title: Stability Results for Steady, Spatially--Periodic Planforms

abstract:
  We consider the symmetry-breaking steady state bifurcation of a
spatially-uniform equilibrium solution of E(2)-equivariant PDEs. We restrict
the space of solutions to those that are doubly-periodic with respect to a
square or hexagonal lattice, and consider the bifurcation problem restricted to
a finite-dimensional center manifold. For the square lattice we assume that the
kernel of the linear operator, at the bifurcation point, consists of 4 complex
Fourier modes, with wave vectors K_1=(a,b), K_2=(-b,a), K_3=(b,a), and
K_4=(-a,b), where a>b>0 are integers. For the hexagonal lattice, we assume that
the kernel of the linear operator consists of 6 complex Fourier modes, also
parameterized by an integer pair (a,b). We derive normal forms for the
bifurcation problems, which we use to compute the linear, orbital stability of
those solution branches guaranteed to exist by the equivariant branching lemma.
These solutions consist of rolls, squares, hexagons, a countable set of rhombs,
and a countable set of planforms that are superpositions of all of the Fourier
modes in the kernel. Since rolls and squares (hexagons) are common to all of
the bifurcation problems posed on square (hexagonal) lattices, this framework
can be used to determine their stability relative to a countable set of
perturbations by varying a and b. For the hexagonal lattice, we analyze the
degenerate bifurcation problem obtained by setting the coefficient of the
quadratic term to zero. The unfolding of the degenerate bifurcation problem
reveals a new class of secondary bifurcations on the hexagons and rhombs
solution branches.


title: Synchronisation in Coupled Sine Circle Maps

abstract:
  We study the spatially synchronized and temporally periodic solutions of a
1-d lattice of coupled sine circle maps. We carry out an analytic stability
analysis of this spatially synchronized and temporally periodic case and obtain
the stability matrix in a neat block diagonal form. We find spatially
synchronized behaviour over a substantial range of parameter space. We have
also extended the analysis to higher spatial periods with similar results.
Numerical simulations for various temporal periods of the synchronized
solution, reveal that the entire structure of the Arnold tongues and the
devil's staircase seen in the case of the single circle map can also be
observed for the synchronized coupled sine circle map lattice. Our formalism
should be useful in the study of spatially periodic behaviour in other coupled
map lattices.


title: Pattern selection of cracks in directionally drying fracture

abstract:
  We study pattern selection of cracks in directionally drying fractures by
analyzing the experimental systems recently devised by C. Allain and L. Limat.
[Phys. Rev. Lett. {\bf 74}, 2981 (1995).] Proposing a simple picture of crack
formation, we clarify the mechanism of how cracks array regularly and find that
the interval between neighboring cracks is proportional to the $2/3$ power of
the cell thickness. This result explains well the experimental data of Allain
and Limat.


title: Stability of periodic arrays of vortices

abstract:
  The stability of periodic arrays of Mallier-Maslowe or Kelvin-Stuart vortices
is discussed. We derive with the energy-Casimir stability method the nonlinear
stability of this solution in the inviscid case as a function of the solution
parameters and of the domain size. We exhibit the maximum size of the domain
for which the vortex street is stable. By adapting a numerical time-stepping
code, we calculate the linear stability of the Mallier-Maslowe solution in the
presence of viscosity and compensating forcing. Finally, the results are
discussed and compared to a recent experiment in fluids performed by Tabeling
et al.~[Europhysics Letters {\bf 3}, 459 (1987)]. Electromagnetically driven
counter-rotating vortices are unstable above a critical electric current, and
give way to co-rotating vortices. The importance of the friction at the bottom
of the experimental apparatus is also discussed.


title: Attracting Manifold for a Viscous Topology Transition

abstract:
  An analytical method is developed describing the approach to a finite-time
singularity associated with collapse of a narrow fluid layer in an unstable
Hele-Shaw flow. Under the separation of time scales near a bifurcation point, a
long-wavelength mode entrains higher-frequency modes, as described by a version
of Hill's equation. In the slaved dynamics, the initial-value problem is solved
explicitly, yielding the time and analytical structure of a singularity which
is associated with the motion of zeroes in the complex plane. This suggests a
general mechanism of singularity formation in this system.


title:
Steady and Oscillatory Side-Band Instabilities in Marangoni Convection
  with Deformable Interface

abstract:
  The stability of Marangoni roll convection in a liquid-gas system with
deformable interface is studied in the case when there is a nonlinear
interaction between two modes of Marangoni instability: long-scale surface
deformations and short-scale convection. Within the framework of a model
derived previously, it is shown that the nonlinear interaction between the two
modes substantially changes the width of the band of stable wave numbers of the
short-scale convection pattern as well as the type of the instability limiting
the band. Depending on the parameters of the system, the instability can be
either long- or short-wave, either monotonic or oscillatory. The stability
boundaries strongly differ from the standard ones and sometimes exclude the
band center. The long-wave limit of the side-band instability is studied in
detail within the framework of the phase approximation. It is shown that the
monotonic instability is always subcritical, while the long-wave oscillatory
instability can be supercritical, leading to the formation of either travelling
or standing waves modulating the pattern wavelength and the surface
deformation.


title:
Secondary Instabilities of Surface Waves on Viscous Fluids in the
  Faraday Instability

abstract:
  Secondary instabilities of Faraday waves show three regimes: (1) As seen
previously, low-viscosity (nu) fluids destabilize first into squares. At higher
driving accelerations a, squares show low-frequency modulations corresponding
to the motion of phase defects, while theory predicts a stationary transverse
amplitude modulation (TAM). (2) High-nu fluids destabilize first to stripes.
Stripes then show an oscillatory TAM whose frequency is incommensurate with the
driving frequency. At higher a, the TAM undergoes a phase instability. At still
higher a, edge dislocations form and fluid droplets are ejected. (3)
Intermediate-nu fluids show a complex coexistence of squares and stripes, as
well as stationary and oscillatory TAM instabilities of the stripes.


title: Existence and Stability of Steady Fronts in Bistable CML

abstract:
  We prove the existence and we study the stability of the kink-like fixed
points in a simple Coupled Map Lattice for which the local dynamics has two
stable fixed points. The condition for the existence allows us to define a
critical value of the coupling parameter where a (multi) generalized
saddle-node bifurcation occurs and destroys these solutions. An extension of
the results to other CML's in the same class is also displayed. Finally, we
emphasize the property of spatial chaos for small coupling.


title: Noise and dynamical pattern selection

abstract:
  In pattern forming systems such as Rayleigh-Benard convection or directional
solidification, a large number of linearly stable, patterned steady states
exist when the basic, simple steady state is unstable. Which of these steady
states will be realized in a given experiment appears to depend on unobservable
details of the system's initial conditions. We show, however, that weak,
Gaussian white noise drives such a system toward a preferred wave number which
depends only on the system parameters and is independent of initial conditions.
We give a prescription for calculating this wave number, analytically near the
onset of instability and numerically otherwise.


title: The Speed of Fronts of the Reaction Diffusion Equation

abstract:
  We study the speed of propagation of fronts for the scalar reaction-diffusion
equation $u_t = u_{xx} + f(u)$\, with $f(0) = f(1) = 0$. We give a new integral
variational principle for the speed of the fronts joining the state $u=1$ to
$u=0$. No assumptions are made on the reaction term $f(u)$ other than those
needed to guarantee the existence of the front. Therefore our results apply to
the classical case $f > 0$ in $(0,1)$, to the bistable case and to cases in
which $f$ has more than one internal zero in $(0,1)$.


title: Two Scale Model for Aggregation and Etching

abstract:
  We propose a dual scale drift-diffusion model for interfacial growth and
etching processes. The two scales are: (i) a depletion layer width surrounding
the aggregate and (ii) a drift length.The interplay between these two
antithetical scales yields a variety of distinct morphologies reported in
electrochemical deposition of metals, viscous fingering in fluids and in porous
silicon formation. Further, our algorithm interpolates between existing growth
models (diffusion limited aggregation, ballistic deposition and Eden) for
limiting values of these variables.


title: Absolutely stable solitons in two-component active systems

abstract:
  As is known, a solitary pulse in the complex cubic Ginzburg-Landau (GL)
equation is unstable. We demonstrate that a system of two linearly coupled GL
equations with gain and dissipation in one subsystem and pure dissipation in
another produces absolutely stable solitons and their bound states. The problem
is solved in a fully analytical form by means of the perturbation theory. The
soliton coexists with a stable trivial state; there is also an unstable soliton
with a smaller amplitude, which is a separatrix between the two stable states.
This model has a direct application in nonlinear fiber optics, describing an
Erbium-doped laser based on a dual-core fiber.


title:
Early stages of ramified growth in quasi-two-dimensional electrochemical
  deposition

abstract:
  I have measured the early stages of the growth of branched metal aggregates
formed by electrochemical deposition in very thin layers. The growth rate of
spatial Fourier modes is described qualitatively by the results of a linear
stability analysis [D.P. Barkey, R.H. Muller, and C.W. Tobias, J. Electrochem.
Soc. {\bf 136}, 2207 (1989)]. The maximum growth rate is proportional to
$(I/c)^\delta$ where $I$ is the current through the electrochemical cell, $c$
the electrolyte concentration, and $\delta = 1.37 \pm 0.08$. Differences
between my results and the theoretical predictions suggest that
electroconvection in the electrolyte has a large influence on the instability
leading to ramified growth.


title:
On the Interaction of Extreme-Low-Frequency Radiation with Living
  Matter's Coherent Spiral States

abstract:
  Conventional physics cannot explain bioeffects caused by ELF fields [1]. A
(double) resonance interaction of ELF radiation with the highly sub-thermal
coherent motion of spiral waves in living matter (LM) is suggested. The
(geo)magnetic cyclotron resonance (CR) absorption by the constituent ions can
drastically change their non-thermal degrees of freedom in the spiral state,
destroying their pattern, which is heuristically assumed to be part of the cell
signaling. Combining ELF irradiation with calcium imaging to unravel the
suggested mechanism, a new instrument in biomedicine might emerge. These
fascinating waves, being part of the bio-machinery, might provide insight in
cancer, embryogenesis, etc., as they are affected by ELF radiation in an as yet
unknown manner [2-6].


title:
Parametric Forcing of Waves with Non-Monotonic Dispersion Relation:
  Domain Structures in Ferrofluids?

abstract:
  Surface waves on ferrofluids exposed to a dc-magnetic field exhibit a
non-monotonic dispersion relation. The effect of a parametric driving on such
waves is studied within suitable coupled Ginzburg-Landau equations. Due to the
non-monotonicity the neutral curve for the excitation of standing waves can
have up to three minima. The stability of the waves with respect to long-wave
perturbations is determined $via$ a phase-diffusion equation. It shows that the
band of stable wave numbers can split up into two or three sub-bands. The
resulting competition between the wave numbers corresponding to the respective
sub-bands leads quite naturally to patterns consisting of multiple domains of
standing waves which differ in their wave number. The coarsening dynamics of
such domain structures is addressed.


title: Local and Global Existence of Multiple Waves Near Formal Approximations

abstract:
  Assuming that a formal approximation of multiple waves has been obtained by
matched asymptotic methods, we derive a {\em Spatial Shadowing lemma} to
construct exact solutions near the formal approximation.
  In Part I, we consider a general singularly perturbed parabolic system. $$
\epsilon u_t + (-\epsilon^2)^m D^{2m}_x u = f(u,\epsilon u_x,\cdots,(\epsilon
D_x)^{2m-1} u,x,\epsilon). $$ We show that if the formal approximation is
precise, there is always an exact solution nearby for at least a short time.
Examples include Cahn-Hilliard equation and viscous profile of conservation
laws.
  In Part II, we show under some more assumptions, the process in Part I can be
repeated to obtain global solutions if the formal approximation is a global
one. Examples include reaction-diffusion equations and phase field equations.


title: Coarse-grained description of thermo-capillary flow

abstract:
  A mesoscopic or coarse-grained approach is presented to study
thermo-capillary induced flows. An order parameter representation of a
two-phase binary fluid is used in which the interfacial region separating the
phases naturally occupies a transition zone of small width. The order parameter
satisfies the Cahn-Hilliard equation with advective transport. A modified
Navier-Stokes equation that incorporates an explicit coupling to the order
parameter field governs fluid flow. It reduces, in the limit of an infinitely
thin interface, to the Navier-Stokes equation within the bulk phases and to two
interfacial forces: a normal capillary force proportional to the surface
tension and the mean curvature of the surface, and a tangential force
proportional to the tangential derivative of the surface tension. The method is
illustrated in two cases: thermo-capillary migration of drops and phase
separation via spinodal decomposition, both in an externally imposed
temperature gradient.


title:
Wave-unlocking transition in resonantly coupled complex Ginzburg-Landau
  equations

abstract:
  We study the effect of spatial frequency-forcing on standing-wave solutions
of coupled complex Ginzburg-Landau equations. The model considered describes
several situations of nonlinear counterpropagating waves and also of the
dynamics of polarized light waves. We show that forcing introduces spatial
modulations on standing waves which remain frequency locked with a
forcing-independent frequency. For forcing above a threshold the modulated
standing waves unlock, bifurcating into a temporally periodic state. Below the
threshold the system presents a kind of excitability.


title: Hydrodynamics of Monolayer Domains at the Air-Water Interface

abstract:
  Molecules at the air-water interface often form inhomogeneous layers in which
domains of different densities are separated by sharp interfaces. Complex
interfacial pattern formation may occur through the competition of short- and
long-range forces acting within the monolayer. The overdamped hydrodynamics of
such interfacial motion is treated here in a general manner that accounts for
dissipation both within the monolayer and in the subfluid. Previous results on
the linear stability of interfaces are recovered and extended, and a
formulation applicable to the nonlinear regime is developed. A simplified
dynamical law valid when dissipation in the monolayer itself is negligible is
also proposed. Throughout the analysis, special attention is paid to the
dependence of the dynamical behavior on a characteristic length scale set by
the ratio of the viscosities in the monolayer and in the subphase.


title: Core Dynamics of Multi-Armed Spirals in Rayleigh-Benard Convection

abstract:
  We report experimental observations of the core dynamics of multi-armed,
rotating spirals in Rayleigh-Benard convection for a fluid with Prandtl number
near one. In addition to the large-scale rotation of the spirals, we found a
cyclic core motion within a central area of radius r = appr. (n lambda/2),
where n is the number of spiral arms ending in the core and lambda is the
wavelength of the pattern. The dynamics of the core was much faster than the
large-scale spiral rotation. We observed multi-armed spirals for which the two
periods were commensurate and others for which they were incommensurate.


title: Stability of Travelling Waves for a Damped Hyperbolic Equation

abstract:
  We consider a nonlinear damped hyperbolic equation in $\real^n$, $1 \le n \le
4$, depending on a positive parameter $\epsilon$. If we set $\epsilon=0$, this
equation reduces to the well-known Kolmogorov-Petrovski-Piskunov equation. We
remark that, after a change of variables, this hyperbolic equation has the same
family of one-dimensional travelling waves as the KPP equation. Using various
energy functionals, we show that, if $\epsilon >0$, these fronts are locally
stable under perturbations in appropriate weighted Sobolev spaces. Moreover,
the decay rate in time of the perturbed solutions towards the front of minimal
speed $c=2$ is shown to be polynomial. In the one-dimensional case, if
$\epsilon < 1/4$, we can apply a Maximum Principle for hyperbolic equations and
prove a global stability result. We also prove that the decay rate of the
perturbated solutions towards the fronts is polynomial, for all $c > 2$.


title: Numerical Proof of Self-Similarity in Burgers' Turbulence

abstract:
  We study the statistical properties of solutions to Burgers' equation, $v_t +
vv_x = \nu v_{xx}$, for large times, when the initial velocity and its
potential are stationary Gaussian processes. The initial power spectral density
at small wave numbers follows a steep power-law $E_0(k) \sim |k|^n$ where the
exponent $n$ is greater than two. We compare results of numerical simulations
with dimensional predictions, and with asymptotic analytical theory. The theory
predicts self-similarity of statistical characteristics of the turbulence, and
also leads to a logarithmic correction to the law of energy decay in comparison
with dimensional analysis. We confirm numerically the existence of
self-similarity for the power spectral density, and the existence of a
logarithmic correction to the dimensional predictions.


title:
General theory of instabilities for patterns with sharp interfaces in
  reaction-diffusion systems

abstract:
  An asymptotic method for finding instabilities of arbitrary $d$-dimensional
large-amplitude patterns in a wide class of reaction-diffusion systems is
presented. The complete stability analysis of 2- and 3-dimensional localized
patterns is carried out. It is shown that in the considered class of systems
the criteria for different types of instabilities are universal. The specific
nonlinearities enter the criteria only via three numerical constants of order
one. The performed analysis explains the self-organization scenarios observed
in the recent experiments and numerical simulations of some concrete
reaction-diffusion systems.


title:
Critical phenomena employed in hydrodynamic problems: A case study of
  Rayleigh-Benard convection

abstract:
  By virtue of Rayleigh-Benard convection, we illustrate the advantages of
combining a hydrodynamic pattern forming instability with a thermodynamic
critical point. This has already lead to many novel unexpected observations and
is further shown to possess opportunities for the study of exciting fundamental
problems in nonequilibrium systems.


title: Convective Fingering of an Autocatalytic Reaction Front

abstract:
  We report experimental observations of the convection-driven fingering
instability of an iodate-arsenous acid chemical reaction front. The front
propagated upward in a vertical slab; the thickness of the slab was varied to
control the degree of instability. We observed the onset and subsequent
nonlinear evolution of the fingers, which were made visible by a {\it p}H
indicator. We measured the spacing of the fingers during their initial stages
and compared this to the wavelength of the fastest growing linear mode
predicted by the stability analysis of Huang {\it et. al.} [{\it Phys. Rev. E},
{\bf 48}, 4378 (1993), and unpublished]. We find agreement with the thickness
dependence predicted by the theory.


title:
Systematic derivation of a rotationally covariant extension of the
  2-dimensional Newell-Whitehead-Segel equation

abstract:
  An extension of the Newell-Whitehead-Segel amplitude equation covariant under
abritrary rotations is derived systematically by the renormalization group
method.


title: Square Patterns and Quasi-patterns in Weakly Damped Faraday Waves

abstract:
  Pattern formation in parametric surface waves is studied in the limit of weak
viscous dissipation. A set of quasi-potential equations (QPEs) is introduced
that admits a closed representation in terms of surface variables alone. A
multiscale expansion of the QPEs reveals the importance of triad resonant
interactions, and the saturating effect of the driving force leading to a
gradient amplitude equation. Minimization of the associated Lyapunov function
yields standing wave patterns of square symmetry for capillary waves, and
hexagonal patterns and a sequence of quasi-patterns for mixed capillary-gravity
waves. Numerical integration of the QPEs reveals a quasi-pattern of eight-fold
symmetry in the range of parameters predicted by the multiscale expansion.


title: Shadowing matching errors for wave-front-like solutions

abstract:
  Consider a singularly perturbed system $$\epsilon u_t=\epsilon^2 u_{xx} +
f(u,x,\epsilon),\quad u\in {\Bbb R}^n,x\in{\Bbb R},t\geq 0. $$ Assume that the
system has a sequence of regular and internal layers occurring alternatively
along the $x$-direction. These ``multiple wave'' solutions can formally be
constructed by matched asymptotic expansions. To obtain a genuine solution, we
derive a {\em Spatial Shadowing Lemma} which assures the existence of an exact
solution that is near the formal asymptotic series provided (1) the residual
errors are small in all the layers, and (2) the matching errors are small along
the lateral boundaries of the adjacent layers. The method should work on some
other systems like $\epsilon u_t=-(-\epsilon^2 D_{xx})^m u+ \dots.$


title: Solitonic-exchange mechanism of surface~diffusion

abstract:
  We study surface diffusion in the framework of a generalized
Frenkel-Kontorova model with a nonconvex transverse degree of freedom. The
model describes a lattice of atoms with a given concentration interacting by
Morse-type forces, the lattice being subjected to a two-dimensional substrate
potential which is periodic in one direction and nonconvex (Morse) in the
transverse direction. The results are used to describe the complicated
exchange-mediated diffusion mechanism recently observed in MD simulations [J.E.
Black and Zeng-Ju Tian, Phys. Rev. Lett. {\bf 71}, 2445-2448(1993)].


title: Mobility and diffusivity in a generalized Frenkel-Kontorova model

abstract:
  Molecular dynamics simulations are used to investigate the atomic mobility
and diffusivity of a generalized Frenkel-Kontorova model which takes into
account anharmonic (exponential) interaction of atoms subjected to a
three-dimensional substrate potential periodic in two dimensions and nonconvex
(Morse) in the third dimension. The numerical results are explained by a
phenomenological theory which treats a system of strongly interacting atoms as
a system of weakly interacting quasiparticles (kinks). Model parameters are
chosen close to those for {\bf K}-{\bf W}(112) adsorption system.


title: Twisted and Nontwisted Bifurcations Induced by Diffusion

abstract:
  We discuss a diffusively perturbed predator-prey system. Freedman and
Wolkowicz showed that the corresponding ODE can have a periodic solution that
bifurcates from a homoclinic loop. When the diffusion coefficients are large,
this solution represents a stable, spatially homogeneous time-periodic solution
of the PDE. We show that when the diffusion coefficients become small, the
spatially homogeneous periodic solution becomes unstable and bifurcates into
spatially nonhomogeneous periodic solutions.
  The nature of the bifurcation is determined by the twistedness of an
equilibrium/homoclinic bifurcation that occurs as the diffusion coefficients
decrease. In the nontwisted case two spatially nonhomogeneous simple periodic
solutions of equal period are generated, while in the twisted case a unique
spatially nonhomogeneous double periodic solution is generated through
period-doubling.
  Key Words: Reaction-diffusion equations; predator-prey systems; homoclinic
bifurcations; periodic solutions.


title: Non-Monotonic Dispersion of Surface Waves in Magnetic Fluids

abstract:
  The dispersion relation of surface waves of a magnetic fluid in a magnetic
field is studied experimentally. We verify the theoretically predicted
existence of a non-monotonic dispersion relation. In particular, we demonstrate
the existence of two different wave numbers occuring at the same frequency in
an annular geometry.


title: The Newell-Whitehead-Segel Equation for Traveling Waves

abstract:
  An equation to describe nearly one-dimensional traveling-waves patterns is
put forward. This is a dispersive generalization of the classical
Newell-Whitehead-Segel (NWS) equation. Transverse stability of plane waves is
considered within the framework of this equation. It is shown that the
dispersion terms drastically alter the stability. A necessary stability
condition is obtained in the form of a transverse Benjamin-Feir criterion. If
this condition is met, a quarter of the plane-wave existence band (in terms of
the squared wave number) is unstable, while three quarters are transversely
stable. Next, linear defects in the form of grain boundaries (GB's) are
studied. An effective Burgers equation is derived from the dispersive NWS
equation, in the framework of which a GB is tantamount to a shock wave. It is
shown that the GB's are generic solutions. Asymmetric GB's are moving at a
constant velocity, which is found. The integrability of the Burgers equation
allows one as well to analyze transient processes and interactions between
parallel GB's. The shock-wave solutions obtained in this work may also find
applications in nonlinear fiber optics.


title: A Simple Model for Faraday Waves

abstract:
  We show that the linear-stability analysis of the birth of Faraday waves on
the surface of a fluid is simplified considerably when the fluid container is
driven by a triangle waveform rather than by a sine wave. The calculation is
simple enough to use in an undergraduate course on fluid dynamics or nonlinear
dynamics. It is also an attractive starting point for a nonlinear analysis.


title: A Variational Principle for Eigenvalue Problems of 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 $\lambda$. We give a variational characterization of
the bifurcating branch $\lambda$ as a function of the amplitude of the
solution. As an application we show how it can be used to obtain simple
approximate closed formulae for the period of large amplitude oscillations.


title: Breathing Spots in a Reaction-Diffusion System

abstract:
  A quasi-2-dimensional stationary spot in a disk-shaped chemical reactor is
observed to bifurcate to an oscillating spot when a control parameter is
increased beyond a critical value. Further increase of the control parameter
leads to the collapse and disappearance of the spot. Analysis of a bistable
activator-inhibitor model indicates that the observed behavior is a consequence
of interaction of the front with the boundary near a parity breaking front
bifurcation.


title:
Weakly Nonlinear Theory of Pattern-Forming Systems with Spontaneously
  Broken Isotropy

abstract:
  Quasi two-dimensional pattern forming systems with spontaneously broken
isotropy represent a novel symmetry class, that is experimentally accessible in
electroconvection of homeotropically aligned liquid crystals. We present a
weakly nonlinear analysis leading to amplitude equations which couple the
short-wavelength patterning mode with the Goldstone mode resulting from the
broken isotropy. The new coefficients in these equations are calculated from
the hydrodynamics. Simulations exhibit a new type of spatio-temporal chaos at
onset. The results are compared with experiments.


title: Direct Hopf Bifurcation in Parametric Resonance of Hybridized Waves

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


title:
Temporal Modulation of Traveling Waves in the Flow Between Rotating
  Cylinders With Broken Azimuthal Symmetry

abstract:
  The effect of temporal modulation on traveling waves in the flows in two
distinct systems of rotating cylinders, both with broken azimuthal symmetry,
has been investigated. It is shown that by modulating the control parameter at
twice the critical frequency one can excite phase-locked standing waves and
standing-wave-like states which are not allowed when the system is rotationally
symmetric. We also show how previous theoretical results can be extended to
handle patterns such as these, that are periodic in two spatial direction.


title:
Self-replication and splitting of domain patterns in reaction-diffusion
  systems with fast inhibitor

abstract:
  An asymptotic equation of motion for the pattern interface in the
domain-forming reaction-diffusion systems is derived. The free boundary problem
is reduced to the universal equation of non-local contour dynamics in two
dimensions in the parameter region where a pattern is not far from the points
of the transverse instabilities of its walls. The contour dynamics is studied
numerically for the reaction-diffusion system of the FitzHugh-Nagumo type. It
is shown that in the asymptotic limit the transverse instability of the
localized domains leads to their splitting and formation of the multidomain
pattern rather than fingering and formation of the labyrinthine pattern.


title: The Exactly Solvable Simplest Model for Queue Dynamics

abstract:
  We present an exactly solvable model for queue dynamics. Our model is very
simple but provides the essential property for such dynamics. As an example,
the model has the traveling cluster solution as well as the homogeneous flow
solution. The model is the limiting case of Optimal Velocity (OV) model, which
is proposed for the car following model to induce traffic jam spontaneously.


title:
Mutually Penetrating Motion of Self-Organized 2D Patterns of
  Soliton-Like Structures

abstract:
  Results of numerical simulations of a recently derived most general
dissipative-dispersive PDE describing evolution of a film flowing down an
inclined plane are presented. They indicate that a novel complex type of
spatiotemporal patterns can exist for strange attractors of nonequilibrium
systems. It is suggested that real-life experiments satisfying the validity
conditions of the theory are possible: the required sufficiently viscous
liquids are readily available.


title: Links, Knots and Knotted Labyrinths in Bistable Systems

abstract:
  The existence of stable links and knots is demonstrated in three-dimensional,
bistable, chemical media. The reaction-diffusion medium segregates into regions
of high and low concentration separated by sharp interfaces. The interfaces
repel at short distances so that domains with various topologies are possible
depending on the initial conditions and system parameters. Front instabilities
can give rise to knotted labyrinthine patterns. A lattice-gas model whose
mean-field limit is the FitzHugh-Nagumo equation is described and implemented
to carry out the simulations.


title:
On a Conjecture of Goriely for the Speed of Fronts of the
  Reaction--Diffusion Equation

abstract:
  In a recent paper Goriely considers the one--dimensional scalar
reaction--diffusion equation $u_t = u_{xx} + f(u)$ with a polynomial reaction
term $f(u)$ and conjectures the existence of a relation between a global
resonance of the hamiltonian system $ u_{xx} + f(u) = 0$ and the asymptotic
speed of propagation of fronts of the reaction diffusion equation. Based on
this conjecture an explicit expression for the speed of the front is given. We
give a counterexample to this conjecture and conclude that additional
restrictions should be placed on the reaction terms for which it may hold.


title:
Nonlinear saturation of electrostatic waves: mobile ions modify trapping
  scaling

abstract:
  The amplitude equation for an unstable electrostatic wave in a multi-species
Vlasov plasma has been derived. The dynamics of the mode amplitude $\rho(t)$ is
studied using an expansion in $\rho$; in particular, in the limit
$\gamma\rightarrow0^+$, the singularities in the expansion coefficients are
analyzed to predict the asymptotic dependence of the electric field on the
linear growth rate $\gamma$. Generically $|E_k|\sim \gamma^{5/2}$, as
$\gamma\rightarrow0^+$, but in the limit of infinite ion mass or for
instabilities in reflection-symmetric systems due to real eigenvalues the more
familiar trapping scaling $|E_k|\sim \gamma^{2}$ is predicted.


title:
Multiple-Time Higher-Order Perturbation Analysis of the Regularized
  Long-Wavelength Equation

abstract:
  By considering the long-wave limit of the regularized long wave (RLW)
equation, we study its multiple-time higher-order evolution equations. As a
first result, the equations of the Korteweg-de Vries hierarchy are shown to
play a crucial role in providing a secularity-free perturbation theory in the
specific case of a solitary-wave solution. Then, as a consequence, we show that
the related perturbative series can be summed and gives exactly the
solitary-wave solution of the RLW equation. Finally, some comments and
considerations are made on the N-soliton solution, as well as on the
limitations of applicability of the multiple scale method in obtaining uniform
perturbative series.


title: Electroconvection in a Suspended Fluid Film: A Linear Stability Analysis

abstract:
  A suspended fluid film with two free surfaces convects when a sufficiently
large voltage is applied across it. We present a linear stability analysis for
this system. The forces driving convection are due to the interaction of the
applied electric field with space charge which develops near the free surfaces.
Our analysis is similar to that for the two-dimensional B\'enard problem, but
with important differences due to coupling between the charge distribution and
the field. We find the neutral stability boundary of a dimensionless control
parameter ${\cal R}$ as a function of the dimensionless wave number ${\kappa}$.
${\cal R}$, which is proportional to the square of the applied voltage, is
analogous to the Rayleigh number. The critical values ${{\cal R}_c}$ and
${\kappa_c}$ are found from the minimum of the stability boundary, and its
curvature at the minimum gives the correlation length ${\xi_0}$. The
characteristic time scale ${\tau_0}$, which depends on a second dimensionless
parameter ${\cal P}$, analogous to the Prandtl number, is determined from the
linear growth rate near onset. ${\xi_0}$ and ${\tau_0}$ are coefficients in the
Ginzburg-Landau amplitude equation which describes the flow pattern near onset
in this system. We compare our results to recent experiments.


title: Scenarios of domain pattern formation in a reaction-diffusion system

abstract:
  We performed an extensive numerical study of a two-dimensional
reaction-diffusion system of the activator-inhibitor type in which domain
patterns can form. We showed that both multidomain and labyrinthine patterns
may form spontaneously as a result of Turing instability. In the stable
homogeneous system with the fast inhibitor one can excite both localized and
extended patterns by applying a localized stimulus. Depending on the parameters
and the excitation level of the system stripes, spots, wriggled stripes, or
labyrinthine patterns form. The labyrinthine patterns may be both connected and
disconnected. In the the stable homogeneous system with the slow inhibitor one
can excite self-replicating spots, breathing patterns, autowaves and
turbulence. The parameter regions in which different types of patterns are
realized are explained on the basis of the asymptotic theory of instabilities
for patterns with sharp interfaces developed by us in Phys. Rev. E. 53, 3101
(1996). The dynamics of the patterns observed in our simulations is very
similar to that of the patterns forming in the ferrocyanide-iodate-sulfite
reaction.


title:
Motion of buoyant particles and coarsening of solid-liquid mixtures in a
  random acceleration field

abstract:
  Flow induced by a random acceleration field (g-jitter) is considered in two
related situations that are of interest for microgravity fluid experiments: the
random motion of an isolated buoyant particle and coarsening of a solid-liquid
mixture. We start by analyzing in detail actual accelerometer data gathered
during a recent microgravity mission, and obtain the values of the parameters
defining a previously introduced stochastic model of this acceleration field.
We then study the motion of a solid particle suspended in an incompressible
fluid that is subjected to such random accelerations. The displacement of the
particle is shown to have a diffusive component if the correlation time of the
stochastic acceleration is finite or zero, and mean squared velocities and
effective diffusion coefficients are obtained explicitly. Finally, the effect
of g-jitter on coarsening of a solid-liquid mixture is considered. Corrections
due to the induced fluid motion are calculated, and estimates are given for
coarsening of Sn-rich particles in a Sn-Pb eutectic fluid, experiment to be
conducted in microgravity in the near future.


title:
Pattern selection in the absolutely unstable regime as a nonlinear
  eigenvalue problem: Taylor vortices in axial flow

abstract:
  A unique pattern selection in the absolutely unstable regime of a driven,
nonlinear, open-flow system is analyzed: The spatiotemporal structures of
rotationally symmetric vortices that propagate downstream in the annulus of the
rotating Taylor-Couette system due to an externally imposed axial through-flow
are investigated for two different axial boundary conditions at the in- and
outlet. Unlike the stationary patterns in systems without through-flow the
spatiotemporal structures of propagating vortices are independent of parameter
history, initial conditions, and system's length. They do, however, depend on
the axial boundary conditions, the driving rate of the inner cylinder and the
through-flow rate. Our analysis of the amplitude equation shows that the
pattern selection can be described by a nonlinear eigenvalue problem with the
frequency being the eigenvalue. Approaching the border between absolute and
convective instability the eigenvalue problem becomes effectively linear and
the selection mechanism approaches that one of linear front propagation.
PACS:47.54.+r,47.20.Ky,47.32.-y,47.20.Ft


title: Numerical Simulation of Three-Dimensional Dendritic Growth

abstract:
  Dendritic crystal growth in a pure undercooled melt is simulated
quantitatively in three dimensions using a phase-field approach. The full
non-axisymmetric morphology of the steady-state dendrite tip and $\sigma^*$ are
determined as a function of anisotropy for a crystal with a cubic symmetry.
Results are compared to experiment and used to critically test solvability
theory.


title: Pattern formation in weakly damped Faraday waves

abstract:
  We present a theoretical study of nonlinear pattern formation in parametric
surface waves for fluids of low viscosity, and in the limit of large aspect
ratio. The analysis is based on a quasi-potential approximation to the
equations governing fluid motion, followed by a multiscale asymptotic expansion
in the distance away from threshold. Close to onset, the asymptotic expansion
yields an amplitude equation which is of gradient form, and allows the explicit
calculation of the functional form of the cubic nonlinearities. In particular,
we find that three-wave resonant interactions contribute significantly to the
nonlinear terms, and therefore are important for pattern selection.
Minimization of the associated Lyapunov functional predicts a primary
bifurcation to a standing wave pattern of square symmetry for
capillary-dominated surface waves, in agreement with experiments. In addition,
we find that patterns of hexagonal and quasi-crystalline symmetry can be
stabilized in certain mixed capillary-gravity waves, even in this case of
single frequency forcing. Quasi-crystalline patterns are predicted in a region
of parameters readily accessible experimentally.


title: Vortex -- Kink Interaction and Capillary Waves in a Vector Superfluid

abstract:
  Interaction of a vortex in a circularly polarized superfluid component of a
2d complex vector field with the phase boundary between superfluid phases with
opposite signs of polarization leads to a resonant excitation of a
``capillary'' wave on the boundary. This leads to energy losses by the
vortex--image pair that has to cause its eventual annihilation.


title: Delay of Vehicle Motion in Traffic Dynamics

abstract:
  We demonstrate that in Optimal Velocity Model (OVM) delay times of vehicles
coming from the dynamical equation of motion of OVM almost explain the order of
delay times observed in actual traffic flows without introducing explicit delay
times. Delay times in various cases are estimated: the case of a leader vehicle
and its follower, a queue of vehicles controlled by traffic lights and
many-vehicle case of highway traffic flow. The remarkable result is that in
most of the situation for which we can make a reasonable definition of a delay
time, the obtained delay time is of order 1 second.


title:
Large-Scale Flow and Spiral Core Instability in Rayleigh-Benard
  Convection

abstract:
  The spiral core instability, observed in large aspect ratio Rayleigh-Benard
convection, is studied numerically in the framework of the Swift-Hohenberg
equation coupled to a large-scale flow. It is shown that the instability leads
to non-trivial core dynamics and is driven by the self-generated vorticity.
Moreover, the recently reported transition from spirals to hexagons near the
core is shown to occur only in the presence of a non-variational nonlinearity,
and is triggered by the spiral core instability. Qualitative agreement between
the simulations and the experiments is demonstrated.


title:
Interaction of Reaction-Diffusion Fronts and Marangoni Flow on the
  Interface of Deep Fluid

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


title:
Synchronization, chaos, and breakdown of collective domain oscillations
  in reaction-diffusion systems

abstract:
  The universal equations describing collective oscillations of the multidomain
patterns of small period in an arbitrary $d$-dimensional reaction-diffusion
system of the activator-inhibitor type are asymptotically derived. It is shown
that not far from the instability leading to the formation of the pulsating
multidomain pattern the oscillations of different domains synchronize. In one
dimension standing and traveling waves of the oscillation phase are realized.
In addition to these, in two dimensions target and spiral waves of the
oscillation phase, as well as spatio-temporal chaos of domain oscillations, are
feasible. Further inside the unstable region the collective oscillations break
down, so the pulsating multidomain pattern transforms into an irregular
pulsating pattern, the uniform self-oscillations, or turbulence. The parameter
regions where these effects occur are analyzed. The effects of the pattern's
disorder are also studied. The conclusions of the analysis are supported by the
numerical simulations of a concrete model. The obtained results explain the
dynamics of Turing patterns observed in the experiments on
chlorite-iodide-malonic acid reaction.


title: Kink Arrays and Solitary Structures in Optically Biased Phase Transition

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


title: Turing patterns and solitary structures under global control

abstract:
  Striped Turing patterns and solitary band and disk structures are constructed
using a three-variable multiscale model with cubic nonlinearity and global
control. The existence and stability conditions of regular structures are
analysed using the equation of motion of curved boundaries between alternative
states of the short-range component. The combined picture of transitions
between striped and spotted patterns with changing level of global control is
in qualitative agreement with the results of the computer experiment by Middya
and Luss


title: Renormalization Group Derivation of Phase Equations

abstract:
  Phase equations describing the evolution of large scale modulation of
spatially periodic patterns in two dimensional systems are derived by employing
the renormalization group method. A general formula for phase diffusion
coefficients is given under certain conditions.


title:
Bifurcating bright and dark solitary waves of the nearly nonlinear
  cubic-quintic Schrodinger equation

abstract:
  The existence of bright and dark multi-bump solitary waves for
Ginzburg-Landau type perturbations of the cubic-quintic Schrodinger equation is
considered. The waves in question are not perturbations of known analytic
solitary waves, but instead arise as a bifurcation from a heteroclinic cycle in
a three dimensional ODE phase space. Using geometric singular perturbation
techniques, regions in parameter space for which 1-bump bright and dark
solitary waves will bifurcate are identified. The existence of N-bump dark
solitary waves is shown via an application of the Exchange Lemma with
Exponentially Small Error. N-bump bright solitary waves are shown to exist as a
consequence of the previous work of S. Maier-Paape and this author.


title:
Dynamical Properties of Multi-Armed Global Spirals in Rayleigh-Benard
  Convection

abstract:
  Explicit formulas for the rotation frequency and the long-wavenumber
diffusion coefficients of global spirals with $m$ arms in Rayleigh-Benard
convection are obtained. Global spirals and parallel rolls share exactly the
same Eckhaus, zigzag and skewed-varicose instability boundaries. Global spirals
seem not to have a characteristic frequency $\omega_m$ or a typical size $R_m$,
but their product $\omega_m R_m$ is a constant under given experimental
conditions. The ratio $R_i/R_j$ of the radii of any two dislocations ($R_i$,
$R_j$) inside a multi-armed spiral is also predicted to be constant. Some of
these results have been tested by our numerical work.


title: Structure of Complex-Periodic and Chaotic Media with Spiral Waves

abstract:
  The spatiotemporal structure of reactive media supporting a solitary spiral
wave is studied for systems where the local reaction law exhibits a
period-doubling cascade to chaos. This structure is considerably more complex
than that of simple period-1 oscillatory media. As one moves from the core of
the spiral wave the local dynamics takes the form of perturbed period-doubled
orbits whose character varies with spatial location relative to the core. An
important feature of these media is the existence of a curve where the local
dynamics is effectively period-1. This curve arises as a consequence of the
necessity to reconcile the conflict between the global topological organization
of the medium induced by the presence of a spiral wave and the topological
phase space structure of local orbits determined by the reaction rate law. Due
to their general topological nature, the phenomena described here should be
observable in a broad class of systems with complex periodic behavior.


title:
Convection in rotating annuli: Ginzburg-Landau equations with tunable
  coefficients

abstract:
  The coefficients of the complex Ginzburg-Landau equations that describe
weakly nonlinear convection in a large rotating annulus are calculated for a
range of Prandtl numbers $\sigma$. For fluids with $\sigma \approx 0.15$, we
show that the rotation rate can tune the coefficients of the corresponding
amplitude equations from regimes where coherent patterns prevail to regimes of
spatio-temporal chaos.


title:
A domain wall between single-mode and bimodal states and its transition
  to dynamical behavior in inhomogeneous systems

abstract:
  We consider domain walls (DW's) between single-mode and bimodal states that
occur in coupled nonlinear diffusion (NLD), real Ginzburg-Landau (RGL), and
complex Ginzburg-Landau (CGL) equations with a spatially dependent coupling
coefficient. Group-velocity terms are added to the NLD and RGL equations, which
breaks the variational structure of these models. In the simplest case of two
coupled NLD equations, we reduce the description of stationary configurations
to a single second-order ordinary differential equation. We demonstrate
analytically that a necessary condition for existence of a stationary DW is
that the group-velocity must be below a certain threshold value. Above this
threshold, dynamical behavior sets in, which we consider in detail. In the CGL
equations, the DW may generate spatio-temporal chaos, depending on the
nonlinear dispersion.


title:
Extended Symbolic Dynamics in Bistable CML: Existence and Stability of
  Fronts

abstract:
  We consider a diffusive Coupled Map Lattice (CML) for which the local map is
piece-wise affine and has two stable fixed points. By introducing a
spatio-temporal coding, we prove the one-to-one correspondence between the set
of global orbits and the set of admissible codes. This relationship is applied
to the study of the (uniform) fronts' dynamics. It is shown that, for any given
velocity in $[-1,1]$, there is a parameter set for which the fronts with that
velocity exist and their shape is unique. The dependence of the velocity of the
fronts on the local map's discontinuity is proved to be a Devil's staircase.
Moreover, the linear stability of the global orbits which do not reach the
discontinuity follows directly from our simple map. For the fronts, this
statement is improved and as a consequence, the velocity of all the propagating
interfaces is computed for any parameter. The fronts are shown to be also
nonlinearly stable under some restrictions on the parameters. Actually, these
restrictions follow from the co-existence of uniform fronts and non-uniformly
travelling fronts for strong coupling. Finally, these results are extended to
some $C^{\infty}$ local maps.


title: Pattern selection as a nonlinear eigenvalue problem

abstract:
  A unique pattern selection in the absolutely unstable regime of driven,
nonlinear, open-flow systems is reviewed. It has recently been found in
numerical simulations of propagating vortex structures occuring in
Taylor-Couette and Rayleigh-Benard systems subject to an externally imposed
through-flow. Unlike the stationary patterns in systems without through-flow
the spatiotemporal structures of propagating vortices are independent of
parameter history, initial conditions, and system length. They do, however,
depend on the boundary conditions in addition to the driving rate and the
through-flow rate. Our analysis of the Ginzburg-Landau amplitude equation
elucidates how the pattern selection can be described by a nonlinear eigenvalue
problem with the frequency being the eigenvalue. Approaching the border between
absolute and convective instability the eigenvalue problem becomes effectively
linear and the selection mechanism approaches that of linear front propagation.
PACS: 47.54.+r,47.20.Ky,47.32.-y,47.20.Ft


title:
Influence of through-flow on linear pattern formation properties in
  binary mixture convection

abstract:
  We investigate how a horizontal plane Poiseuille shear flow changes linear
convection properties in binary fluid layers heated from below. The full linear
field equations are solved with a shooting method for realistic top and bottom
boundary conditions. Through-flow induced changes of the bifurcation thresholds
(stability boundaries) for different types of convective solutions are deter-
mined in the control parameter space spanned by Rayleigh number, Soret coupling
(positive as well as negative), and through-flow Reynolds number. We elucidate
the through-flow induced lifting of the Hopf symmetry degeneracy of left and
right traveling waves in mixtures with negative Soret coupling. Finally we
determine with a saddle point analysis of the complex dispersion relation of
the field equations over the complex wave number plane the borders between
absolute and convective instabilities for different types of perturbations in
comparison with the appropriate Ginzburg-Landau amplitude equation
approximation. PACS:47.20.-k,47.20.Bp, 47.15.-x,47.54.+r


title: Influence of boundaries on pattern selection in through-flow

abstract:
  The problem of pattern selection in absolutely unstable open flow systems is
investigated by considering the example of Rayleigh-B\'{e}nard convection. The
spatiotemporal structure of convection rolls propagating downstream in an
externally imposed flow is determined for six different inlet/outlet boundary
conditions. Results are obtained by numerical simulations of the Navier-Stokes
equations and by comparison with the corresponding Ginzburg-Landau amplitude
equation. A unique selection process is observed being a function of the
control parameters and the boundary conditions but independent of the history
and the system length. The problem can be formulated in terms of a nonlinear
eigen/boundary value problem where the frequency of the propagating pattern is
the eigenvalue. PACS: 47.54.+r, 47.20.Bp, 47.27.Te, 47.20.Ky


title: Stationary planar domain walls of a classical spin chain

abstract:
  Domain walls of a discrete model of an anisotropic ferromagnet are studied.
They can be described by sequences of two reversible mappings. Competition
between the length scale of spatial structures and the lattice constant leads
to a rich diversity of domain wall solutions related in a bifurcation scenario.


title: Direct Transition to Spatiotemporal Chaos in Low Prandtl Number Fluids

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

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


title:
An analytical stability theory for Faraday waves and the observation of
  the harmonic surface response

abstract:
  We present an analytical stability theory for the onset of the Faraday
instability, applying over a wide frequency range between shallow water gravity
and deep water capillary waves. For sufficiently thin fluid layers the surface
is predicted to occur in harmonic rather than subharmonic resonance with the
forcing. An experimental confirmation of this result is given. PACS: 47.20.Ma,
47.20.Gv, 47.15.Cb


title: Nonlinear Modulation of Travelling Rolls in Magnetoconvection

abstract:
  Modulational dynamics of oscillatory travelling rolls in magnetoconvection is
studied near the onset of a Hopf bifurcation. Using weakly nonlinear analysis,
we derive an envelope equation of oscillatory travelling rolls in the plane
perpendicular to an ambient vertical magnetic field.The envelope equation is
the Davey-Stewartson (DS) equation with complex coefficients, from which we
obtain criteria for the modulational (Benjamin-Feir) instability of oscillatory
travelling rolls.


title: Solitary vortex couples in viscoelastic Couette flow

abstract:
  We report experimental observation of a localized structure, which is of a
new type for dissipative systems. It appears as a solitary vortex couple
("diwhirl") in Couette flow with highly elastic polymer solutions. A unique
property of the diwhirls is that they are stationary, in contrast to the usual
localized wave structures in both Hamiltonian and dissipative systems which are
stabilized by wave dispersion. It is also a new object in fluid dynamics - a
couple of vortices that build a single entity somewhat similar to a magnetic
dipole. The diwhirls arise as a result of a purely elastic instability through
a hysteretic transition at negligible Reynolds numbers. It is suggested that
the vortex flow is driven by the same forces that cause the Weissenberg effect.
The diwhirls have a striking asymmetry between the inflow and outflow, which is
also an essential feature of the suggested elastic instability mechanism.


title: Solvable Optimal Velocity Models and Asymptotic Trajectory

abstract:
  In the Optimal Velocity Model proposed as a new version of Car Following
Model, it has been found that a congested flow is generated spontaneously from
a homogeneous flow for a certain range of the traffic density. A
well-established congested flow obtained in a numerical simulation shows a
remarkable repetitive property such that the velocity of a vehicle evolves
exactly in the same way as that of its preceding one except a time delay $T$.
This leads to a global pattern formation in time development of vehicles'
motion, and gives rise to a closed trajectory on $\Delta x$-$v$
(headway-velocity) plane connecting congested and free flow points. To obtain
the closed trajectory analytically, we propose a new approach to the pattern
formation, which makes it possible to reduce the coupled car following
equations to a single difference-differential equation (Rondo equation). To
demonstrate our approach, we employ a class of linear models which are exactly
solvable. We also introduce the concept of ``asymptotic trajectory'' to
determine $T$ and $v_B$ (the backward velocity of the pattern), the global
parameters associated with vehicles' collective motion in a congested flow, in
terms of parameters such as the sensitivity $a$, which appeared in the original
coupled equations.


title: Vortex Dynamics in Dissipative Systems

abstract:
  We derive the exact equation of motion for a vortex in two- and three-
dimensional non-relativistic systems governed by the Ginzburg-Landau equation
with complex coefficients. The velocity is given in terms of local gradients of
the magnitude and phase of the complex field and is exact also for arbitrarily
small inter-vortex distances. The results for vortices in a superfluid or a
superconductor are recovered.

categories: patt-sol chao-dyn cond-mat.supr-con hep-th nlin.CD nlin.PS physics.chem-ph physics.flu-dyn


title: The Dynamics of Curved Fronts: Beyond Geometry

abstract:
  We derive a new set of kinematic equations for front motion in
two-dimensional bistable media. The equations generalize the geometric approach
by complementing the equation for the front curvature with an order parameter
equation associated with a nonequilibrium Ising-Bloch bifurcation. The
resulting equations capture the core structure of spiral waves and spontaneous
spiral-wave nucleation.


title:
A Mechanism for Spatio-Temporal Disorder in Bistable Reaction-Diffusion
  Systems

abstract:
  In bistable systems, the stability of front structures often influences the
dynamics of extended patterns. We show how the combined effect of an
instability to curvature modulations and proximity to a pitchfork front
bifurcation leads to spontaneous nucleation of spiral waves along the front.
This effect is demonstrated by direct simulations of a FitzHugh-Nagumo (FHN)
model and by simulations of order parameter equations for the front velocity
and curvature. Spontaneous spiral-wave nucleation often results in a state of
spatio-temporal disorder involving repeated events of spiral wave nucleation,
domain spliting and spiral wave annihilation.


title:
Geometric Stability Analysis for Periodic Solutions of the
  Swift-Hohenberg Equation

abstract:
  In this paper we describe invariant geometrical ~structures in the phase
space of the Swift-Hohenberg equation in a neighborhood of its periodic
stationary states. We show that in spite of the fact that these states are only
marginally stable (i.e., the linearized problem about these states has
continuous spectrum extending all the way up to zero), there exist finite
dimensional invariant manifolds in the phase space of this equation which
determine the long-time behavior of solutions near these stationary solutions.
In particular, using this point of view, we obtain a new demonstration of
Schneider's recent proof that these states are nonlinearly stable.

categories: patt-sol chao-dyn funct-an math.FA nlin.CD nlin.PS physics.flu-dyn


title:
Sources and sinks separating domains of left- and right-traveling waves:
  Experiment versus amplitude equations

abstract:
  In many pattern forming systems that exhibit traveling waves, sources and
sinks occur which separate patches of oppositely traveling waves. We show that
simple qualitative features of their dynamics can be compared to predictions
from coupled amplitude equations. In heated wire convection experiments, we
find a discrepancy between the observed multiplicity of sources and theoretical
predictions. The expression for the observed motion of sinks is incompatible
with any amplitude equation description.


title:
Optical Solitary Waves in the Higher Order Nonlinear Schrodinger
  Equation

abstract:
  We study solitary wave solutions of the higher order nonlinear Schrodinger
equation for the propagation of short light pulses in an optical fiber. Using a
scaling transformation we reduce the equation to a two-parameter canonical
form. Solitary wave (1-soliton) solutions exist provided easily met inequality
constraints on the parameters in the equation are satisfied. Conditions for the
existence of N-soliton solutions (N>1) are determined; when these conditions
are met the equation becomes the modified KdV equation. A proper subset of
these conditions meet the Painleve plausibility conditions for integrability.


title:
Worm Structure in Modified Swift-Hohenberg Equation for
  Electroconvection

abstract:
  A theoretical model for studying pattern formation in electroconvection is
proposed in the form of a modified Swift-Hohenberg equation. A localized state
is found in two dimension, in agreement with the experimentally observed
``worm" state. The corresponding one dimensional model is also studied, and a
novel stationary localized state due to nonadiabatic effect is found. The
existence of the 1D localized state is shown to be responsible for the
formation of the two dimensional ``worm" state in our model.


title: Bistability of Slow and Fast Traveling Waves in Fluid Mixtures

abstract:
  The appearence of a new type of fast nonlinear traveling wave states in
binary fluid convection with increasing Soret effect is elucidated and the
parameter range of their bistability with the common slower ones is evaluated
numerically. The bifurcation behavior and the significantly different
spatiotemporal properties of the different wave states - e.g. frequency, flow
structure, and concentration distribution - are determined and related to each
other and to a convenient measure of their nonlinearity. This allows to derive
a limit for the applicability of small amplitude expansions. Additionally an
universal scaling behavior of frequencies and mixing properties is found.
  PACS: 47.20.-k, 47.10.+g, 47.20.Ky


title:
Energy thresholds for discrete breathers in one-, two- and
  three-dimensional lattices

abstract:
  Discrete breathers are time-periodic, spatially localized solutions of
equations of motion for classical degrees of freedom interacting on a lattice.
They come in one-parameter families. We report on studies of energy properties
of breather families in one-, two- and three-dimensional lattices. We show that
breather energies have a positive lower bound if the lattice dimension of a
given nonlinear lattice is greater than or equal to a certain critical value.
These findings could be important for the experimental detection of discrete
breathers.


title: Diffusive Boundary Layers in the Free-Surface Excitable Medium Spiral

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


title: Pattern Formation from Defect Chaos --- A Theory of Chevrons

abstract:
  For over 25 years it is known that the roll structure of electroconvection
(EC) in the dielectric regime in planarly aligned nematic liquid crystals has,
after a transition to defect chaos, the tendency to form chevron structures. We
show, with the help of a coarse-grained model, that this effect can generally
be expected for systems with spontaneously broken isotropy, that is lifted by a
small external perturbation. The linearized model as well as a nonlinear
extension are compared to simulations of a system of coupled amplitude
equations which generate chevrons out of defect chaos. The mechanism of chevron
formation is similar to the development of Turing patterns in reaction
diffusion systems.


title:
Synchronization of globally-coupled phase oscillators: singularities and
  scaling for general couplings

abstract:
  The onset of collective behavior in a population of globally coupled
oscillators with randomly distributed frequencies is studied for phase
dynamical models with arbitrary coupling; the effect of a stochastic temporal
variation in the frequencies is also included. The Fokker-Planck equation for
the coupled Langevin system is reduced to a kinetic equation for the oscillator
distribution function. Instabilities of the phase-incoherent state are studied
by center manifold reduction to the amplitude dynamics of the unstable modes.
Depending on the coupling, the coefficients in the normal form can be singular
in the limit of weak instability when the diffusive effect of the noise is
neglected. A detailed analysis of these singularities to all orders in the
normal form expansion is presented. Physically, the singularities are
interpreted as predicting an altered scaling of the entrained component near
the onset of synchronization.


title:
Travelling waves in electroconvection of the nematic Phase 5: A test of
  the weak electrolyte model

abstract:
  We investigated travelling waves appearing as the primary pattern-forming
instability in the nematic Phase 5 (Merck) in the planar geometry in order to
test the recently developed weak electrolyte model of ac-driven
electroconvection [M. Treiber and L. Kramer, Mol. Cryst. Liq. Cryst 261, 311
(1995)]. Travelling waves are observed over the full conductive range of
applied frequencies for three cells of different layer thickness d. We also
measured the elastic constants, the electric conductivity, and the dielectric
constant. The other parameters of Phase 5 are known, apart from the (relatively
unimportant) viscosity $\alpha_1$ and the two parameters of the weak
electrolyte model that are proportional to the geometric mean of the
mobilities, and the recombination rate, respectively. Assuming a sufficiently
small recombination rate, the predicted dependence of the frequency of the
travelling waves at onset (Hopf frequency) on d and on the applied frequency
agreed quantitatively with the experiments, essentially without fit parameters.
The absolute value of the Hopf frequency implies that the geometric mean of the
mobilities amounts to $1.1 \times 10^{-10} m^2/(Vs)$.


title:
Predicting Non-linear Cellular Automata Quickly by Decomposing Them into
  Linear Ones

abstract:
  We show that a wide variety of non-linear cellular automata (CAs) can be
decomposed into a quasidirect product of linear ones. These CAs can be
predicted by parallel circuits of depth O(log^2 t) using gates with binary
inputs, or O(log t) depth if ``sum mod p'' gates with an unbounded number of
inputs are allowed. Thus these CAs can be predicted by (idealized) parallel
computers much faster than by explicit simulation, even though they are
non-linear.
  This class includes any CA whose rule, when written as an algebra, is a
solvable group. We also show that CAs based on nilpotent groups can be
predicted in depth O(log t) or O(1) by circuits with binary or ``sum mod p''
gates respectively.
  We use these techniques to give an efficient algorithm for a CA rule which,
like elementary CA rule 18, has diffusing defects that annihilate in pairs.
This can be used to predict the motion of defects in rule 18 in O(log^2 t)
parallel time.

categories: patt-sol adap-org comp-gas nlin.AO nlin.CG nlin.PS


title:
Pattern formation in weakly damped parametric surface waves driven by
  two frequency components

abstract:
  A quasi-potential approximation to the Navier-Stokes equation for low
viscosity fluids is developed to study pattern formation in parametric surface
waves driven by a force that has two frequency components. A bicritical line
separating regions of instability to either one of the driving frequencies is
explicitly obtained, and compared with experiments involving a frequency ratio
of 1/2. The procedure for deriving standing wave amplitude equations valid near
onset is outlined for an arbitrary frequency ratio following a multiscale
asymptotic expansion of the quasi-potential equations. Explicit results are
presented for subharmonic response to a driving force of frequency ratio 1/2,
and used to study pattern selection. Even though quadratic terms are prohibited
in this case, hexagonal or triangular patterns are found to be stable in a
relatively large parameter region, a fact that is in qualitative agreement with
experimental results.


title:
Order Parameter Equations for Front Transitions: Planar and Circular
  Fronts

abstract:
  Near a parity breaking front bifurcation, small perturbations may reverse the
propagation direction of fronts. Often this results in nonsteady asymptotic
motion such as breathing and domain breakup. Exploiting the time scale
differences of an activator-inhibitor model and the proximity to the front
bifurcation, we derive equations of motion for planar and circular fronts. The
equations involve a translational degree of freedom and an order parameter
describing transitions between left and right propagating fronts.
Perturbations, such as a space dependent advective field or uniform curvature
(axisymmetric spots), couple these two degrees of freedom. In both cases this
leads to a transition from stationary to oscillating fronts as the parity
breaking bifurcation is approached. For axisymmetric spots, two additional
dynamic behaviors are found: rebound and collapse.


title:
Stability criterion for bright solitary waves of the perturbed
  cubic-quintic Schroedinger equation

abstract:
  The stability of the bright solitary wave solution to the perturbed
cubic-quintic Schroedinger equation is considered. It is shown that in a
certain region of parameter space these solutions are unstable, with the
instability being manifested as a small positive eigenvalue. Furthermore, it is
shown that in the complimentary region of parameter space there are no small
unstable eigenvalues. The proof involves a novel calculation of the Evans
function, which is of interest in its own right. As a consequence of the
eigenvalue calculation, it is additionally shown that N-bump bright solitary
waves bifurcate from the primary wave.


title: Renormalization Group Method and Reductive Perturbation Method

abstract:
  It is shown that the renormalization group method does not necessarily
eliminate all secular terms in perturbation series to partial differential
equations and a functional subspace of renormalizable secular solutions
corresponds to a choice of scales of independent variables in the reductive
perturbation method.


title: Amplitude equations and pattern selection in Faraday waves

abstract:
  We present a systematic nonlinear theory of pattern selection for parametric
surface waves (Faraday waves), not restricted to fluids of low viscosity. A
standing wave amplitude equation is derived from the Navier-Stokes equations
that is of gradient form. The associated Lyapunov function is calculated for
different regular patterns to determine the selected pattern near threshold.
For fluids of large viscosity, the selected wave pattern consists of parallel
stripes. At lower viscosity, 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. The regions of stability of the various patterns are in
quantitative agreement with recent experiments conducted in large aspect ratio
systems.


title:
Transition from Spatiotemporal Chaos to Ideal Straight Rolls in
  Rayleigh-Benard Convection

abstract:
  For Rayleigh-Benard convection in a square cell with a fluid of Prandtl
number one, we report experimental results on the transition between a
stationary pattern of ideal straight rolls (ISR) and the spatiotemporal chaotic
state of spiral defect chaos (SDC). In contrast to experiments in circular
geometries, we found ISR states below a particular value of the control
parameter and SDC states above this value. By characterizing the pattern with a
global measure, the pattern entropy, we found that the transition from SDC to
ISR showed similarities to phase transitions in equilibrium finite-size
systems.


title:
Competition between Spiral-Defect Chaos and Rolls in Rayleigh-Benard
  Convection

abstract:
  We present experimental results for pattern formation in Rayleigh-Benard
convection of a fluid with a Prandtl number, Pr~ 4. We find that the
spiral-defect-chaos (SDC) attractor which exists for Pr~1 has become unstable.
Gradually increasing the temperature difference from below to well above its
critical value no longer leads to SDC. A sudden jump of temperature difference
from below to above onset causes convection to grow from thermal fluctuations
and does yield SDC. However, the SDC is a transient; it coarsens and forms a
single cell-filling spiral which then drifts toward the cell wall and
disappears.


title: Streamer Propagation as a Pattern Formation Problem: Planar Fronts

abstract:
  Streamers often constitute the first stage of dielectric breakdown in strong
electric fields: a nonlinear ionization wave transforms a non-ionized medium
into a weakly ionized nonequilibrium plasma. New understanding of this old
phenomenon can be gained through modern concepts of (interfacial) pattern
formation. As a first step towards an effective interface description, we
determine the front width, solve the selection problem for planar fronts and
calculate their properties. Our results are in good agreement with many
features of recent three-dimensional numerical simulations.


title: Propagation and Structure of Planar Streamer Fronts

abstract:
  Streamers often constitute the first stage of dielectric breakdown in strong
electric fields: a nonlinear ionization wave transforms a non-ionized medium
into a weakly ionized nonequilibrium plasma. New understanding of this old
phenomenon can be gained through modern concepts of (interfacial) pattern
formation. As a first step towards an effective interface description, we
determine the front width, solve the selection problem for planar fronts and
calculate their properties. Our results are in good agreement with many
features of recent three-dimensional numerical simulations.
  In the present long paper, you find the physics of the model and the
interfacial approach further explained. As a first ingredient of this approach,
we here analyze planar fronts, their profile and velocity. We encounter a
selection problem, recall some knowledge about such problems and apply it to
planar streamer fronts. We make analytical predictions on the selected front
profile and velocity and confirm them numerically.
  (abbreviated abstract)


title: Weakly Nonlinear Analysis of Electroconvection in a Suspended Fluid Film

abstract:
  It has been experimentally observed that weakly conducting suspended films of
smectic liquid crystals undergo electroconvection when subjected to a large
enough potential difference. The resulting counter-rotating vortices form a
very simple convection pattern and exhibit a variety of interesting nonlinear
effects. The linear stability problem for this system has recently been solved.
The convection mechanism, which involves charge separation at the free surfaces
of the film, is applicable to any sufficiently two-dimensional fluid. In this
paper, we derive an amplitude equation which describes the weakly nonlinear
regime, by starting from the basic electrohydrodynamic equations. This regime
has been the subject of several recent experimental studies. The lowest order
amplitude equation we derive is of the Ginzburg-Landau form, and describes a
forward bifurcation as is observed experimentally. The coefficients of the
amplitude equation are calculated and compared with the values independently
deduced from the linear stability calculation.


title: Hexagonal patterns in finite domains

abstract:
  In many mathematical models for pattern formation, a regular hexagonal
pattern is stable in an infinite region. However, laboratory and numerical
experiments are carried out in finite domains, and this imposes certain
constraints on the possible patterns. In finite rectangular domains, it is
shown that a regular hexagonal pattern cannot occur if the aspect ratio is
rational. In practice, it is found experimentally that in a rectangular region,
patterns of irregular hexagons are often observed. This work analyses the
geometry and dynamics of irregular hexagonal patterns. These patterns occur in
two different symmetry types, either with a reflection symmetry, involving two
wavenumbers, or without symmetry, involving three different wavenumbers. The
relevant amplitude equations are studied to investigate the detailed
bifurcation structure in each case. It is shown that hexagonal patterns can
bifurcate subcritically either from the trivial solution or from a pattern of
rolls. Numerical simulations of a model partial differential equation are also
presented to illustrate the behaviour.


title:
Nonlinear Dynamics of a Single Ferrofluid-Peak in an Oscillating
  Magnetic Field

abstract:
  If a magnetic field normal to the surface of a magnetic fluid is increased
beyond a critical value a spontaneous deformation of the surface arises (normal
field instability). The instability is subcritical and leads to peaks of a
characteristic shape. We investigate the neighborhood of this instability
experimentally under the influence of a temporal modulation of the magnetic
field. We use a small vessel, where only one peak arises. The modulation can
either be stabilizing or destabilizing, depending on the frequency and
amplitude. We observe a cascade of odd-numbered response-periods up to period
11, and also a domain of even-numbered periods. We propose a minimal model
involving a cutoff-condition which captures the essence of the experimental
observations.
  PACS: 47.20.-k, 47.20.Ky, 75.50.Mm
  Keywords: magnetic fluid; nonlinear oscillator; subharmonic response; surface
instability;


title: Derivation of Amplitude Equations by Renormalization Group Method

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


title: A discrete phi^4 system without Peierls-Nabarro barrier

abstract:
  A discrete phi^4 system is proposed which preserves the topological lower
bound on the kink energy. Existence of static kink solutions saturating this
lower bound and occupying any position relative to the lattice is proved.
Consequently, kinks of the model experience no Peierls-Nabarro barrier, and can
move freely through the lattice without being pinned. Numerical simulations
reveal that kink dynamics in this system is significantly less dissipative than
that of the conventional discrete phi^4 system, so that even on extremely
coarse lattices the kink behaves much like its continuum counterpart. It is
argued, therefore, that this is a natural discretization for the purpose of
numerically studying soliton dynamics in the continuum phi^4 model.


title: Spinodal Decomposition in Binary Gases

abstract:
  We carried out three-dimensional simulations, with about 1.4 million
particles, of phase segregation in a low density binary fluid mixture,
described mesoscopically by energy and momentum conserving Boltzmann-Vlasov
equations. Using a combination of Direct Simulation Monte Carlo(DSMC) for the
short range collisions and a version of Particle-In-Cell(PIC) evolution for the
smooth long range interaction, we found dynamical scaling after the ratio of
the interface thickness(whose shape is described approximately by a hyperbolic
tangent profile) to the domain size is less than ~0.1. The scaling length R(t)
grows at late times like t^alpha, with alpha=1 for critical quenches and
alpha=1/3 for off-critical ones. We also measured the variation of temperature,
total particle density and hydrodynamic velocity during the segregation
process.


title: On the Origin of Traveling Pulses in Bistable Systems

abstract:
  The interaction between a pair of Bloch fronts forming a traveling domain in
a bistable medium is studied. A parameter range beyond the nonequilibrium
Ising-Bloch bifurcation is found where traveling domains collapse. Only beyond
a second threshold the repulsive front interactions become strong enough to
balance attractive interactions and asymmetries in front speeds, and form
stable traveling pulses. The analysis is carried out for the forced complex
Ginzburg-Landau equation. Similar qualitative behavior is found in the bistable
FitzHugh-Nagumo model.


title:
Resonances of dynamical checkerboard states in Josephson arrays with
  self-inductance

abstract:
  We study the dynamics of fully frustrated, underdamped Josephson arrays.
Experiments reveal remarkable similarities among the dc current-voltage
characteristics of several kinds of square and triangular arrays, where two
resonant voltages are observed. Simulations indicate that a dynamical
checkerboard solution underlies these similarities. By assuming such a
solution, we reduce the governing equations to three coupled pendulum
equations, and thereby calculate the voltages of the intrinsic resonances
analytically.


title: Localized and Cellular Patterns in a Vibrated Granular Layer

abstract:
  We propose a phenomenological model for pattern formation in a vertically
vibrated layer of granular material. This model exhibits a variety of stable
cellular patterns including standing rolls and squares as well as localized
objects (oscillons and worms), similar to recent experimental
observations(Umbanhowar et al., 1996). The model is an amplitude equation for
the parametrical instability coupled to the mass conservation law. The
structure and dynamics of the solutions resemble closely the properties of
localized and cellular patterns observed in the experiments.


title:
Liquefaction of a Horizontally Vibrated Granular Bed: Friction, Dilation
  and Segregation

abstract:
  We present experimental observations of the onset of flow (liquefaction) for
horizontally vibrated granular materials. As the acceleration increases above
certain value, the top layer of granular material liquefies, while the
remainder of the layer moves with the shaker in solid body motion. With
increasing acceleration, more of the layer becomes fluidized. The initial
bifurcation is backward, and the amount of hysteresis depends mainly on
frictional properties of the granular media. A small amount of fluidization by
gas flow lifts the hysteresis. Modest differences in the frictional properties
of otherwise identical particles leads to rapid segregation.


title: Instabilities of periodic orbits with spatio-temporal symmetries

abstract:
  Motivated by recent analytical and numerical work on two- and
three-dimensional convection with imposed spatial periodicity, we analyse three
examples of bifurcations from a continuous group orbit of spatio-temporally
symmetric periodic solutions of partial differential equations. Our approach is
based on centre manifold reduction for maps, and is in the spirit of earlier
work by Iooss (1986) on bifurcations of group orbits of spatially symmetric
equilibria. Two examples, two-dimensional pulsating waves (PW) and
three-dimensional alternating pulsating waves (APW), have discrete
spatio-temporal symmetries characterized by the cyclic groups Z_n, n=2 (PW) and
n=4 (APW). These symmetries force the Poincare return map M to be the nth
iterate of a map G: M=G^n. The group orbits of PW and APW are generated by
translations in the horizontal directions and correspond to a circle and a
two-torus, respectively. An instability of pulsating waves can lead to
solutions that drift along the group orbit, while bifurcations with Floquet
multiplier +1 of alternating pulsating waves do not lead to drifting solutions.
The third example we consider, alternating rolls, has the spatio-temporal
symmetry of alternating pulsating waves as well as being invariant under
reflections in two vertical planes. When the bifurcation breaks these
reflections, the map G has a "two-symmetry", as analysed by Lamb (1996). This
leads to a doubling of the marginal Floquet multiplier and the possibility of
bifurcation to two distinct types of drifting solutions.


title:
Compacton Solutions in a Class of Generalized Fifth Order Korteweg-de
  Vries Equations

abstract:
  We study a class of generalized fifth order Korteweg-de Vries (KdV) equations
which are derivable from a Lagrangian L(p,m,n,l) which has variable powers of
the first and second derivatives of the field with powers given by the
parameters p,m,n,l. The resulting field equation has solitary wave solutions of
both the usual (non-compact) and compact variety ("compactons"). For the
particular case that p=m=n+l, the solitary wave solutions have compact support
and the feature that their width is independent of the amplitude. We discuss
the Hamiltonian structure of these theories and find that mass, momentum, and
energy are conserved. We find in general that these are not completely
integrable systems. Numerical simulations show that an arbitrary compact
initial wave packet whose width is wider than that of a compacton breaks up
into several compactons all having the same width. The scattering of two
compactons is almost elastic, with the left over wake eventually turning into
compacton-anticompacton pairs. When there are two different compacton solutions
for a single set of parameters the wider solution is stable, and this solution
is a minimum of the Hamiltonian.

categories: patt-sol hep-th nlin.PS nlin.SI solv-int


title: Discrete Breathers

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


title: Onset of oscillatory instabilities under stochastic modulation

abstract:
  We study the effect of external stochastic modulation on a system with O(2)
symmetry that exhibits a Hopf or oscillatory instability in the absence of
modulation. The study includes a random component in both the control parameter
of the bifurcation and in the modulation amplitude. Stability boundaries are
computed by either solving the stationary Fokker-Planck equation in the
vicinity of the center manifold of the underlying deterministic system whenever
possible, or by direct numerical solution otherwise. If the modulation
amplitude has a stochastic component, the primary bifurcation is always to
standing waves at a value of the control parameter that depends on the
intensity of the fluctuations. More precisely, and to contrast our results with
the case of a deterministic periodic forcing, the onset of instability in the
standing wave regime is shifted from its deterministic location, and the region
of primary bifurcation to traveling waves disappears yielding instead standing
waves at negative values of the control parameter.


title: Spiral Dynamics in Pattern-Forming Systems: Mean Flow Effects

abstract:
  Mean flow effects are discussed for two different pattern-forming systems:
Rayleigh-Benard convection and Faraday instability in viscous fluid. In both
systems spirals are observed in certain parameter regions. In the
Rayleigh-Benard convection, the spiral core instability and subsequent
generation of up- and downflow hexagons are shown to occur due to the mean flow
generated by the curved rolls near the core. In the Faraday instability, the
mean flow which is generated by rapidly decaying surface waves near the wall,
causes wavenumber frustration which leads to a rigid-body spiral rotation. In
both cases we use phenomenological Swift-Hohenberg-type equations for the order
parameter coupled to a large-scale mean flow. Numerical simulations are
compared to recently reported experimental results.


title:
Phase Segregation Dynamics In Particle Systems with Long Range
  Interactions II: Interface motion

abstract:
  We study properties of the solutions of a family of second order
integro-differential equations, which describe the large scale dynamics of a
class of microscopic phase segregation models with particle conserving
dynamics. We first establish existence and uniqueness as well as some
properties of the instantonic solutions. Then we concentrate on formal
asymptotic (sharp interface) limits. We argue that the obtained interface
evolution laws (a Stefan-like problem and the Mullins-Sekerka solidification
model) coincide with the ones which can be obtained in the analogous limits
from the Cahn-Hilliard equation, the fourth order PDE which is the standard
macroscopic model for phase segregation with one conservation law.


title: A Lubrication Model of Coating Flows over a Curved Substrate in Space

abstract:
  Consider the three-dimensional flow of a viscous Newtonian fluid upon an
abitrarily curved substrate when the fluid film is thin as occurs in many
draining, coating and biological flows. We derive a model of the dynamics of
the film, the model being expressed in terms of the film thickness and the
curvature tensor of the substrate. The model accurately includes the effects of
the curvature of the substrate, via a physical multiple-scale approach, and
gravity and inertia, via more rigorous centre manifold techniques. Numerical
simulations exhibit some generic features of the dynamics of such thin fluid
films on substrates with complex curvature.


title:
Dynamical Reduction of Discrete Systems Based on the Renormalization
  Group Method

abstract:
  The renormalization group (RG) method is extended for global asymptotic
analysis of discrete systems. We show that the RG equation in the discretized
form leads to difference equations corresponding to the Stuart-Landau or
Ginzburg-Landau equations. We propose a discretization scheme which leads to a
faithful discretization of the reduced dynamics of the original differential
equations.

categories: patt-sol chao-dyn cond-mat hep-th nlin.CD nlin.PS nlin.SI solv-int


title:
Selection of the Saffman-Taylor Finger Width in the Absence of Surface
  Tension: an Exact Result

abstract:
  Using our exact time-depending solutions, we solve the Saffman-Taylor finger
selection problem in the absence of surface tension by showing that an
arbitrary interface in a Hele-Shaw cell evolves to a single uniformly advancing
finger occupying one half of the channel width.
  This result contradicts the generally accepted belief that surface tension is
indispensable for the selection of the one-half-width finger.


title: Unicellular algal growth: A biomechanical approach to cell wall dynamics

abstract:
  We present a model for unicellular algal growth as motivated by several
experiments implicating the importance of calcium ions and ``loosening''
enzymes in morphogenesis. A growing cell at rest in a diffusive calcium
solution is viewed as an elastic shell on short timescales. For a given turgor
pressure, we calculate the stressed shapes of the wall elements whose elastic
properties are determined by Young's modulus and the thickness of the wall. The
local enzyme concentration then determines the rate at which the unstressed
shape of a wall element relaxes toward its stressed shape. The local wall
thickness is calculated from the calcium-mediated addition of material and
thinning due to elongation. We use this model to calculate growth rates for
small perturbations to a circular cell. We find an instability related to
modulations of the wall thickness, leading to growth rates which peak at a
finite wave number.


title: The branching structure of diffusion-limited aggregates

abstract:
  I analyze the topological structures generated by diffusion-limited
aggregation (DLA), using the recently developed "branched growth model". The
computed bifurcation number B for DLA in two dimensions is B ~ 4.9, in good
agreement with the numerically obtained result of B ~ 5.2. In high dimensions,
B -> 3.12; the bifurcation ratio is thus a decreasing function of
dimensionality. This analysis also determines the scaling properties of the
ramification matrix, which describes the hierarchy of branches.


title: On Envelope Dynamics In 2D Faraday Waves

abstract:
  A weakly nonlinear model for two-dimensional Faraday waves over infinite
depth is derived and studied. Sideband instability of monochromatic standing
waves as well as non-monochromatic solutions are studied analytically.
Persistent irregular regimes are found numerically.