title: Point symmetries in the Hartree-Fock approach: Symmetry-breaking schemes

abstract:
  We analyze breaking of symmetries that belong to the double point group
D2h(TD) (three mutually perpendicular symmetry axes of the second order,
inversion, and time reversal). Subgroup structure of the D2h(TD) group
indicates that there can be as much as 28 physically different, broken-symmetry
mean-field schemes --- starting with solutions obeying all the symmetries of
the D2h(TD) group, through 26 generic schemes in which only a non-trivial
subgroup of D2h(TD) is conserved, down to solutions that break all of the
D2h(TD) symmetries. Choices of single-particle bases and the corresponding
structures of single-particle hermitian operators are discussed for several
subgroups of D2h(TD).


title:
The new mechanism for intermediate- and short-range nucleon-nucleon
  interaction

abstract:
  Arguments against the traditional Yukawa-type approach to $NN$ intermediate-
and short-range interaction due to scalar-isoscalar meson exchange are
presented. Instead of the Yukawa mechanism for intermediate-range attraction
some new approach based on formation of the symmetric six-quark bag in the
state $|(0s)^6[6]_X,L=0>$ dressed due to strong coupling to $\pi$, $\sigma$ and
$\rho$ fields are suggested. These new mechanism offers a strong
intermediate-range attraction which replaces the effective $\sigma$-exchange
(or excitation of two isobars in the intermediate state) in traditional force
models. A similar mechanism with vector $\rho$-meson production in the
intermediate six-quark state is expected to lead to a strong short-range
spin-orbital nonlocal interaction in the $NN$ system, which may resolve the
long-standing puzzle of the spin-orbit force in baryons and in two-baryon
systems. Illustrative examples are developed which demonstrate clearly how well
the suggested new model can reproduce $NN$ data. Strong interrelations have
been shown to exist between the proposed microscopic model and the
one-component Moscow $NN$ potential developed by the authors previously and
also with some hybrid models and the one-term separable Tabakin potential. The
new implications of the proposed model for nuclear physics are discussed.


title: Implementing PCAC in Nonperturbative Models of Pion Production

abstract:
  Traditional few-body descriptions of pion production use integral equations
to sum the strong interactions nonperturbatively. Although much physics is
thereby included, there has not been a practical way of incorporating the
constraints of chiral symmetry into such approaches. Thus the traditional
few-body descriptions fail to reflect the underlying theory of strong
interactions, QCD, which is largely chirally symmetric. In addition, the lack
of chiral symmetry in the few-body approaches means that their predictions of
pion production are in principle not consistent with the partial conservation
of axial current (PCAC), a fact that has especially large consequences at low
energies. We discuss how the recent introduction of the ``gauging of equations
method'' can be used to include PCAC into traditional few-body descriptions and
thereby solve this long standing problem


title: Positron Tunnelling through the Coulomb Barrier of Superheavy Nuclei

abstract:
  We study beams of medium-energy electrons and positrons which obey the Dirac
equation and scatter from nuclei with $Z > 100.$ At small distances the
potential is modelled to be that of a charged sphere. A large peak is found in
the probability of positron penetration to the origin for $Z \approx 184.$ This
may be understood as an example of Klein tunnelling through the Coulomb
barrier: it is the analogue of the Klein Paradox for the Coulomb potential.


title:
Multidimensional Pattern Formation Has an Infinite Number of Constants
  of Motion

abstract:
  Extending our previous work on 2D growth for the Laplace equation we study
here {\it multidimensional} growth for {\it arbitrary elliptic} equations,
describing inhomogeneous and anisotropic pattern formations processes. We find
that these nonlinear processes are governed by an infinite number of
conservation laws. Moreover, in many cases {\it all dynamics of the interface
can be reduced to the linear time--dependence of only one ``moment" $M_0$}
which corresponds to the changing volume while {\it all higher moments, $M_l$,
are constant in time. These moments have a purely geometrical nature}, and thus
carry information about the moving shape. These conserved quantities (eqs.~(7)
and (8) of this article) are interpreted as coefficients of the multipole
expansion of the Newtonian potential created by the mass uniformly occupying
the domain enclosing the moving interface. Thus the question of how to recover
the moving shape using these conserved quantities is reduced to the classical
inverse potential problem of reconstructing the shape of a body from its
exterior gravitational potential. Our results also suggest the possibility of
controlling a moving interface by appropriate varying the location and strength
of sources and sinks.


title:
Discrete Dynamical Models Showing Pattern Formation in Subaqueous
  Bedforms

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


title:
Rotating Rayleigh-B\'enard Convection: Aspect Ratio Dependence of the
  Initial Bifurcations

abstract:
  The initial bifurcations in rotating Rayleigh-B\'enard convection are studied
in the range of dimensionless rotation rate $0 < \Omega < 2150$ for an
aspect-ratio-2.5 cylindrical cell. We used simultaneous optical shadowgraph,
heat transport and local temperature measurements to determine the stability
and characteristics of the azimuthally-periodic wall convection state. We also
show that the second transition corresponds to the onset of bulk convection.
Our results for critical Rayleigh numbers, precession frequencies and critical
mode numbers agree well with theoretical results. The dynamics of the wall
convection state can be described by a complex Ginzburg-Landau amplitude
equation.


title: Asymptotic Behavior of Excitable Cellular Automata

abstract:
  We study two families of excitable cellular automata known as the
Greenberg-Hastings Model (GHM) and the Cyclic Cellular Automaton (CCA). Each
family consists of local deterministic oscillating lattice dynamics, with
parallel discrete-time updating, parametrized by the range of interaction, the
"shape" of its neighbor set, threshold value for contact updating, and number
of possible states per site. GHM and CCA are mathematically tractable
prototypes for the spatially distributed periodic wave activity of so-called
excitable media observed in diverse disciplines of experimental science.
Earlier work by Fisch, Gravner, and Griffeath studied the ergodic behavior of
these excitable cellular automata on Z^2, and identified two distinct (but
closely-related) elaborate phase portraits as the parameters vary. In
particular, they noted the emergence of asymptotic phase diagrams (and
Euclidean dynamics) in a well-defined threshold-range scaling limit. In this
study we present several rigorous results and some experimental findings
concerning various phase transitions in the asymptotic diagrams, focusing on
evaluating the limiting threshold cutoff for existence of the spirals that
characterize many excitable media. For mathematical expediency our main results
are formulated in terms of spo(p), the cutoff for existence of stable periodic
objects that arise as spiral cores.


title: Solitons and Long Josephson Junctions

abstract:
  Magnetic flux quanta, of value h/2e, in long Josephson junctions behave as
(quasi) solitons. Fluxon dynamical states are well described by a perturbed
sine-Gordon equation model, with boundary conditions determined by the junction
geometry and by externally applied magnetic fields, and they give rise to
readily measurable physical phenomena, such as step structure in
current-voltage characteristics and microwave radiation emission. Devices based
on fluxon propagation offer potentially interesting applications as oscillators
and amplifiers - as well as digital applica- tions, described elsewhere in this
volume - in high performance integrated superconductive circuits.


title: Threshold Growth Dynamics

abstract:
  We study the asymptotic shape of the occupied region for monotone
deterministic dynamics in d-dimensional Euclidean space parametrized by a
threshold theta, and a Borel set N with positive and finite Lebesgue measure.
If A_n denotes the occupied set of the dynamics at integer time n, then A_n+1
is obtained by adjoining any point x for which the volume of overlap between
x+N and A_n exceeds theta. Except in some degenerate cases, we prove that A_n
converges to a unique limiting "shape" L starting from any bounded initial
region that is suitably large. Moreover, L is computed as the polar transform
for 1/w, where w is an explicit width function that depends on N and theta. It
is further shown that L describes the limiting shape of wave fronts for certain
cellular automaton growth rules related to lattice models of excitable media,
as the threshold and range of interaction increase suitably. In the case of 2-d
box neighborhoods, these limiting shapes are calculated and the dependence of
their anisotropy on theta is examined. Other specific two- and three-
dimensional examples are also discussed in some detail.


title: Metastability in the Greenberg-Hastings Model

abstract:
  The Greenberg-Hastings Model (GHM) is a family of multitype cellular automata
that emulate excitable media, exhibiting the nucleation and spiral formation
characteristic of such complex systems. In this paper we study the asymptotic
frequency of nucleation in 2-d GHM dynamics as the number k of types, or
colors, becomes large. Starting from uniform product measure over colors, and
assuming that the excitation threshold is not too large, the box size needed
for formation of a spiral core is shown to grow exponentially in k. By
exploiting connections with percolation theory we find that a box of size
exp{(.23+-.06)k} is required in the nearest neighbor, threshold 1 case. In
contrast, GHM rules obey power law nucleation scaling when started from a
suitable non-uniform product measure over the k colors; this effect is driven
by critical percolation. Finally, we present some analogous results for a
Random GHM, an interacting Markovian system closely related to the epidemic
with regrowth of Durrett and Neuhauser.


title: Threshold-Range Scaling of Excitable Cellular Automata

abstract:
  Each cell of a two-dimensional lattice is painted one of k colors, arranged
in a "color wheel." The colors advance (0 to k-1 mod k) either automatically or
by contact with at least a threshold number of successor colors in a prescribed
local neighborhood. Discrete-time parallel systems of this sort in which color
0 updates by contact and the rest update automatically are called
Greenberg-Hastings (GH) rules. A system in which all colors update by contact
is called a cyclic cellular automaton (CCA). Started from appropriate initial
conditions these models generate periodic traveling waves. Started from random
configurations the same rules exhibit complex self-organization, typically
characterized by nucleation of locally periodic "ram's horns" or spirals.
Corresponding random processes give rise to a variety of "forest fire"
equilibria that display large-scale stochastic wave fronts. This article
describes a framework, theoretically based, but relying on extensive
interactive computer graphics experimentation, for investigation of the complex
dynamics shared by excitable media in a broad spectrum of scientific contexts.
By focusing on simple mathematical prototypes we obtain a better understanding
of the basic organizational principles underlying spatially-distributed
oscillating systems.


title:
Domain Walls in Non-Equilibrium Systems and the Emergence of Persistent
  Patterns

abstract:
  Domain walls in equilibrium phase transitions propagate in a preferred
direction so as to minimize the free energy of the system. As a result, initial
spatio-temporal patterns ultimately decay toward uniform states. The absence of
a variational principle far from equilibrium allows the coexistence of domain
walls propagating in any direction. As a consequence, *persistent* patterns may
emerge. We study this mechanism of pattern formation using a non-variational
extension of Landau's model for second order phase transitions. PACS numbers:
05.70.Fh, 42.65.Pc, 47.20.Ky, 82.20Mj


title: Complex Patterns in a Simple System

abstract:
  Numerical simulations of a simple reaction--diffusion model reveal a
surprising variety of irregular spatio--temporal patterns. These patterns arise
in response to finite--amplitude perturbations. Some of them resemble the
steady irregular patterns discussed by Lee et al. Others consist of spots which
grow until they reach a critical size at which time they divide in two. If, in
some region, the spots become overcrowded, all the spots in that region decay
into the uniform background.


title: Solitons and 1/f Noise in Molecular Chains

abstract:
  Davydov's model of solitons in alpha-helix protein chains is shown to display
features of self-organized criticality (SOC), i.e., power law behaviour of
correlations in space and 1/f-noise, as a consequence of considering random
peptide group displacements from their (periodic) equilibrium positions along a
chain. This may shed light on a basic mechanism leading to obtain flicker noise
in alpha-helix protein chains and to predict a SOC regime in biomolecular
structures from first principles. We believe our treatment of 1/f noise to be
of some relevance to recent findings due to Voss on DNA [Phys. Rev. Lett. 68,
3805 (1992)].


title:
Pattern formation during Rayleigh-B\'enard convection in non-Boussinesq
  fluids

abstract:
  Motivated by recent experimental studies of Bodenschatz et al. [E.
Bodenschatz, J.R. de Bruyn, G. Ahlers and D.S. Cannell, Phys. Rev. Lett. {\bf
67}, 3078 (1991) ], we present a numerical study of a generalized two
dimensional Swift-Hohenberg equation to model pattern formation in
Rayleigh-B\'enard convection in a non-Boussinesq fluid. It is shown that many
of the features observed in these experiments can be reproduced by this
generalized model that explicitly includes non-Boussinesq and mean flow
effects. The spontaneous formation of hexagons, rolls, and a rotating spiral
pattern is studied, as well as the transitions and competition among them. Mean
flow, non-Boussinesq effects, the geometric shape of the lateral wall, and
sidewall forcing are all shown to be crucial in the formation of the rotating
spirals. We also study nucleation and growth of hexagonal patterns and find
that the front velocity in this two dimensional model is consistent with the
prediction of marginal stability theory for one dimensional fronts.


title: An integrable shallow water equation with peaked solitons

abstract:
  We derive a new completely integrable dispersive shallow water equation that
is biHamiltonian and thus possesses an infinite number of conservation laws in
involution. The equation is obtained by using an asymptotic expansion directly
in the Hamiltonian for Euler's equations in the shallow water regime. The
soliton solution for this equation has a limiting form that has a discontinuity
in the first derivative at its peak.


title: Localized States in Discrete Nonlinear Schr\"{o}dinger Equations

abstract:
  A new 1-D discrete nonlinear Schr\"{o}dinger (NLS) Hamiltonian is introduced
which includes the integrable Ablowitz-Ladik system as a limit. The symmetry
properties of the system are studied. The relationship between intrinsic
localized states and the soliton of the Ablowitz-Ladik NLS is discussed. It is
pointed out that a staggered localized state can be viewed as a particle of a
{\em negative} effective mass. It is shown that staggered localized states can
exist in the discrete dark NLS. The motion of localized states and
Peierls-Nabarro pinning are studied.


title: Spiral defect chaos in a model of Rayleigh-Benard convection

abstract:
  A numerical solution of a generalized Swift-Hohenberg equation in two
dimensions reveals the existence of a spatio-temporal chaotic state comprised
of a large number of rotating spirals. This state is observed for a reduced
Rayleigh number $\epsilon=0.25$. The power spectrum of the state is isotropic,
and the spatial correlation function decays exponentially, with an estimated
decay length $\xi \approx 2.5 \lambda_{c}$, where $\lambda_{c}$ is the critical
wavelength near the onset of convection. Our study suggests that this spiral
defect state occurs for low Prandtl numbers and large aspect ratios.


title: Spiral Defect Chaos in Large Aspect Ratio Rayleigh-Benard Convection

abstract:
  We report experiments on convection patterns in a cylindrical cell with a
large aspect ratio. The fluid had a Prandtl number of approximately 1. We
observed a chaotic pattern consisting of many rotating spirals and other
defects in the parameter range where theory predicts that steady straight rolls
should be stable. The correlation length of the pattern decreased rapidly with
increasing control parameter so that the size of a correlated area became much
smaller than the area of the cell. This suggests that the chaotic behavior is
intrinsic to large aspect ratio geometries.


title:
Solitary Waves and Compactons in a class of Generalized Korteweg-DeVries
  Equations

abstract:
  We study the class of generalized Korteweg-DeVries equations derivable from
the Lagrangian: $ L(l,p) = \int \left( \frac{1}{2} \vp_{x} \vp_{t} - {
{(\vp_{x})^{l}} \over {l(l-1)}} + \alpha(\vp_{x})^{p} (\vp_{xx})^{2} \right)
dx, $ where the usual fields $u(x,t)$ of the generalized KdV equation are
defined by $u(x,t) = \vp_{x}(x,t)$. This class contains compactons, which are
solitary waves with compact support, and when $l=p+2$, these solutions have the
feature that their width is independent of the amplitude. We consider the
Hamiltonian structure and integrability properties of this class of KdV
equations. We show that many of the properties of the solitary waves and
compactons are easily obtained using a variational method based on the
principle of least action. Using a class of trial variational functions of the
form $u(x,t) = A(t) \exp \left[-\beta (t) \left|x-q(t) \right|^{2n} \right]$ we
find soliton-like solutions for all $n$, moving with fixed shape and constant
velocity, $c$. We show that the velocity, mass, and energy of the variational
travelling wave solutions are related by $ c = 2 r E M^{-1}$, where $ r =
(p+l+2)/(p+6-l)$, independent of $n$.\newline \newline PACS numbers: 03.40.Kf,
47.20.Ky, Nb, 52.35.Sb


title:
Pattern Formation in Dissipative Nonvariational Systems: The Effects of
  Front Bifurcations

abstract:
  Patterns in reaction-diffusion systems often contain two spatial scales; a
long scale determined by a typical wavelength or domain size, and a short scale
pertaining to front structures separating different domains. Such patterns
naturally develop in bistable and excitable systems, but may also appear far
beyond Hopf and Turing bifurcations. The global behavior of domain patterns
strongly depends on the fronts' inner structures. In this paper we study a
symmetry breaking front bifurcation expected to occur in a wide class of
reaction-diffusion systems, and the effects it has on pattern formation and
pattern dynamics. We extend previous works on this type of front bifurcation
and clarify the relations among them. We show that the appearance of front
multiplicity beyond the bifurcation point allows the formation of persistent
patterns rather than transient ones. In a different parameter regime, we find
that the front bifurcation outlines a transition from oscillating (or
breathing) patterns to traveling ones. Near a boundary we find that fronts
beyond the bifurcation can reflect, while those below it either bind to the
boundary or disappear.


title: Phase Transitions In Two-Dimensional Traffic Flow Models

abstract:
  We introduce two simple two-dimensional lattice models to study traffic flow
in cities. We have found that a few basic elements give rise to the
characteristic phase diagram of a first-order phase transition from a freely
moving phase to a jammed state, with a critical point. The jammed phase
presents new transitions corresponding to structural transformations of the
jam. We discuss their relevance in the infinite size limit.


title: Nonequilibrium Phenomena in Liquid Crystals

abstract:
  This paper summarizes a talk presented at the April NATO ASI on
Spatiotemporal Chaos in Complex Fluids, in Santa Fe, NM. The paper gives
reasons that make complex fluids good material systems for conducting
experiments on pattern formation and other nonequilibrium phenomena. Much of
the discussion focuses on the different phenomena observed in solidification
and how the increasing complexity of fluid systems decreases the velocity scale
for achieving "rapid" solidification. Five systems are compared to illustrate
this point: simple fluids, simple alloys, thermotropic liquid crystals,
lyotropic liquid crystals, and diblock copolymers. Finally, an example is given
of the kinds of transitions that may be observed in rapid solidification.


title:
A New Class of Nonsingular Exact Solutions for Laplacian Pattern
  Formation

abstract:
  We present a new class of exact solutions for the so-called {\it Laplacian
Growth Equation} describing the zero-surface-tension limit of a variety of 2D
pattern formation problems. Contrary to common belief, we prove that these
solutions are free of finite-time singularities (cusps) for quite general
initial conditions and may well describe real fingering instabilities. At long
times the interface consists of N separated moving Saffman-Taylor fingers, with
``stagnation points'' in between, in agreement with numerous observations. This
evolution resembles the N-soliton solution of classical integrable PDE's.


title: Discrete Breathers and Energy Localization in Nonlinear Lattices

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


title:
Dissipative Boussinesq System of Equations in the B\'enard-Marangoni
  Phenomenon

abstract:
  By using the long-wave approximation, a system of coupled evolution equations
for the bulk velocity and the surface perturbations of a B\'enard-Marangoni
system is obtained. It includes nonlinearity, dispersion and dissipation, and
it can be interpreted as a dissipative generalization of the usual Boussinesq
system of equations. As a particular case, a strictly dissipative version of
the Boussinesq system is obtained. Finnaly, some speculations are made on the
nature of the physical phenomena described by this system of equations.


title: A quantitative measurement of spatial order in ventricular fibrillation

abstract:
  As an objective measurement of spatial order in ventricular fibrillation
(VF), spatial correlation functions and their characteristic lengths were
estimated from epicardial electrograms of pigs in VF. The correlation length of
the VF in pigs was found to be approximately 4-10 mm, varying as fibrillation
progressed. The degree of correlation decreased in the first 4 seconds after
fibrillation then increased over the next minute. The correlation length is
much smaller than the scale of the heart, suggesting that many independent
regions of activity exist on the epicardium at any one time. On the other hand,
the correlation length is 4 to 10 times the interelectrode spacing, indicating
that some coherence is present. These results imply that the heart behaves
during VF as a high-dimensional, but not random, system involving many spatial
degrees of freedom, which may explain the lack of convergence of fractal
dimension estimates reported in the literature. Changes in the correlation
length also suggest that VF reorganizes slightly in the first minute after an
initial breakdown in structure.


title:
One parameter family of Compacton Solutions in a class of Generalized
  Korteweg-DeVries Equations

abstract:
  We study the generalized Korteweg-DeVries equations derivable from the
Lagrangian: $ L(l,p) = \int \left( \frac{1}{2} \varphi_{x} \varphi_{t} - {
{(\varphi_{x})^{l}} \over {l(l-1)}} + \alpha(\varphi_{x})^{p}
(\varphi_{xx})^{2} \right) dx, $ where the usual fields $u(x,t)$ of the
generalized KdV equation are defined by $u(x,t) = \varphi_{x}(x,t)$. For $p$ an
arbitrary continuous parameter $0< p \leq 2 ,l=p+2$ we find compacton solutions
to these equations which have the feature that their width is independent of
the amplitude. This generalizes previous results which considered $p=1,2$. For
the exact compactons we find a relation between the energy, mass and velocity
of the solitons. We show that this relationship can also be obtained using a
variational method based on the principle of least action.


title: Nonradial Solutions of a Semilinear Elliptic Equation in Two Dimensions

abstract:
  : We establish existence of an infinite family of exponentially-decaying
non-radial $C^2$ solutions to the equation $\Delta u + f(u) = 0$ on $R^2$ for a
large class of nonlinearities $f$. These solutions have the form $u(r,\theta
)=e^{i m\theta }w(r)$, where $r$ and $\theta$ are polar coordinates, $m$ is an
integer, and $w:[0,\infty ) \to R$ is exponentially decreasing far from the
origin. We prove there is a solution with each prescribed number of nodes.


title:
Disordered Regimes of the one-dimensional complex Ginzburg-Landau
  equation

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


title: Thermodynamic Limit Of The Ginzburg-Landau Equations

abstract:
  We investigate the existence of a global semiflow for the complex
Ginzburg-Landau equation on the space of bounded functions in unbounded domain.
This semiflow is proven to exist in dimension 1 and 2 for any parameter values
of the standard cubic Ginzburg-Landau equation. In dimension 3 we need some
restrictions on the parameters but cover nevertheless some part of the
Benjamin-Feijer unstable domain.


title: Temporal Forcing of Small-Amplitude Waves in Anisotropic Systems

abstract:
  We investigate the effect of resonant temporal forcing on an anisotropic
system that exhibits a Hopf bifurcation to obliquely traveling waves in the
absence of this forcing. We find that the forcing can excite various
phase-locked standing-wave structures: rolls, rectangles and cross rolls. At
onset, at most one of the two - rolls or rectangles - is stable. The cross
rolls can arise in a secondary bifurcation and can be stable. Experimentally,
they would appear as a periodic switching between a structure in which the
`zig'-component dominates and one with dominating `zag'-structure. Since there
are two symmetry-related states of this kind one may expect disordered
structures to arise due to the break-up of the pattern into domains. The
results are consistent with recent experiments on electro-convection in nematic
liquid crystals by de la Torre and Rehberg. We also apply the general analysis
to a model of the behavior near a Lifshitz point, where the angle of
obliqueness vanishes. This analysis indicates that phase-locked standing
rectangles are always unstable in this parameter regime.


title:
Exact Solutions of the One-Dimensional Quintic Complex Ginzburg-Landau
  Equation

abstract:
  Exact solitary wave solutions of the one-dimensional quintic complex
Ginzburg-Landau equation are obtained using a method derived from the
Painlev\'e test for integrability. These solutions are expressed in terms of
hyperbolic functions, and include the pulses and fronts found by van Saarloos
and Hohenberg. We also find previously unknown sources and sinks. The emphasis
is put on the systematic character of the method which breaks away from
approaches involving somewhat ad hoc Ans\"atze.


title:
Amplitude Expansions for Instabilities in Populations of
  Globally-Coupled Oscillators

abstract:
  We analyze the nonlinear dynamics near the incoherent state in a mean-field
model of coupled oscillators. The population is described by a Fokker-Planck
equation for the distribution of phases, and we apply center-manifold reduction
to obtain the amplitude equations for steady-state and Hopf bifurcation from
the equilibrium state with a uniform phase distribution. When the population is
described by a native frequency distribution that is reflection-symmetric about
zero, the problem has circular symmetry. In the limit of zero extrinsic noise,
although the critical eigenvalues are embedded in the continuous spectrum, the
nonlinear coefficients in the amplitude equation remain finite in contrast to
the singular behavior found in similar instabilities described by the
Vlasov-Poisson equation. For a bimodal reflection-symmetric distribution, both
types of bifurcation are possible and they coincide at a codimension-two Takens
Bogdanov point. The steady-state bifurcation may be supercritical or
subcritical and produces a time-independent synchronized state. The Hopf
bifurcation produces both supercritical stable standing waves and supercritical
unstable travelling waves. Previous work on the Hopf bifurcation in a bimodal
population by Bonilla, Neu, and Spigler and Okuda and Kuramoto predicted stable
travelling waves and stable standing waves, respectively. A comparison to these
previous calculations shows that the prediction of stable travelling waves
results from a failure to include all unstable modes.


title: D4-symmetric Maps with Hidden Euclidean Symmetry

abstract:
  Bifurcation problems in which periodic boundary conditions (PBC) or Neumann
boundary conditions (NBC) are imposed often involve partial differential
equations that have Euclidean symmetry. In this case posing the bifurcation
problem with either PBC or NBC on a finite domain can lead to a symmetric
bifurcation problem for which the manifest symmetries of the domain do not
completely characterize the constraints due to symmetry on the bifurcation
equations. Additional constraints due to the Euclidean symmetry of the
equations can have a crucial influence on the structure of the bifurcation
equations. An example is the bifurcation of standing waves on the surface of
fluid layer. The Euclidean symmetry of an infinite fluid layer constrains the
bifurcation of surface waves in a finite container with square cross section
because the waves satisfying PBC or NBC can be shown to lie in certain
finite-dimensional fixed point subspaces of the infinite-dimensional problem.
These constraints are studied by analyzing the finite-dimensional vector fields
obtained on these subspaces by restricting the bifurcation equations for the
infinite layer. Particular emphasis is given to determining which bifurcations
might reveal observable effects of the rotational symmetry of the infinite
layer. It turns out that a necessary condition for this possibility to arise is
that the subspace for PBC must carry a reducible representation of the
normalizer subgroup acting on the subspace. This condition can be met in
different ways in both codimension-one and codimension-two bifurcations.


title:
On the existence of monotonic fronts for a class of physical problems
  described by the equation $\lambda u''' + u' = f(u)$

abstract:
  We obtain an upper bound on the value of $\lambda$ for which monotonic front
solutions of the equation $\lambda u''' + u' = f(u)$ with $\lambda > 0$ may
exist.


title: Oscillatory instability of crack propagations in quasi-static fracture

abstract:
  Crack propagations in quasi-static fracture are studied theoretically. The
Griffith theory is applied to discuss a crack extension condition and motion of
crack tips in straight propagations. Stability of the straight propagations is
investigated based on the simple assumption that a curvature near the crack tip
is determined by a singular shear stress. It is shown that straight
propagations become unstable under ceatain conditions. Combined with boundary
effects, this instability leads to an oscillatory propagation of cracks. The
critical conditions are calculated quantitatively.


title: Multidimensional Localized Solitons

abstract:
  Recently it has been discovered that some nonlinear evolution equations in
2+1 dimensions, which are integrable by the use of the Spectral Transform,
admit localized (in the space) soliton solutions. This article briefly reviews
some of the main results obtained in the last five years thanks to the renewed
interest in soliton theory due to this discovery. The theoretical tools needed
to understand the unexpected richness of behaviour of multidimensional
localized solitons during their mutual scattering are furnished. Analogies and
especially discrepancies with the unidimensional case are stressed.


title: Properties Of Solutions Of The Kpi Equation

abstract:
  The Kadomtsev--Petviashvili I (KPI) is considered as a useful laboratory for
experimenting new theoretical tools able to handle the specific features of
integrable models in $2+1$ dimensions. The linearized version of the KPI
equation is first considered by solving the initial value problem for different
classes of initial data. Properties of the solutions in different cases are
analyzed in details. The obtained results are used as a guideline for studying
the properties of the solution $u(t,x,y)$ of the Kadomtsev--Petviashvili I
(KPI) equation with given initial data $u(0,x,y)$ belonging to the Schwartz
space. The spectral theory associated to KPI is studied in the space of the
Fourier transform of the solutions. The variables $p=\{p_1,p_2\}$ of the
Fourier space are shown to be the most convenient spectral variables to use for
spectral data. Spectral data are shown to decay rapidly at large $p$ but to be
discontinuous at $p=0$. Direct and inverse problems are solved with special
attention to the behaviour of all the quantities involved in the neighborhood
of $t=0$ and $p=0$. It is shown in particular that the solution $u(t,x,y)$ has
a time derivative discontinuous at $t=0$ and that at any $t\not=0$ it does not
belong to the Schwartz space no matter how small in norm and rapidly decaying
at large distances the initial data are chosen.


title: $D_4\dot{+} T^2$ Mode Interactions and Hidden Rotational Symmetry

abstract:
  Bifurcation problems in which periodic boundary conditions or Neumann
boundary conditions are imposed often involve partial differential equations
that have Euclidean symmetry. As a result the normal form equations for the
bifurcation may be constrained by the ``hidden'' Euclidean symmetry of the
equations, even though this symmetry is broken by the boundary conditions. The
effects of such hidden rotation symmetry on $D_4\dot{+} T^2$ mode interactions
are studied by analyzing when a $D_4\dot{+} T^2$ symmetric normal form
$\tilde{F}$ can be extended to a vector field ${\rm \cal F}$ with Euclidean
symmetry. The fundamental case of binary mode interactions between two
irreducible representations of $D_4\dot{+} T^2$ is treated in detail. Necessary
and sufficient conditions are given that permit $\tilde{F}$ to be extended when
the Euclidean group ${\rm \cal E}(2)$ acts irreducibly. When the Euclidean
action is reducible, the rotations do not impose any constraints on the normal
form of the binary mode interaction. In applications, this dependence on the
representation of ${\rm \cal E}(2)$ implies that the effects of hidden
rotations are not present if the critical eigenvalues are imaginary.
Generalization of these results to more complicated mode interactions is
discussed.


title:
Secondary Instabilities and Spatiotemporal Chaos in Parametric Surface
  Waves

abstract:
  A two dimensional model is introduced to study pattern formation, secondary
instabilities and the transition to spatiotemporal chaos (weak turbulence) in
parametric surface waves. The stability of a periodic standing wave state above
onset is studied against Eckhaus, zig-zag and transverse amplitude modulations
(TAM) as a function of the control parameter $\varepsilon$ and the detuning. A
mechanism leading to a finite threshold for the TAM instability is identified.
Numerical solutions of the model are in agreement with the stability diagram,
and also reveal the existence of a transition to spatiotemporal chaotic states
at a finite $\varepsilon$. Power spectra of temporal fluctuations in the
chaotic state are broadband, decaying as a power law of the frequency
$\omega^{-z}$ with $z \approx 4.0$.

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


title: Solitons in the Camassa-Holm Shallow Water Equation

abstract:
  We study the class of shallow water equations of Camassa and Holm derived
from the Lagrangian: $ L= \int \left( \frac{1}{2} (\varphi_{xxx}-\varphi_{x}
)\varphi_{t} - {1 \over 2} {(\varphi_{x})^{3}} - {1 \over
2}\varphi_{x}(\varphi_{xx})^{2} - {1 \over 2} \kappa \varphi_{x}^{2} \right)
dx, $


title: Dual Fronts Propagating into an Unstable State

abstract:
  The interface between an unstable state and a stable state usually develops a
single confined front travelling with constant velocity into the unstable
state. Recently, the splitting of such an interface into {\em two} fronts
propagating with {\em different} velocities was observed numerically in a
magnetic system. The intermediate state is unstable and grows linearly in time.
We first establish rigorously the existence of this phenomenon, called ``dual
front,'' for a class of structurally unstable one-component models. Then we use
this insight to explain dual fronts for a generic two-component
reaction-diffusion system, and for the magnetic system.


title: Multiple Front Propagation into Unstable States

abstract:
  The dynamics of transient patterns formed by front propagation in extended
nonequilibrium systems is considered. Under certain circumstances, the state
left behind a front propagating into an unstable homogeneous state can be an
unstable periodic pattern. It is found by a numerical solution of a model of
the Fr\'eedericksz transition in nematic liquid crystals that the mechanism of
decay of such periodic unstable states is the propagation of a second front
which replaces the unstable pattern by a another unstable periodic state with
larger wavelength. The speed of this second front and the periodicity of the
new state are analytically calculated with a generalization of the marginal
stability formalism suited to the study of front propagation into periodic
unstable states. PACS: 47.20.Ky, 03.40.Kf, 47.54.+r


title:
Hydrothermal Surface-Wave Instability and the Kuramoto-Sivashinsky
  Equation

abstract:
  We consider a system formed by an infinite viscous liquid layer with a
constant horizontal temperature gradient, and a basic nonlinear bulk velocity
profile. In the limit of long-wavelength and large nondimensional surface
tension, we show that hydrothermal surface-wave instabilities may give rise to
disturbances governed by the Kuramoto-Sivashinsky equation. A possible
connection to hot-wire experiments is also discussed.


title: From Labyrinthine Patterns to Spiral Turbulence

abstract:
  A new mechanism for spiral vortex nucleation in nongradient reaction
diffusion systems is proposed. It involves two key ingredients: An Ising-Bloch
type front bifurcation and an instability of a planar front to transverse
perturbations. Vortex nucleation by this mechanism plays an important role in
inducing a transition from labyrinthine patterns to spiral turbulence. PACS
numbers: 05.45.+b, 82.20.Mj


title: Soliton solutions of the Hamiltonian DSI and DSIII equations

abstract:
  By introducing generalized Backlund Transformations depending on arbitrary
functions, wave and localized soliton solutions of the Davey- Stewartson
equations are generated. Moreover explicit soliton solutions of the Hamiltonian
DSI and DSIII equations are obtained.


title:
Domain Structures and Zig-Zag Patterns Modeled by a Fourth-Order
  Ginzburg-Landau Equation

abstract:
  Domain walls between spatially periodic patterns with different wave numbers,
can arise in pattern-forming systems with a neutral curve that has a double
minimum. Within the framework of the phase equation, the interaction of such
walls is purely attractive. Thus, they annihilate each other and, if the total
phase is not conserved, the final state will be periodic with a single wave
number. Here we study the stability of arrays of domain walls (domain
structures) in a fourth-order Ginzburg-Landau equation. We find a discrete set
of domain structures which differ in the quantized lengths of the domains. They
are stable even without phase conservation, as, for instance, in the presence
of a subcritical ramp in the control parameter. We attribute this to the
spatially oscillatory behavior of the wave number which should lead to an
oscillatory interaction between the domain walls. These results are expected to
shed also some light on the stability of two-dimensional zig-zag structures.


title:
Kink stability, propagation, and length scale competition in the
  periodically modulated sine-Gordon equation

abstract:
  We have examined the dynamical behavior of the kink solutions of the
one-dimensional sine-Gordon equation in the presence of a spatially periodic
parametric perturbation. Our study clarifies and extends the currently
available knowledge on this and related nonlinear problems in four directions.
First, we present the results of a numerical simulation program which are not
compatible with the existence of a radiative threshold, predicted by earlier
calculations. Second, we carry out a perturbative calculation which helps
interpret those previous predictions, enabling us to understand in depth our
numerical results. Third, we apply the collective coordinate formalism to this
system and demonstrate numerically that it accurately reproduces the observed
kink dynamics. Fourth, we report on a novel occurrence of length scale
competition in this system and show how it can be understood by means of linear
stability analysis. Finally, we conclude by summarizing the general physical
framework that arises from our study.


title:
Front Structures in a Real Ginzburg-Landau Equation Coupled to a Mean
  Field

abstract:
  Localized traveling wave trains or pulses have been observed in various
experiments in binary mixture convection. For strongly negative separation
ratio, these pulse structures can be described as two interacting fronts of
opposite orientation. An analytical study of the front solutions in a real
Ginzburg-Landau equation coupled to a mean field is presented here as a first
approach to the pulse solution. The additional mean field becomes important
when the mass diffusion in the mixture is small as is the case in liquids.
Within this framework it can lead to a hysteretic transition between slow and
fast fronts when the Rayleigh number is changed.


title: Structure And Dynamics Of Modulated Traveling Waves In Cellular Flames

abstract:
  We describe spatial and temporal patterns in cylindrical premixed flames in
the cellular regime, $Le < 1$, where the Lewis number $Le$ is the ratio of
thermal to mass diffusivity of a deficient component of the combustible
mixture. A transition from stationary, axisymmetric flames to stationary
cellular flames is predicted analytically if $Le$ is decreased below a critical
value. We present the results of numerical computations to show that as $Le$ is
further decreased traveling waves (TWs) along the flame front arise via an
infinite-period bifurcation which breaks the reflection symmetry of the
cellular array. Upon further decreasing $Le$ different kinds of periodically
modulated traveling waves (MTWs) as well as a branch of quasiperiodically
modulated traveling waves (QPMTWs) arise. These transitions are accompanied by
the development of different spatial and temporal symmetries including period
doublings and period halvings. We also observe the apparently chaotic temporal
behavior of a disordered cellular pattern involving creation and annihilation
of cells. We analytically describe the stability of the TW solution near its
onset+ using suitable phase-amplitude equations. Within this framework one of
the MTW's can be identified as a localized wave traveling through an underlying
stationary, spatially periodic structure. We study the Eckhaus instability of
the TW and find that in general they are unstable at onset in infinite systems.
They can, however, become stable for larger amplitudes.


title: SUSY-Based Variational Method for the Anharmonic Oscillator

abstract:
  Using a newly suggested algorithm of Gozzi, Reuter, and Thacker for
calculating the excited states of one dimensional systems, we determine
approximately the eigenvalues and eigenfunctions of the anharmonic oscillator,
described by the Hamiltonian $H= p^2/2 + g x^4$. We use ground state
post-gaussian trial wave functions of the form $\Psi(x) =
  N{\rm{exp}}[-b |x|^{2n}]$, where $n$ and $b$ are continuous variational
parameters. This algorithm is based on the hierarchy of Hamiltonians related by
supersymmetry (SUSY) and the factorization method. We find that our two
parameter family of trial wave functions yields excellent energy eigenvalues
and wave functions for the first few levels of the anharmonic oscillator.


title: Domain Structures in Fourth-Order Phase and Ginzburg-Landau Equations

abstract:
  In pattern-forming systems, competition between patterns with different wave
numbers can lead to domain structures, which consist of regions with differing
wave numbers separated by domain walls. For domain structures well above
threshold we employ the appropriate phase equation and obtain detailed
qualitative agreement with recent experiments. Close to threshold a
fourth-order Ginzburg-Landau equation is used which describes a steady
bifurcation in systems with two competing critical wave numbers. The existence
and stability regime of domain structures is found to be very intricate due to
interactions with other modes.
  In contrast to the phase equation the Ginzburg-Landau equation allows a
spatially oscillatory interaction of the domain walls. Thus, close to threshold
domain structures need not undergo the coarsening dynamics found in the phase
equation far above threshold, and can be stable even without phase
conservation. We study their regime of stability as a function of their
(quantized) length. Domain structures are related to zig-zags in
two-dimensional systems. The latter are therefore expected to be stable only
when quenched far enough beyond the zig-zag instability.


title:
On the validity of the linear speed selection mechanism for fronts of
  the nonlinear diffusion equation

abstract:
  We consider the problem of the speed selection mechanism for the one
dimensional nonlinear diffusion equation $u_t = u_{xx} + f(u)$. It has been
rigorously shown by Aronson and Weinberger that for a wide class of functions
$f$, sufficiently localized initial conditions evolve in time into a monotonic
front which propagates with speed $c^*$ such that $2 \sqrt{f'(0)} \leq c^* < 2
\sqrt{\sup(f(u)/u)}$. The lower value $c_L = 2 \sqrt{f'(0)}$ is that predicted
by the linear marginal stability speed selection mechanism. We derive a new
lower bound on the the speed of the selected front, this bound depends on $f$
and thus enables us to assess the extent to which the linear marginal selection
mechanism is valid.


title:
Asymptotic expansion for layer solutions of a singularly perturbed
  reaction-diffusion system

abstract:
  For a singularly perturbed system of reaction--diffusion equations, assuming
that the 0th order solutions in regular and singular regions are all stable, we
construct matched asymptotic expansions for formal solutions to any desired
order in $\epsilon$. The formal solution shows that there is an invariant
manifold of wave-front-like solutions that attracts other nearby solutions.
With an additional assumption on the sign of the wave speed, the
wave-front-like solutions converge slowly to stable stationary solutions on
that manifold.


title:
New exact fronts for the nonlinear diffusion equation with quintic
  nonlinearities

abstract:
  We consider travelling wave solutions of the reaction diffusion equation with
quintic nonlinearities $u_t = u_{xx} + \mu u (1 -u )
  ( 1 +\alpha u + \beta u^2 +\gamma u^3)$. If the parameters $\alpha , \beta$
and $\gamma$ obey a special relation, then the criterion for the existence of a
strong heteroclinic connection can be expressed in terms of two of these
parameters. If an additional restriction is imposed, explicit front solutions
can be obtained. The approach used can be extended to polynomials whose highest
degree is odd.


title: Kinks in the Presence of Rapidly Varying Perturbations

abstract:
  Dynamics of sine-Gordon kinks in the presence of rapidly varying periodic
perturbations of different physical origins is described analytically and
numerically. The analytical approach is based on asymptotic expansions, and it
allows to derive, in a rigorous way, an effective nonlinear equation for the
slowly varying field component in any order of the asymptotic procedure as
expansions in the small parameter $\omega^{-1}$, $\omega$ being the frequency
of the rapidly varying ac driving force. Three physically important examples of
such a dynamics, {\em i.e.}, kinks driven by a direct or parametric ac force,
and kinks on rotating and oscillating background, are analysed in detail. It is
shown that in the main order of the asymptotic procedure the effective equation
for the slowly varying field component is {\em a renormalized sine-Gordon
equation} in the case of the direct driving force or rotating (but phase-locked
to an external ac force) background, and it is {\em the double sine-Gordon
equation} for the parametric driving force. The properties of the kinks
described by the renormalized nonlinear equations are analysed, and it is
demonstrated analytically and numerically which kinds of physical phenomena may
be expected in dealing with the renormalized, rather than the unrenormalized,
nonlinear dynamics. In particular, we predict several qualitatively new effects
which include, {\em e.g.}, the perturbation-induced internal oscillations of
the $2\pi-$kink in a parametrically driven sine-Gordon model, and generation of
kink motion by a pure ac driving force on a rotating background.


title: Inverse ac Josephson Effect for a Fluxon in a Long Modulated Junction

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


title: Multidimensional Dynamical Systems Accepting the Normal Shift

abstract:
  The dynamical systems of the form $\ddot\bold r=\bold F (\bold r,\dot\bold
r)$ in $\Bbb R^n$ accepting the normal shift are considered. The concept of
weak normality for them is introduced. The partial differential equations for
the force field $\bold F(\bold r,\dot\bold r)$ of the dynamical systems with
weak and complete normality are derived.

categories: patt-sol alg-geom astro-ph math.AG nlin.PS


title: KdV shock-like waves as invariant solutions of KdV equation symmetries

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


title:
Whitham deformations partially saturating the modulational instability
  in the nonlinear Schrodinger equation

abstract:
  In the framework of Gurevich and Pitaevskii approach [1] we construct
modulated by Whitham [2] solution of nonlinear Shrodinger (NS) equation
partially saturating the modulational instability. This solution describes new
scenario of monochromatic wave evolution in NS equation which leads to
generation of new phase and oscillation region.


title:
Example of shock wave in unstaible medium: The focusing nonlinear
  Schrodinger equation

abstract:
  Dissipationless shock waves in modulational unstable one-dimensional medium
are investigated on the simplest example of integrable focusing nonlinear
Schr\''odinger (NS) equation. Our approach is based on the construction of
special exact solution of the Whitham-NS system, which ''partially saturates''
the modulational instability.


title: Dynamic Front Transitions and Spiral-Vortex Nucleation

abstract:
  This is a study of front dynamics in reaction diffusion systems near
Nonequilibrium Ising-Bloch bifurcations. We find that the relation between
front velocity and perturbative factors, such as external fields and curvature,
is typically multivalued. This unusual form allows small perturbations to
induce dynamic transitions between counter-propagating fronts and nucleate
spiral vortices. We use these findings to propose explanations for a few
numerical and experimental observations including spiral breakup driven by
advective fields, and spot splitting.


title:
Normal Form, Symmetry and Infinite Dimensional Lie Algebra for System of
  Ode's

abstract:
  The normal form for a system of ode's is constructed from its polynomial
symmetries of the linear part of the system, which is assumed to be
semi-simple. The symmetries are shown to have a simple structure such as
invariant function times symmetries of degree one called basic symmetries. We
also show that the set of symmetries naturally forms an infinite dimensional
Lie algebra graded by the degree of invariant polynomials. This implies that if
this algebra is non-commutative then the method of multiple scales with more
than two scaling variables fails to apply.


title: New Sets of Kink Bearing Hamiltonians

abstract:
  Given a kink bearing Hamiltonian, Isospectral Hamiltonian approach is used in
generating new sets of Hamiltonains which also admit kink solutions. We use
Sine-Gordon model as a example and explicitly work out the new sets of
potentials and the solutions.


title:
Complex Patterns in Reaction-Diffusion Systems: A Tale of Two Front
  Instabilities

abstract:
  Two front instabilities in a reaction-diffusion system are shown to lead to
the formation of complex patterns. The first is an instability to transverse
modulations that drives the formation of labyrinthine patterns. The second is a
Nonequilibrium Ising-Bloch (NIB) bifurcation that renders a stationary planar
front unstable and gives rise to a pair of counterpropagating fronts. Near the
NIB bifurcation the relation of the front velocity to curvature is highly
nonlinear and transitions between counterpropagating fronts become feasible.
Nonuniformly curved fronts may undergo local front transitions that nucleate
spiral-vortex pairs. These nucleation events provide the ingredient needed to
initiate spot splitting and spiral turbulence. Similar spatio-temporal
processes have been observed recently in the ferrocyanide-iodate-sulfite
reaction.


title: The Korteweg-de Vries Hierarchy and Long Water-Waves

abstract:
  By using the multiple scale method with the simultaneous introduction of
multiple times, we study the propagation of long surface-waves in a shallow
inviscid fluid. As a consequence of the requirements of scale invariance and
absence of secular terms in each order of the perturbative expansion, we show
that the Korteweg-de Vries hierarchy equations do appear in the description of
such waves. Finally, we show that this procedure of eliminating secularities is
closely related to the renormalization technique introduced by Kodama and
Taniuti.


title:
Universal trapping scaling on the unstable manifold for a collisionless
  electrostatic mode

abstract:
  An amplitude equation for an unstable mode in a collisionless plasma is
derived from the dynamics on the two-dimensional unstable manifold of the
equilibrium. The mode amplitude $\rho(t)$ decouples from the phase due to the
spatial homogeneity of the equilibrium, and the resulting one-dimensional
dynamics is analyzed using an expansion in $\rho$. As the linear growth rate
$\gamma$ vanishes, the expansion coefficients diverge; a rescaling
$\rho(t)\equiv\gamma^2\,r(\gamma t)$ of the mode amplitude absorbs these
singularities and reveals that the mode electric field exhibits trapping
scaling $|E_1|\sim\gamma^2$ as $\gamma\rightarrow0$. The dynamics for $r(\tau)$
depends only on the phase $e^{i\xi}$ where $d\epsilon_{{k}}
/dz=|{\epsilon_{{k}}}|e^{-i\xi/2}$ is the derivative of the dielectric as
$\gamma\rightarrow0$.


title:
On the self-similar solutions of normality equation in two-dimensional
  case

abstract:
  The integrability in quadratures of normality equation for spatially
homogeneous dynamical systems in two-dimensional space is shown. The classical
symmetries of this equation are calculated and the corresponding self-similar
solutions are found.


title:
Variational Characterization of the Speed of Propagation of Fronts for
  the Nonlinear Diffusion Equation

abstract:
  We give an integral variational characterization for the speed of fronts of
the nonlinear diffusion equation $u_t = u_{xx} + f(u)$ with $f(0)=f(1)=0$, and
$f>0$ in $(0,1)$, which permits, in principle, the calculation of the exact
speed for arbitrary $f$.


title:
Bound Pairs of Fronts in a Real Ginzburg-Landau Equation Coupled to a
  Mean Field

abstract:
  Motivated by the observation of localized traveling-wave states (`pulses') in
convection in binary liquid mixtures, the interaction of fronts is investigated
in a real Ginzburg-Landau equation which is coupled to a mean field. In that
system the Ginzburg-Landau equation describes the traveling-wave amplitude and
the mean field corrsponds to a concentration mode which arises due to the
slowness of mass diffusion. For single fronts the mean field can lead to a
hysteretic transition between slow and fast fronts. Its contribution to the
interaction between fronts can be attractive as well as repulsive and depends
strongly on their direction of propagation. Thus, the concentration mode leads
to a new localization mechanism, which does not require any dispersion in
contrast to that operating in the nonlinear Schr\"odinger equation. Based on
this mechanism alone, pairs of fronts in binary-mixture convection are expected
to form {\it stable} pulses if they travel {\it backward}, i.e. opposite to the
phase velocity. For positive velocities the interaction becomes attractive and
destabilizes the pulses. These results are in qualitative agreement with recent
experiments. Since the new mechanism is very robust it is expected to be
relevant in other systems as well in which a wave is coupled to a mean field.


title: Hole Solutions in the 1d Complex Ginzburg-Landau Equation

abstract:
  The cubic Complex Ginzburg-Landau Equation (CGLE) has a one parameter family
of traveling localized source solutions. These so called 'Nozaki-Bekki holes'
are (dynamically) stable in some parameter range, but always structually
unstable: A perturbation of the equation in general leads to a (positive or
negative) monotonic acceleration or an oscillation of the holes. This confirms
that the cubic CGLE has an inner symmetry. As a consequence small perturbations
change some of the qualitative dynamics of the cubic CGLE and enhance or
suppress spatio-temporal intermittency in some parameter range. An analytic
stability analysis of holes in the cubic CGLE and a semianalytical treatment of
the acceleration instability in the perturbed equation is performed by using
matching and perturbation methods. Furthermore we treat the asymptotic
hole-shock interaction. The results, which can be obtained fully analytically
in the nonlinear Schroedinger limit, are also used for the quantitative
description of modulated solutions made up of periodic arrangements of
traveling holes and shocks.


title:
From the defocusing nonlinear Schroedinger to the complex
  Ginzburg-Landau equation

abstract:
  Perturbation approaches developed so far for the dark soliton solutions of
the (fully integrable) defocusing nonlinear Schroedinger equation cannot
describe the dynamics resulting from dissipative perturbations of the
Ginzburg-Landau type. Here spatially slowly decaying changes of the background
wavenumber occur which requires the use of matching technics. It is shown how
the perturbation selects a 1 or 2-parameter subfamily from the 3-parameter
family of dark solitons of the nonlinear Schroedinger equation. The dynamics of
the perturbed system can then be described analytically as motion within this
selected subfamily yielding interesting scenarios. Interaction with shocks
occurring in the complex Ginzburg-Landau equation can be included in a straight
forward way.


title: Jam phases in two-dimensional cellular automata model of traffic flow

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


title: Bipolar Electrodiffusion model for Electroconvection in Nematics

abstract:
  The common description of the electrical behavior of a nematic liquid crystal
as an anisotropic dielectric medium with (weak) ohmic conductivity is extended
to an electrodiffusion model with two active ionic species. Under appropriate,
but rather general conditions the additional effects can lead to a distinctive
change of the threshold behavior of the electrohydrodynamic instability, namely
to travelling patterns instead of static ones. This may explain the
experimentally observed phenomena.


title:
A Variational Principle for the Asymptotic Speed of Fronts of the
  Density Dependent Diffusion--Reaction Equation

abstract:
  We show that the minimal speed for the existence of monotonic fronts of the
equation $u_t = (u^m)_{xx} + f(u)$ with $f(0) = f(1) = 0$, $m >1$ and $f>0$ in
$(0,1)$ derives from a variational principle. The variational principle allows
to calculate, in principle, the exact speed for arbitrary $f$. The case $m=1$
when $f'(0)=0$ is included as an extension of the results.


title:
Front propagation into unstable and metastable states in Smectic C*
  liquid crystals: linear and nonlinear marginal stability analysis

abstract:
  We discuss the front propagation in ferroelectric chiral smectics (SmC*)
subjected to electric and magnetic fields applied parallel to smectic layers.
The reversal of the electric field induces the motion of domain walls or fronts
that propagate into either an unstable or a metastable state. In both regimes,
the front velocity is calculated exactly. Depending on the field, the speed of
a front propagating into the unstable state is given either by the so-called
linear marginal stability velocity or by the nonlinear marginal stability
expression. The cross-over between these two regimes can be tuned by a magnetic
field. The influence of initial conditions on the velocity selection problem
can also be studied in such experiments. SmC$^*$ therefore offers a unique
opportunity to study different aspects of front propagation in an experimental
system.


title:
On the validity of mean-field amplitude equations for counterpropagating
  wavetrains

abstract:
  We rigorously establish the validity of the equations describing the
evolution of one-dimensional long wavelength modulations of counterpropagating
wavetrains for a hyperbolic model equation, namely the sine-Gordon equation. We
consider both periodic amplitude functions and localized wavepackets. For the
localized case, the wavetrains are completely decoupled at leading order, while
in the periodic case the amplitude equations take the form of mean-field
(nonlocal) Schr\"odinger equations rather than locally coupled partial
differential equations. The origin of this weakened coupling is traced to a
hidden translation symmetry in the linear problem, which is related to the
existence of a characteristic frame traveling at the group velocity of each
wavetrain. It is proved that solutions to the amplitude equations dominate the
dynamics of the governing equations on asymptotically long time scales. While
the details of the discussion are restricted to the class of model equations
having a leading cubic nonlinearity, the results strongly indicate that
mean-field evolution equations are generic for bimodal disturbances in
dispersive systems with \O(1) group velocity.


title: Defect Dynamics for Spiral Chaos in Rayleigh-Benard Convection

abstract:
  A theory of the novel spiral chaos state recently observed in Rayleigh-Benard
convection is proposed in terms of the importance of invasive defects i.e
defects that through their intrinsic dynamics expand to take over the system.
The motion of the spiral defects is shown to be dominated by wave vector
frustration, rather than a rotational motion driven by a vertical vorticity
field. This leads to a continuum of spiral frequencies, and a spiral may rotate
in either sense depending on the wave vector of its local environment. Results
of extensive numerical work on equations modelling the convection system
provide some confirmation of these ideas.


title: Observation of Conservations Laws in Diffusion Limited Aggregation

abstract:
  We repeat the numerical experiments for diffusion limited aggregation (DLA)
and show that there is a potentially infinite set of conserved quantities for
the long time asymptotics. We connect these observations with the exact
integrability of the continuum limit of the DLA (quasi-static Stefan problem).
The conserved quantities of the Stefan problem (harmonic moments) when
discretized are our conserved quantities. These numerical experiments show that
the exact integrability of the Stefan problem may be continued beyond the
formation of cusps in the moving boundary.


title:
The Swift-Hohenberg Equation Requires Non-Local Modifications to Model
  Spatial Pattern Evolution of Physical Problems

abstract:
  I argue that ``good'' mathematical models of spatio-temporal dynamics in
two-dimensions require non-local operators in the nonlinear terms.
Consequently, the often used Swift-Hohenberg equation requires modification as
it is purely local. My aim here is to provoke more critical examination of the
rationale for using the Swift-Hohenberg equations as a reliable model of the
spatial pattern evolution in specific physical systems.


title: Periodicity of cylindrical linear cellular automata

abstract:
  Periodicity and relaxation are investigated for the trajectories of the
states in cylindrical linear cellular automata. The time evolutions are
described with matrices. The eigenvalue analysis is applied to obtain the
maximum values of period and relaxation. The translational invariance
suppresses the maximum period and relaxation.


title: Penta-Hepta Defect Motion in Hexagonal Patterns

abstract:
  Structure and dynamics of penta-hepta defects in hexagonal patterns is
studied in the framework of coupled amplitude equations for underlying plane
waves. Analytical solution for phase field of moving PHD is found in the far
field, which generalizes the static solution due to Pismen and Nepomnyashchy
(1993). The mobility tensor of PHD is calculated using combined analytical and
numerical approach. The results for the velocity of PHD climbing in slightly
non-optimal hexagonal patterns are compared with numerical simulations of
amplitude equations. Interaction of penta-hepta defects in optimal hexagonal
patterns is also considered.


title:
Amplitude equations near pattern forming instabilities for strongly
  driven ferromagnets

abstract:
  A transversally driven isotropic ferromagnet being under the influence of a
static external and an uniaxial internal anisotropy field is studied. We
consider the dissipative Landau-Lifshitz equation as the fundamental equation
of motion and treat it in $1+1$~dimensions. The stability of the spatially
homogeneous magnetizations against inhomogeneous perturbations is analyzed.
Subsequently the dynamics above threshold is described via amplitude equations
and the dependence of their coefficients on the physical parameters of the
system is determined explicitly. We find soft- and hard-mode instabilities,
transitions between sub- and supercritical behaviour, various bifurcations of
higher codimension, and present a series of explicit bifurcation diagrams. The
analysis of the codimension-2 point where the soft- and hard-mode instabilities
coincide leads to a system of two coupled Ginzburg-Landau equations.


title: Multistability in networks of weakly coupled bistable units

abstract:
  We study the stationary states of networks consisting of weakly coupled
bistable units. We prove the existence of a high multiplicity of stable steady
states in networks with very general inter-unit dynamics. We present a method
for estimating the critical coupling strength below which these stationary
states persist in the network. In some cases, the presence of time-independent
localized states in the system can be regarded as a `propagation failure'
phenomenon. We analyse this type of behaviour in the case of diffusive networks
whose elements are described by one or two variables and give concrete
examples.


title:
Metastability in Fluctuation Driven First-order Transitions: Nucleation
  of Lamellar Phases

abstract:
  The nucleation of a lamellar phase from a supercooled homogeneous phase in a
fluctuation driven first-order transition is studied, based on a
phenomenological free energy due to Brazovskii. The absence of phase
coexistence in the corresponding mean-field approximation makes application of
the standard droplet theory of homogeneous nucleation problematic. A
self-consistent coarse-graining procedure is introduced to overcome this
difficulty, and the barrier height for nucleation of a critical droplet is
estimated in the weak-coupling limit. Contrary to earlier estimates the
critical droplet shape is shown to be anisotropic in general. Some effects of
distortions and defects in the lamellar structure are considered and are shown
to affect the critical droplet only very near coexistence where the probability
of nucleation vanishes. The coarse-graining procedure introduced here follows
from a novel application of the momentum-shell renormalization group method to
systems in the Brazovskii class. Possible applications of the theory to the
microphase separation transition in diblock copolymers and to
Rayleigh-B\'{e}nard convection are briefly discussed.


title: Obtaining Breathers in Nonlinear Hamiltonian Lattices

abstract:
  We present a numerical method for obtaining high-accuracy numerical solutions
of spatially localized time-periodic excitations on a nonlinear Hamiltonian
lattice. We compare these results with analytical considerations of the spatial
decay. We show that nonlinear contributions have to be considered, and obtain
very good agreement between the latter and the numerical results. We discuss
further applications of the method and results.


title:
INVERSE SCATTERING TRANSFORM ANALYSIS OF STOKES-ANTI-STOKES STIMULATED
  RAMAN SCATTERING

abstract:
  Zakharov-Shabat--Ablowitz-Kaup-Newel-Segur representation for
Stokes-anti-Stokes stimulated Raman scattering is proposed. Periodical waves,
solitons and self-similarity solutions are derived. Transient and bright
threshold solitons are discussed.


title:
Thermally Induced Fluctuations Below the Onset of Rayleigh-B\'enard
  Convection

abstract:
  We report quantitative experimental results for the intensity of
noise-induced fluctuations below the critical temperature difference $\Delta
T_c$ for Rayleigh-B\'enard convection. The structure factor of the fluctuating
convection rolls is consistent with the expected rotational invariance of the
system. In agreement with predictions based on stochastic hydrodynamic
equations, the fluctuation intensity is found to be proportional to
$1/\sqrt{-\epsilon}$ where $\epsilon \equiv \Delta T / \Delta T_c -1$. The
noise power necessary to explain the measurements agrees with the prediction
for thermal noise. (WAC95-1)


title: MULTIPLE FRONT PROPAGATION IN A POTENTIAL NON-GRADIENT SYSTEM

abstract:
  A classification of dynamical systems in terms of their variational
properties is reviewed. Within this classification, front propagation is
discussed in a non-gradient relaxational potential flow. The model is motivated
by transient pattern phenomena in nematics. A front propagating into an
unstable homogenous state leaves behind an unstable periodic pattern, which
decays via a second front and a second periodic state. An interface between
unstable periodic states is shown to be a source of propagating fronts in
opposite directions.


title: Points, Walls and Loops in Resonant Oscillatory Media

abstract:
  In an experiment of oscillatory media, domains and walls are formed under the
parametric resonance with a frequency double the natural one. In this bi-stable
system, %phase jumps $\pi$ by crossing walls. a nonequilibrium transition from
Ising wall to Bloch wall consistent with prediction is confirmed
experimentally. The Bloch wall moves in the direction determined by its
chirality with a constant speed. As a new type of moving structure in
two-dimension, a traveling loop consisting of two walls and Neel points is
observed.


title:
Attractive Interaction Between Pulses in a Model for Binary-Mixture
  Convection

abstract:
  Recent experiments on convection in binary mixtures have shown that the
interaction between localized waves (pulses) can be repulsive as well as {\it
attractive} and depends strongly on the relative {\it orientation} of the
pulses. It is demonstrated that the concentration mode, which is characteristic
of the extended Ginzburg-Landau equations introduced recently, allows a natural
understanding of that result. Within the standard complex Ginzburg-Landau
equation this would not be possible.


title: Solitary Waves under the Influence of a Long-Wave Mode

abstract:
  The dynamics of solitons of the nonlinear Schr\"odinger equation under the
influence of dissipative and dispersive perturbations is investigated. In
particular a coupling to a long-wave mode is considered using extended
Ginzburg-Landau equations. The study is motivated by the experimental
observation of localized wave trains (`pulses') in binary-liquid convection.
These pulses have been found to drift exceedingly slowly. The perturbation
analysis reveals two distinct mechanisms which can lead to a `trapping' of the
pulses by the long-wave concentration mode. The are given by the effect of the
concentration mode on the local growth rate and on the frequency of the wave.
The latter, dispersive mechanism has not been recognized previously, despite
the fact that it dominates over the dissipative contribution within the
perturbation theory. A second unexpected result is that the pulse can be
accelerated by the concentration mode despite the reduced growth rate ahead of
the pulse. The dependence of the pulse velocity on the Rayleigh number is
discussed, and the hysteretic `trapping' transitions suggested by the
perturbation theory are confirmed by numerical simulations, which also reveal
oscillatory behavior of the pulse velocity in the vicinity of the transition.
The derivation and reconstitution of the extended Ginzburg-Landau equations is
discussed in detail.


title: Aggregation Patterns in Stressed Bacteria

abstract:
  We study the formation of spot patterns seen in a variety of bacterial
species when the bacteria are subjected to oxidative stress due to hazardous
byproducts of respiration. Our approach consists of coupling the cell density
field to a chemoattractant concentration as well as to nutrient and waste
fields. The latter serves as a triggering field for emission of
chemoattractant. Important elements in the proposed model include the
propagation of a front of motile bacteria radially outward form an initial
site, a Turing instability of the uniformly dense state and a reduction of
motility for cells sufficiently far behind the front. The wide variety of
patterns seen in the experiments is explained as being due the variation of the
details of the initiation of the chemoattractant emission as well as the
transition to a non-motile phase.


title: Global modes for the complex Ginzburg-Landau equation

abstract:
  Linear global modes, which are time-harmonic solutions with vanishing
boundary conditions, are analysed in the context of the complex Ginzburg-Landau
equation with slowly varying coefficients in doubly infinite domains. The most
unstable modes are shown to be characterized by the geometry of their Stokes
line network: they are found to generically correspond to a configuration with
two turning points issued from opposite sides of the real axis which are either
merged or connected by a common Stokes line. A region of local absolute
instability is also demonstrated to be a necessary condition for the existence
of unstable global modes.


title: Phase Slips and the Eckhaus Instability

abstract:
  We consider the Ginzburg-Landau equation, $ \partial_t u= \partial_x^2 u + u
- u|u|^2 $, with complex amplitude $u(x,t)$. We first analyze the phenomenon of
phase slips as a consequence of the {\it local} shape of $u$. We next prove a
{\it global} theorem about evolution from an Eckhaus unstable state, all the
way to the limiting stable finite state, for periodic perturbations of Eckhaus
unstable periodic initial data. Equipped with these results, we proceed to
prove the corresponding phenomena for the fourth order Swift-Hohenberg
equation, of which the Ginzburg-Landau equation is the amplitude approximation.
This sheds light on how one should deal with local and global aspects of phase
slips for this and many other similar systems.


title:
Boundary-induced wavelength selection in a one-dimensional
  pattern-forming system

abstract:
  We have measured the stability boundary for steady electrically-driven
convective flow in thin, freely suspended films of Smectic-A liquid crystal.
The thinness and layered structure of the films supress two- and
three-dimensional instabilities of the convection pattern. As the voltage
applied across the film, or the length of the film, is varied, convective
vortices are created or destroyed to keep the wave number of the pattern within
a stable range. The range of stable wave numbers increases linearly with the
dimensionless control parameter $\epsilon$, for small $\epsilon$, and the
vortices always appear and disappear at the ends of the film. These results are
consistent with a mechanism for boundary-induced wavelength selection proposed
by Cross {\it et al.} [Phys. Rev. Lett. {\bf 45}, 898 (1980)].


title:
Defect--Defect Correlation Functions, Generic Scale Invariance, and the
  Complex Ginzburg--Landau Equation

abstract:
  Motivated by the idea of developing a ``hydrodynamic'' description of
spatiotemporal chaos, we have investigated the defect--defect correlation
functions in the defect turbulence regime of the two--dimensional, anisotropic
complex Ginzburg--Landau equation. We compare our results with the predictions
of generic scale invariance. Using the topological nature of the defects, we
prove that defect--defect correlations cannot decay as slowly as predicted by
generic scale invariance. We also present results on the fluctuations of the
amplitude field $A$.


title:
Convection in Binary Fluid Mixtures. I. Extended Traveling Wave and
  Stationary States. (Physical Review E, in press)

abstract:
  Nonlinear convection structures are investigated in quantitative detail as a
function of Rayleigh number for several negative and positive Soret coupling
strengths (separation ratios) and different Lewis and Prandtl numbers
characterizing different mixtures. A finite difference method was used to solve
the full hydrodynamic field equations in a range of experimentally accessible
parameters. We elucidate the important role that the concentration field plays
in the nonlinear states of stationary overturning convection (SOC) and of
traveling wave (TW) convection. Structural differences in the concentration
boundary layers and of the concentration plumes in TW's and SOC's and their
physical consequences are discussed. These properties show that the states con-
sidered here are indeed strongly nonlinear, as expected from the magnitude of
advection and diffusion in the concentration balance. The bifurcation behaviour
of the states is analysed using different order parameters such as flow
intensity, Nusselt number, a newly defined mixing parameter characterized by
the variance of the concentration field, and the TW frequency. For further
comparison with experiments, light intensity distributions are determined that
can be observed in side-view shadowgraphs. Structural analyses of all fields
are made using colour coded isoplots, vertical and lateral field profiles, and
lateral Fourier decompositions. Transport properties of TWs are also discussed,
in particular the mean lateral concentration current that is caused by the
phase difference between concentration wave and velocity wave and that is
roughly proportional to the TW frequency. This current plays an important role
in the structural dynamics and stability of spatially-localized traveling-wave
convection (cf. accompanying paper).


title:
Convection in Binary Fluid Mixtures. II. Localized Traveling Waves.
  (Physical Review E, in press)

abstract:
  Nonlinear, spatially localized structures of traveling convection rolls are
investigated in quantitative detail as a function of Rayleigh number for two
different Soret coupling strengths (separation ratios) with Lewis and Prandtl
numbers characterizing ethanol-water mixtures. A finite-difference method was
used to solve the full hydrodynamic field equations numerically. Structure and
dynamics of these localized traveling waves (LTW) are dominated by the
concentration field. Like in the spatially extended convective states ( cf.
accompanying paper), the Soret-induced concentration variations strongly
influence, via density changes, the buoyancy forces that drive convection. The
spatio-temporal properties of this feed-back mechanism, involving boundary
layers and concentration plumes, show that LTW's are strongly nonlinear states.
Light intensity distributions are determined that can be observed in side-view
shadowgraphs. Detailed analyses of all fields are made using colour-coded
isoplots, among others. In the frame comoving with their drift velocity, LTW's
display a nontrivial spatio-temporal symmetry consisting of time-translation by
half an oscillation period combined with vertical reflection through the
horizontal midplane of the layer. A time-averaged concentration current is
driven by a phase difference between the waves of concentration and vertical
velocity in the bulk of the LTW state. The associated large-scale concentration
redistribution stabilizes the LTW and controls its drift velocity into the
quiescent fluid by generating a buoyancy-reducing concentration "barrier" ahead
of the leading LTW front. The selection of the width of the LTW's is
investigated and comparisons with experiments are presented.


title: Dynamic Domain Walls in Strongly Driven Ferromagnets

abstract:
  A multiple-time scaling analysis of the dissipative, transversely driven
Landau-Lifshitz equation in presence of exchange, shape demagnetisation and
week anisotropy fields is performed for a dynamic domain state. Stationary
solutions of the resulting equations explain the spatiotemporal structure of
the walls and are in agreement with previous simulations.


title:
Mean Field Theory of the Morphology Transition in Stochastic Diffusion
  Limited Growth

abstract:
  We propose a mean-field model for describing the averaged properties of a
class of stochastic diffusion-limited growth systems. We then show that this
model exhibits a morphology transition from a dense-branching structure with a
convex envelope to a dendritic one with an overall concave morphology. We have
also constructed an order parameter which describes the transition
quantitatively. The transition is shown to be continuous, which can be verified
by noting the non-existence of any hysteresis.


title: Influence of the Dufour effect on convection in binary gas mixtures

abstract:
  Linear and nonlinear properties of convection in binary fluid layers heated
from below are investigated, in particular for gas parameters. A Galerkin
approximation for realistic boundary conditions that describes stationary and
oscillatory convection in the form of straight parallel rolls is used to
determine the influence of the Dufour effect on the bifurcation behaviour of
convective flow intensity, vertical heat current, and concentration mixing. The
Dufour--induced changes in the bifurcation topology and the existence regimes
of stationary and traveling wave convection are elucidated. To check the
validity of the Galerkin results we compare with finite--difference numerical
simulations of the full hydrodynamical field equations. Furthermore, we report
on the scaling behaviour of linear properties of the stationary instability.


title: Coexisting Pulses in a Model for Binary-Mixture Convection

abstract:
  We address the striking coexistence of localized waves (`pulses') of
different lengths which was observed in recent experiments and full numerical
simulations of binary-mixture convection. Using a set of extended
Ginzburg-Landau equations, we show that this multiplicity finds a natural
explanation in terms of the competition of two distinct, physical localization
mechanisms; one arises from dispersion and the other from a concentration mode.
This competition is absent in the standard Ginzburg-Landau equation. It may
also be relevant in other waves coupled to a large-scale field.


title: Coupled Maps on Trees

abstract:
  We study coupled maps on a Cayley tree, with local (nearest-neighbor)
interactions, and with a variety of boundary conditions. The homogeneous state
(where every lattice site has the same value) and the node-synchronized state
(where sites of a given generation have the same value) are both shown to occur
for particular values of the parameters and coupling constants. We study the
stability of these states and their domains of attraction. As the number of
sites that become synchronized is much higher compared to that on a regular
lattice, control is easier to effect. A general procedure is given to deduce
the eigenvalue spectrum for these states. Perturbations of the synchronized
state lead to different spatio-temporal structures. We find that a mean-field
like treatment is valid on this (effectively infinite dimensional) lattice.


title: Domain Coarsening in Systems Far from Equilibrium

abstract:
  The growth of domains of stripes evolving from random initial conditions is
studied in numerical simulations of models of systems far from equilibrium such
as Rayleigh-Benard convection. The scaling of the size of the domains deduced
from the inverse width of the Fourier spectrum is studied for both potential
and nonpotential models. The morphology of the domains and the defect
structures are however quite different in the two cases, and evidence is
presented for a second length scale in the nonpotential case.


title:
Two-phase binary fluids and immiscible fluids described by an order
  parameter

abstract:
  A unified framework for coupled Navier-Stokes/Cahn-Hilliard equations is
developed using, as a basis, a balance law for microforces in conjunction with
constitutive equations consistent with a mechanical version of the second law.
As a numerical application of the theory, we consider the kinetics of
coarsening for a binary fluid in two space-dimensions.


title: A Kink Soliton Characterizing Traffic Congestion

abstract:
  We study traffic congestion by analyzing a one dimensional traffic flow
model. Developing an asymptotic method to investigate the long time behavior
near a critical point, we derive the modified Korteweg-de Vries (mKdV) equation
as the lowest order model. There are an infinite number of kink solitons to the
mKdV equation, while it has been found by numerical simulations that the kink
pattern arising in traffic congestion is uniquely determined irrespective of
initial conditions. In order to resolve this selection problem, we consider
higher order corrections to the mKdV equation and find that there is a kink
soliton which can deform continuously with the perturbation represented by the
addition of these corrections. With numerical confirmation, we show that this
continuously deformable kink soliton characterizes traffic congestion. We also
discuss the relationship between traffic congestion and slugging phenomenon in
granular flow.


title: Cancer Detection via Determination of Fractal Cell Dimension

abstract:
  We utilize the fractal dimension of the perimeter surface of cell sections as
a new observable to characterize cells of different types. We propose that it
is possible to distinguish cancerous from healthy cells with the aid of this
new approach. As a first application we show that it is possible to perform
this distinction between patients with hairy-cell lymphocytic leukemia and
those with normal blood lymphocytes.


title: Long-Wavelength Instability in Surface-Tension-Driven Benard Convection

abstract:
  Laboratory studies reveal a deformational instability that leads to a drained
region (dry spot) in an initially flat liquid layer (with a free upper surface)
heated uniformly from below. This long-wavelength instability supplants
hexagonal convection cells as the primary instability in viscous liquid layers
that are sufficiently thin or are in microgravity. The instability occurs at a
temperature gradient 34% smaller than predicted by linear stability theory.
Numerical simulations show a drained region qualitatively similar to that seen
in the experiment.


title: Onset of Surface-Tension-Driven Benard Convection

abstract:
  Experiments with shadowgraph visualization reveal a subcritical transition to
a hexagonal convection pattern in thin liquid layers that have a free upper
surface and are heated from below. The measured critical Marangoni number (84)
and observation of hysteresis (3%) agree with theory. In some experiments,
imperfect bifurcation is observed and is attributed to deterministic forcing
caused in part by the lateral boundaries in the experiment.


title: Hexagons, Kinks and Disorder in Oscillated Granular Layers

abstract:
  Experiments on vertically oscillated granular layers in an evacuated
container reveal a sequence of well-defined pattern bifurcations as the
container acceleration is increased. Period doublings of the layer center of
mass motion and a parametric wave instability interact to produce hexagons and
more complicated patterns composed of distinct spatial domains of different
relative phase separated by kinks (phase discontinuities). Above a critical
acceleration, the layer becomes disordered in both space and time.


title: Front Stability in Mean Field Models of Diffusion Limited Growth

abstract:
  We present calculations of the stability of planar fronts in two mean field
models of diffusion limited growth. The steady state solution for the front can
exist for a continuous family of velocities, we show that the selected velocity
is given by marginal stability theory. We find that naive mean field theory has
no instability to transverse perturbations, while a threshold mean field theory
has such a Mullins-Sekerka instability. These results place on firm theoretical
ground the observed lack of the dendritic morphology in naive mean field theory
and its presence in threshold models. The existence of a Mullins-Sekerka
instability is related to the behavior of the mean field theories in the
zero-undercooling limit.


title:
Boussinesq Solitary-Wave as a Multiple-Time Solution of the Korteweg-de
  Vries Hierarchy

abstract:
  We study the Boussinesq equation from the point of view of a multiple-time
reductive perturbation method. As a consequence of the elimination of the
secular producing terms through the use of the Korteweg--de Vries hierarchy, we
show that the solitary--wave of the Boussinesq equation is a solitary--wave
satisfying simultaneously all equations of the Korteweg--de Vries hierarchy,
each one in an appropriate slow time variable.


title:
A Geometrical Formulation of the Renormalization Group Method for Global
  Analysis II: Partial Differential Equations

abstract:
  It is shown that the renormalization group (RG) method for global analysis
can be formulated in the context of the classical theory of envelopes: Several
examples from partial differential equations are analyzed. The amplitude
equations which are usually derived by the reductive perturbation theory are
shown to be naturally derived as the equations describing the envelopes of the
local solutions obtained in the perturbation theory.

categories: patt-sol chao-dyn cond-mat funct-an math.FA nlin.CD nlin.PS


title: Controlling domain patterns far from equilibrium

abstract:
  A high degree of control over the structure and dynamics of domain patterns
in nonequilibrium systems can be achieved by applying nonuniform external
fields near parity breaking front bifurcations. An external field with a linear
spatial profile stabilizes a propagating front at a fixed position or induces
oscillations with frequency that scales like the square root of the field
gradient. Nonmonotonic profiles produce a variety of patterns with controllable
wavelengths, domain sizes, and frequencies and phases of oscillations.


title:
The Role of the Korteweg-de Vries Hierarchy in the N-Soliton Dynamics of
  the Shallow Water Wave Equation

abstract:
  We apply a multiple-time version of the reductive perturbation method to
study long waves as governed by the shallow water wave model equation. As a
consequence of the requirement of a secularity-free perturbation theory, we
show that the well known N-soliton dynamics of the shallow water wave equation,
in the particular case of $\alpha=2 \beta$, can be reduced to the N-soliton
solution that satisfies simultaneously all equations of the Korteweg-de Vries
hierarchy.


title:
Tangent Bifurcation of Band Edge Plane Waves, Dynamical Symmetry
  Breaking and Vibrational Localization

abstract:
  We study tangent bifurcation of band edge plane waves in nonlinear
Hamiltonian lattices. The lattice is translationally invariant. We argue for
the breaking of permutational symmetry by the new bifurcated periodic orbits.
The case of two coupled oscillators is considered as an example for the
perturbation analysis, where the symmetry breaking can be traced using Poincare
maps. Next we consider a lattice and derive the dependence of the bifurcation
energy on the parameters of the Hamiltonian function in the limit of large
system sizes. A necessary condition for the occurence of the bifurcation is the
repelling of the band edge plane wave's frequency from the linear spectrum with
increasing energy. We conclude that the bifurcated orbits will consequently
exponentially localize in the configurational space.


title:
Reductive Perturbation Method, Multiple-Time Solutions and the KdV
  Hierarchy

abstract:
  We apply a multiple-time version of the reductive perturbation method to
study long waves as governed by the Boussinesq model equation. By requiring the
absence of secular producing terms in each order of the perturbative scheme, we
show that the solitary-wave of the Boussinesq equation can be written as a
solitary-wave satisfying simultaneously all equations of the KdV hierarchy,
each one in a different slow time variable. We also show that the conditions
for eliminating the secularities are such that they make the perturbation
theory compatible with the linear theory coming from the Boussinesq equation.