling is below a critical value. When the
coupling crosses the critical value the system undergoes a depinning
instability via a saddle-node bifurcation, and the wave begins to move. We
obtain the universal scaling for the mean wave velocity just above threshold.


title: Higher order Shapiro steps in ac-driven Josephson junctions

abstract:
  We demonstrate that the well known phase-locking mechanism leading to Shapiro
steps in ac-driven Josephson junctions is always accompanied by a higher order
phase-locking mechanism similar to that of the parametrically driven pendulum.
This effect, resulting in a $\pi$-periodic effective potential for the phase,
manifests itself clearly in the parameter regions where the usual Shapiro steps
are expected to vanish.


title: Dynamics of Wetting Fronts in Porous Media

abstract:
  We propose a new phenomenological approach for describing the dynamics of
wetting front propagation in porous media. Unlike traditional models, the
proposed approach is based on dynamic nature of the relation between capillary
pressure and medium saturation. We choose a modified phase-field model of
solidification as a particular case of such dynamic relation. We show that in
the traveling wave regime the results obtained from our approach reproduce
those derived from the standard model of flow in porous media. In more general
case, the proposed approach reveals the dependence of front dynamics upon the
flow regime.


title:
Helicoidal instability of a scroll vortex in three-dimensional
  reaction-diffusion systems

abstract:
  We study the dynamics of scroll vortices in excitable reaction-diffusion
systems analytically and numerically. We demonstrate that intrinsic
three-dimensional instability of a straight scroll leads to the formation of
helicoidal structures. This behavior originates from the competition between
the scroll curvature and unstable core dynamics. We show that the obtained
instability persists even beyond the meander core instability of
two-dimensional spiral wave.


title: Soliton motion in a parametrically ac-driven damped Toda lattice

abstract:
  We demonstrate that a staggered parametric ac driving term can support stable
progressive motion of a soliton in a Toda lattice with friction, while an
unstaggered drivng force cannot. A physical context of the model is that of a
chain of anharmonically coupled particles adsorbed on a solid surface of a
finite size. The ac driving force models a standing acoustic wave excited on
the surface. Simulations demonstrate that the state left behind the moving
soliton, with the particles shifted from their equilibrium positions, gradually
relaxes back to the equilibrium state that existed before the passage of the
soliton. Perturbation theory predicts that the ac-driven soliton exists if the
amplitude of the drive exceeds a certain threshold. The analytical prediction
for the threshold is in reasonable agreement with that found numerically.
Collisions between two counter propagating solitons were also simulated,
demonstrating that the collisions are, essentially fully elastic.


title: Stability of Propagating Fronts in Damped Hyperbolic Equations

abstract:
  We consider the damped hyperbolic equation in one space dimension $\epsilon
u_{tt} + u_t = u_{xx} + F(u)$, where $\epsilon$ is a positive, not necessarily
small parameter. We assume that $F(0)=F(1)=0$ and that $F$ is concave on the
interval $[0,1]$. Under these assumptions, our equation has a continuous family
of monotone propagating fronts (or travelling waves) indexed by the speed
parameter $c \ge c_*$. Using energy estimates, we first show that the
travelling waves are locally stable with respect to perturbations in a weighted
Sobolev space. Then, under additional assumptions on the non-linearity, we
obtain global stability results using a suitable version of the hyperbolic
Maximum Principle. Finally, in the critical case $c = c_*$, we use self-similar
variables to compute the exact asymptotic behavior of the perturbations as $t
\to +\infty$. In particular, setting $\epsilon = 0$, we recover several
stability results for the travelling waves of the corresponding parabolic
equation.


title:
Parametrically forced sine-Gordon equation and domain walls dynamics in
  ferromagnets

abstract:
  A parametrically forced sine-Gordon equation with a fast periodic {\em
mean-zero} forcing is considered. It is shown that $\pi$-kinks represent a
class of solitary-wave solutions of the equation. This result is applied to
quasi-one-dimensional ferromagnets with an easy plane anisotropy, in a rapidly
oscillating magnetic field. In this case the $\pi$-kink solution we have
introduced corresponds to the uniform ``true'' domain wall motion, since the
magnetization directions on opposite sides of the wall are anti-parallel. In
contrast to previous work, no additional anisotropy is required to obtain a
true domain wall. Numerical simulations showed good qualitative agreement with
the theory.


title: A Potential of Interaction between Two- and Three-Dimensional Solitons

abstract:
  A general method to find an effective potential of interaction between far
separated 2D and 3D solitons is elaborated, including the case of 2D vortex
solitons. The method is based on explicit calculation of the overlapping term
in the full Hamiltonian of the system (_without_ assuming that the ``tail'' of
each soliton is not affected by its interaction with the other soliton, and, in
fact,_without_ knowing the exact form of the solution for an isolated soliton -
the latter problem is circumvented by reducing a bulk integral to a surface
one). The result is obtained in an explicit form that does not contain an
artificially introduced radius of the overlapping region. The potential applies
to spatial and spatiotemporal solitons in nonlinear optics, where it may help
to solve various dynamical problems: collisions, formation of bound states
(BS's), etc. In particular, an orbiting BS of two solitons is always unstable.
In the presence of weak dissipation and gain, the effective potential can also
be derived, giving rise to bound states similar to those recently studied in 1D
models.


title: Standing and Moving Gap Solitons in Resonantly Absorbing Gratings

abstract:
  We present hitherto unknown forms of soliton dynamics in the forbidden
frequency gap of a Bragg reflector, modified by periodic layers of
near-resonant two-level systems (TLS). Remarkably, even extremely low TLS
densities create an allowed band within the forbidden gap. This spectrum gives
rise, for any Bragg reflectivity, to a vast family of stable gap solitons, both
standing and moving, having a unique analytic form, an arbitrary pulse area,
and inelastic collision properties. These findings suggest new possibilities of
transmission control, noise filtering or ``dynamical cavities'' (self-traps)
for both weak and strong signal pulses.


title: $\pi$-kinks in strongly ac driven sine-Gordon systems

abstract:
  We demonstrate that $\pi$-kinks exist in non-parametrically ac driven
sine-Gordon systems if the ac drive is sufficiently fast. It is found that, at
a critical value of the drive amplitude, there are two stable and two unstable
equilibria in the sine-Gordon phase. The pairwise symmetry of these equilibria
implies the existence of a one-parameter family of $\pi$-kink solutions in the
reduced system. In the dissipative case of the ac driven sine-Gordon systems,
corresponding to Josephson junctions, the velocity is selected by the balance
between the perturbations. The results are derived from a perturbation analysis
and verified by direct numerical simulations.


title:
A Transverse Oscillation Arising From Spatial Soliton Formation in
  Nonlinear Optical Cavities

abstract:
  A new type of transverse instability in dispersively nonlinear optical
cavities, called the optical whistle, is discussed. This instability occurs in
the mean-field, soliton forming limit when the cavity is driven with a
finite-width Gaussian beam, and gives rise to oscillation, period doubling, and
chaos. It is also seen that bistability is strongly affected due to the
oscillation within the upper transmission branch. The phenomenon is interpreted
as a mode-mismatch in the soliton formation process and is believed to have
broad applicability.


title: Localized Structures in Pattern-Forming Systems

abstract:
  A number of mechanisms that lead to the confinement of patterns to a small
part of a translationally symmetric pattern-forming system are reviewed:
nonadiabatic locking of fronts, global coupling and conservation laws,
dispersion, and coupling to additional slow modes $via$ gradients. Various
connections with experimental results are made.


title:
Parametrically Excited Surface Waves: Two-Frequency Forcing, Normal Form
  Symmetries, and Pattern Selection

abstract:
  Motivated by experimental observations of exotic standing wave patterns in
the two-frequency Faraday experiment, we investigate the role of normal form
symmetries in the pattern selection problem. With forcing frequency components
in ratio m/n, where m and n are co-prime integers, there is the possibility
that both harmonic and subharmonic waves may lose stability simultaneously,
each with a different wavenumber. We focus on this situation and compare the
case where the harmonic waves have a longer wavelength than the subharmonic
waves with the case where the harmonic waves have a shorter wavelength. We show
that in the former case a normal form transformation can be used to remove all
quadratic terms from the amplitude equations governing the relevant resonant
triad interactions. Thus the role of resonant triads in the pattern selection
problem is greatly diminished in this situation. We verify our general results
within the example of one-dimensional surface wave solutions of the
Zhang-Vinals model of the two-frequency Faraday problem. In one-dimension, a
1:2 spatial resonance takes the place of a resonant triad in our investigation.
We find that when the bifurcating modes are in this spatial resonance, it
dramatically effects the bifurcation to subharmonic waves in the case of
forcing frequencies are in ratio 1/2; this is consistent with the results of
Zhang and Vinals. In sharp contrast, we find that when the forcing frequencies
are in ratio 2/3, the bifurcation to (sub)harmonic waves is insensitive to the
presence of another spatially-resonant bifurcating mode.


title: Dynamics of axial separation in long rotating drums

abstract:
  We propose a continuum description for the axial separation of granular
materials in a long rotating drum. The model, operating with two local
variables, concentration difference and the dynamic angle of repose, describes
both initial transient traveling wave dynamics and long-term segregation of the
binary mixture. Segregation proceeds through ultra-slow logarithmic coarsening.


title: Controlled Dynamics of Interfaces in a Vibrated Granular Layer

abstract:
  We present experimental study of a topological excitation, {\it interface},
in a vertically vibrated layer of granular material. We show that these
interfaces, separating regions of granular material oscillation with opposite
phases, can be shifted and controlled by a very small amount of an additional
subharmonic signal, mixed with the harmonic driving signal. The speed and the
direction of interface motion depends sensitively on the phase and the
amplitude of the subharmonic driving.


title:
Instabilities and Spatio-Temporal Chaos in Hexagon Patterns with
  Rotation

abstract:
  The dynamics of hexagon patterns in rotating systems are investigated within
the framework of modified Swift-Hohenberg equations that can be considered as
simple models for rotating convection with broken up-down symmetry, e.g.
non-Boussinesq Rayleigh-Benard or Marangoni convection. In the weakly nonlinear
regime a linear stability analysis of the hexagons reveals long- and short-wave
instabilities, which can be steady or oscillatory. The oscillatory short-wave
instabilities can lead to stable hexagon patterns that are periodically
modulated in space and time, or to a state of spatio-temporal chaos with a
Fourier spectrum that precesses on average in time. The chaotic state can
exhibit bistability with the steady hexagon pattern. There exist regimes in
which the steady hexagon patterns are unstable at all wavenumbers.


title:
Toda Lattice Solutions of Differential-Difference Equations for
  Dissipative Systems

abstract:
  In a certain class of differential-difference equations for dissipative
systems, we show that hyperbolic tangent model is the only the nonlinear system
of equations which can admit some particular solutions of the Toda lattice. We
give one parameter family of exact solutions, which include as special cases
the Toda lattice solutions as well as the Whitham's solutions in the Newell's
model. Our solutions can be used to describe temporal-spatial density patterns
observed in the optimal velocity model for traffic flow.


title:
Thermoconvection in magnetized ferrofluids: the influence of boundaries
  with finite heat conductivity

abstract:
  Realistic boundaries of finite heat conductivity Realistic boundaries of
finite heat conductivity for thermoconvection in a Rayleigh-B\'enard setup with
magnetized ferrofluids are investigated. A linear stability analysis of the
conductive state is performed with a shooting method. It shows that the
critical wave number is for any magnetic field stronly influenced by the
conductivity of the boundaries. Linear as well as nonlinear coefficients of a
Ginzburg Landau amplitude equation for convection shortly above the onset are
evaluated as functions of the magnetic Rayleigh number, the boundary
conductivities, and the fluid Prandtl number.


title:
Subharmonic bifurcation cascade of pattern oscillations caused by
  winding number increasing entrainment

abstract:
  Convection structures in binary fluid mixtures are investigated for positive
Soret coupling in the driving regime where solutal and thermal contributions to
the buoyancy forces compete. Bifurcation properties of stable and unstable
stationary square, roll, and crossroll (CR) structures and the oscillatory
competition between rolls and squares are determined numerically as a function
of fluid parameters. A novel type of subharmonic bifurcation cascade (SC) where
the oscillation period grows in integer steps as $n (2\pi)/(\omega)$ is found
and elucidated to be an entrainment process.


title: Self-organized Vortex State in Two-dimensional Dictyostelium Dynamics

abstract:
  We present results of experiments on the dynamics of Dictyostelium discoideum
in a novel set-up which constraints cell motion to a plane. After aggregation,
the amoebae collect into round ''pancake" structures in which the cells rotate
around the center of the pancake. This vortex state persists for many hours and
we have explicitly verified that the motion is not due to rotating waves of
cAMP. To provide an alternative mechanism for the self-organization of the
Dictyostelium cells, we have developed a new model of the dynamics of
self-propelled deformable objects. In this model, we show that cohesive energy
between the cells, together with a coupling between the self-generated
propulsive force and the cell's configuration produces a self-organized vortex
state. The angular velocity profiles of the experiment and of the model are
qualitatively similar. The mechanism for self-organization reported here can
possibly explain similar vortex states in other biological systems.


title: Discreteness-Induced Oscillatory Instabilities of Dark Solitons

abstract:
  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.


title: Towards a global classification of excitable reaction-diffusion systems

abstract:
  Patterns in reaction-diffusion systems near primary bifurcations can be
studied locally and classified by means of amplitude equations. This is not
possible for excitable reaction-diffusion systems. In this Letter we propose a
global classification of two variable excitable reaction-diffusion systems. In
particular, we claim that the topology of the underlying two-dimensional
homogeneous dynamics can be used to organize the system's behavior.


title: Faraday waves on a viscoelastic liquid

abstract:
  We investigate Faraday waves on a viscoelastic liquid. Onset measurements and
a nonlinear phase diagram for the selected patterns are presented. By virtue of
the elasticity of the material a surface resonance synchronous to the external
drive competes with the usual subharmonic Faraday instability. Close to the
bicriticality the nonlinear wave interaction gives rise to a variety of novel
surface states: Localised patches of hexagons, hexagonal superlattices,
coexistence of hexagons and lines. Theoretical stability calculations and
qualitative resonance arguments support the experimental observations.


title:
Spatial- and spatio-temporal pattern formation in optically driven
  discrete systems

abstract:
  Dynamical coherent structure (pattern) formation in the Klein-Gordon lattice
excited by periodic external field near the optical resonance is studied. It is
shown that besides spatial patterns discovered recently (V.M.Burlakov,
Phys.Rev.Lett. 80, 3988 (1998)) the spatio-temporal patterns can be generated
in the lattice in rather broad region of excitation parameters. An influence of
cubic anharmonicity in the lattice potential and that of random noise on the
pattern formation is examined.


title:
A study of a hamiltonian model for martensitic phase transformations
  including microkinetic energy

abstract:
  How can a system in a macroscopically stable state explore energetically more
favorable states, which are far away from the current equilibrium state? Based
on continuum mechanical considerations we derive a Boussinesq-type equation
which models the dynamics of martensitic phase transformations. The solutions
of the system, which we refer to as microkinetically regularized wave equation
exhibit strong oscillations after short times, thermalization can be confirmed.
That means that macroscopic fluctuations of the solutions decay at the benefit
of microscopic fluctuations. First analytical and numerical results on the
propagation of phase boundaries and thermalization effects are presented.
Despite the fact that model is conservative, it exhibits the hysteretic
behavior. Such a behavior is usually interpreted in macroscopic models in terms
of dissipative threshold which the driving force has to overcome to ensure that
the phase transformation proceeds. The threshold value depends on the amount of
the transformed phase as it is observed in known experiments. Secondly we
investigate the dynamics of oscillatory solutions. We present a formalism based
on Young measures, which allows us to describe the effective dynamics of
rapidly fluctuating solutions. The new method enables us to derive a numerical
scheme for oscillatory solutions based on particle methods.


title: Optical Gap Solitons in Nonresonant Quadratic Media

abstract:
  We demonstrate an important role of the process of optical rectification in
the theory of nonlinear wave propagation in quadratically nonlinear [or
$\chi^{(2)}$ ] periodic optical media. We derive a novel physical model for gap
solitons in $\chi^{(2)}$ nonlinear Bragg gratings.


title:
Dynamic properties of solitons in the Frenkel-Kontorova Model.
  Application to incommensurate CDW conductors

abstract:
  An impact of kink-type solitons on infrared lattice vibrations is studied for
incommensurate Frenkel-Kontorova model. It is shown that the vibration of
particles involved into the kink formation is very similar to that in a gap
mode around the force constant defect. IR phonon mode intensity is found to
possess a universal dependence on the system parameters and the kink
concentration. It is argued that the giant IR peak observed in a number of
incommensurate charge density wave conductors can be explained in terms of
dynamic charge transfer stimulated by kinks.


title: The exact solutions of differential equation with delay

abstract:
  The exact solutions of the first order differential equation with delay are
derived. The equation has been introduced as a model of traffic flow. The
solution describes the traveling cluster of jam, which is characterized by
Jacobi's elliptic function. We also obtain the family of solutions of such type
of equations.


title: Moving lattice kinks and pulses: an inverse method

abstract:
  We develop a general mapping from given kink or pulse shaped travelling-wave
solutions including their velocity to the equations of motion on
one-dimensional lattices which support these solutions. We apply this mapping -
by definition an inverse method - to acoustic solitons in chains with nonlinear
intersite interactions, to nonlinear Klein-Gordon chains, to reaction-diffusion
equations and to discrete nonlinear Schr\"odinger systems. Potential functions
can be found in at least a unique way provided the pulse shape is reflection
symmetric and pulse and kink shapes are at least $C^2$ functions. For kinks we
discuss the relation of our results to the problem of a Peierls-Nabarro
potential and continuous symmetries. We then generalize our method to higher
dimensional lattices for reaction-diffusion systems. We find that increasing
also the number of components easily allows for moving solutions.


title: Induced Coherence and Stable Soliton Spiraling

abstract:
  We develop a theory of soliton spiraling in a bulk nonlinear medium and
reveal a new physical mechanism: periodic power exchange via induced coherence,
which can lead to stable spiraling and the formation of dynamical two-soliton
states. Our theory not only explains earlier observations, but provides a
number of predictions which are also verified experimentally. Finally, we show
theoretically and experimentally that soliton spiraling can be controled by the
degree of mutual initial coherence.


title: Nonlinear Schr\"odinger solitons scattering off an interface

abstract:
  We integrate the one-dimensional nonlinear Schr\"odinger equation numerically
for solitons moving in external potentials. In particular, we study the
scattering off an interface separating two regions of constant potential
modeled by a linear ramp. Transmission coefficients and inelasticities are
computed as functions of the potential difference and the slope of the ramp.
Our data show that the ramp's slope has a strong influence revealing unexpected
windows of reflection in a transmission regime. The transmission coefficients
for very small potential differences are compared with the theoretical
predictions derived by perturbation theory. Also the time evolution of the
solitary waves after the scattering is studied. We observed that they in
general behave like solitons with an amplified amplitude. Due to this, they
oscillate. The oscillation period is measured and compared with theoretical
predictions.
  PACS number(s): 03.40.Kf, 0.3.65.Ge, 42.65.Tg, 42.81.Dp


title: Scaling Variables and Stability of Hyperbolic Fronts

abstract:
  We consider the damped hyperbolic equation (1) \epsilon u_{tt} + u_t = u_{xx}
+ F(u), x \in R, t \ge 0, where \epsilon is a positive, not necessarily small
parameter. We assume that F(0) = F(1) = 0 and that F is concave on the interval
[0,1]. Under these hypotheses, Eq.(1) has a family of monotone travelling wave
solutions (or propagating fronts) connecting the equilibria u=0 and u=1. This
family is indexed by a parameter c \ge c_* related to the speed of the front.
In the critical case c=c_*, we prove that the travelling wave is asymptotically
stable with respect to perturbations in a weighted Sobolev space. In addition,
we show that the perturbations decay to zero like t^{-3/2} as t \to +\infty and
approach a universal self-similar profile, which is independent of \epsilon, F
and of the initial data. In particular, our solutions behave for large times
like those of the parabolic equation obtained by setting \epsilon = 0 in
Eq.(1). The proof of our results relies on careful energy estimates for the
equation (1) rewritten in self-similar variables x/\sqrt{t}, \log t.