title:
Resonance Effects in the Interaction of NLS Solitons with Potential
  Wells

abstract:
  The interaction of nonlinear Schr\"{o}dinger (NLS) solitons with potential
wells with variable shapes is investigated numerically. For fixed initial
velocities below the threshold for transmission, the outcome pattern as a
function of the width of the potential yields periodically repeating regions of
trapping, transmission and reflection. The observed effects are explained by an
excitation and a following resonant deexcitation of amplitude (shape)
oscillations of the solitons at the boundaries of the well, associated with
radiation modes.


title: Solitons in a system of three linearly coupled fiber gratings

abstract:
  We introduce a model of three parallel-coupled nonlinear waveguiding cores
equipped with Bragg gratings (BGs), which form an equilateral triangle. The
objective of the work is to investigate solitons and their stability in this
system. New results are also obtained for the earlier investigated dual-core
system. Families of symmetric and antisymmetric solutions are found
analytically, extending beyond the spectral gap in both the dual- and tri-core
systems. Moreover, these families persist in the case (strong coupling between
the cores) when there is no gap in the system's linear spectrum. Three
different types of asymmetric solitons are found in the tri-core system. They
exist only inside the spectral gap, but asymmetric solitons with nonvanishing
tails are found outside the gap as well. The symmetric solitons are stable up
to points at which two types of asymmetric solitons bifurcate from them. Beyond
the bifurcation, one type of the asymmetric solitons is stable, and the other
is not. Then, they swap their stability. In both the dual- and tri-core
systems, the stability region of the symmetric solitons extends far beyond the
gap, persisting in the case when the system has no gap at all. The whole
stability region of antisymmetric solitons is located outside the gap. Thus,
solitons in multi-core BGs can be observed experimentally in a much broader
frequency band than in the single-core one, and in a wider parameter range than
it could be expected.


title:
Is relaxation to equilibrium hindered by transient dissipative
  structures in closed systems?

abstract:
  Dissipative structures are generally observed when a system relaxes from a
far from equilibrium state. To address the reverse question given by the title,
we investigate the relaxation process in a closed chemical reaction-diffusion
system which can potentially form Turing-like patterns during the transient. We
find that when certain conditions are fulfilled the relaxation process is
indeed drastically hindered, once the pattern is formed. This slowing-down is
shown to be due to stepwise relaxation, where each plateau in the relaxation
process corresponds to residence at a certain spatial pattern. Universality of
the phenomena as well as their biological relevance is briefly discussed.


title: Universal R operator with Jordanian deformation of conformal symmetry

abstract:
  The Jordanian deformation of $sl(2)$ bi-algebra structure is studied in view
of physical applications to breaking of conformal symmetry in the high energy
asymptotics of scattering. Representations are formulated in terms of
polynomials, generators in terms of differential operators. The deformed $R$
operator with generic representations is analyzed in spectral and integral
forms.


title: Stabilized two-dimensional vector solitons

abstract:
  In this letter we introduce the concept of stabilized vector solitons as
nonlinear waves constructed by addition of mutually incoherent Townes solitons
that are stabilized under the effect of a periodic modulation of the
nonlinearity. We analyze the stability of this new kind of structures and
describe their behavior and formation in Manakov-like interactions. Potential
applications of our results in Bose-Einstein condensation and nonlinear optics
are also discussed.


title:
The characterization of two-component (2+1)-dimensional integrable
  systems of hydrodynamic type

abstract:
  We obtain the necessary and sufficient conditions for a two-component
(2+1)-dimensional system of hydrodynamic type to possess infinitely many
hydrodynamic reductions. These conditions are in involution, implying that the
systems in question are locally parametrized by 15 arbitrary constants. It is
proved that all such systems possess three conservation laws of hydrodynamic
type and, therefore, are symmetrizable in Godunov's sense. Moreover, all such
systems are proved to possess a scalar pseudopotential which plays the role of
the `dispersionless Lax pair'. We demonstrate that the class of two-component
systems possessing a scalar pseudopotential is in fact identical with the class
of systems possessing infinitely many hydrodynamic reductions, thus
establishing the equivalence of the two possible definitions of the
integrability. Explicit linearly degenerate examples are constructed.


title: Effects of forcing in three dimensional turbulent flows

abstract:
  We present the results of a numerical investigation of three-dimensional
homogeneous and isotropic turbulence, stirred by a random forcing with a power
law spectrum, $E_f(k)\sim k^{3-y}$. Numerical simulations are performed at
different resolutions up to $512^3$. We show that at varying the spectrum slope
$y$, small-scale turbulent fluctuations change from a {\it forcing independent}
to a {\it forcing dominated} statistics. We argue that the critical value
separating the two behaviours, in three dimensions, is $y_c=4$. When the
statistics is forcing dominated, for $y<y_c$, we find dimensional scaling, i.e.
intermittency is vanishingly small. On the other hand, for $y>y_c$, we find the
same anomalous scaling measured in flows forced only at large scales. We
connect these results with the issue of {\it universality} in turbulent flows.


title:
Energy funneling in a bent chain of Morse oscillators with long-range
  coupling

abstract:
  A bent chain of coupled Morse oscillators with long-range dispersive
interaction is considered. Moving localized excitations may be trapped in the
bending region. Thus chain geometry acts like an impurity. An energy funneling
effect is observed in the case of random initial conditions.


title: Stability of vortex solitons in a photorefractive optical lattice

abstract:
  Stability of off-site vortex solitons in a photorefractive optical lattice is
analyzed. It is shown that such solitons are linearly unstable in both the high
and low intensity limits. In the high-intensity limit, the vortex looks like a
familiar ring vortex, and it suffers oscillatory instabilities. In the
low-intensity limit, the vortex suffers both oscillatory and Vakhitov-Kolokolov
instabilities. However, in the moderate-intensity regime, the vortex becomes
stable if the lattice intensity or the applied voltage is above a certain
threshold value. Stability regions of vortices are also determined at typical
experimental parameters.


title: Periodic orbit theory of strongly anomalous transport

abstract:
  We establish a deterministic technique to investigate transport moments of
arbitrary order. The theory is applied to the analysis of different kinds of
intermittent one-dimensional maps and the Lorentz gas with infinite horizon:
the typical appearance of phase transitions in the spectrum of transport
exponents is explained.


title: Geometrical resonance in spatiotemporal systems

abstract:
  We generalize the concept of geometrical resonance to perturbed sine-Gordon,
Nonlinear Schrödinger and Complex Ginzburg-Landau equations. Using this
theory we can control different dynamical patterns. For instance, we can
stabilize breathers and oscillatory patterns of large amplitudes successfully
avoiding chaos. On the other hand, this method can be used to suppress
spatiotemporal chaos and turbulence in systems where these phenomena are
already present. This method can be generalized to even more general
spatiotemporal systems.


title: Effective Hamiltonian for traveling discrete breathers in the FPU chain

abstract:
  For the Fermi-Pasta-Ulam chain, an effective Hamiltonian is constructed,
describing the motion of approximate, weakly localized discrete breathers
traveling along the chain. The velocity of these moving and localized
vibrations can be estimated analytically as the group velocity of the
corresponding wave packet. The Peierls-Nabarro barrier is estimated for
strongly localized discrete breathers.


title: Poisson Pencils, Integrability, and Separation of Variables

abstract:
  In this paper we will review a recently introduced method for solving the
Hamilton-Jacobi equations by the method of Separation of Variables. This method
is based on the notion of pencil of Poisson brackets and on the bihamiltonian
approach to integrable systems. We will discuss how separability conditions can
be intrinsically characterized within such a geometrical set-up, the definition
of the separation coordinates being encompassed in the \bih structure itself.
We finally discuss these constructions studying in details a particular
example, based on a generalization of the classical Toda Lattice.


title: Large scale inhomogeneity of inertial particles in turbulent flow

abstract:
  Preferential concentration of inertial particles in turbulent flow is studied
by high resolution direct numerical simulations of two-dimensional turbulence.
The formation of network-like regions of high particle density, characterized
by a length scale which depends on the Stokes number of inertial particles, is
observed. At smaller scales, the size of empty regions appears to be
distributed according to a scaling law.


title: History-dependent synchronization in the Burridge-Knopoff model

abstract:
  A three-blocks Burridge-Knopoff model is investigated. The dimensionless
velocity-dependent friction force $F(v)\propto (1+av)^{-1}$ is linearized
around $a=0$. In this way, the model is transformed into a six-dimensional
mapping $\mathbf{x}(t_{n})\to \mathbf{x}(t_{n+1})$, where $t_n$ are time
moments when a block starts to move or stops. Between these moments, the
equations of motion are integrable. For $a<0.1$, the motion is quasiperiodic or
periodic, depending on the initial conditions. For the periodic solution, we
observe a synchronization of the motion of the lateral blocks. For $a>0.1$, the
motion becomes chaotic. These results are true for the linearized mapping,
linearized numerical and non-linearized numerical solutions.


title:
Antisymmetric solitons and their interactions in strongly
  dispersion-managed fiber-optic systems

abstract:
  By means of the variational approximation (VA), a system of ordinary
differential equations (ODEs) is derived to describe the propagation of
antisymmetric solitons in a multi-channel (WDM) optical fiber link subject to
strong dispersion management. Results are reported for a prototypical model
including two channels. Using the VA technique, conditions for stable
propagation of the antisymmetric dispersion-managed (ASDM) solitons in one
channel are found, and complete and incomplete collisions between the solitons
belonging to the different channels are investigated. In particular, it is
shown that formation of a bound inter-channel state of two ASDM solitons is
possible under certain conditions (but may be easily avoided). The VA
predictions for the single- and two-channel systems are compared with direct
simulations of the underlying partial differential equations. In most cases,
the agreement is very good, but in some cases (very closely spaced channels)
the collision may destroy the ASDM solitons. The timing-jitter suppression
factor (JSF) for the ASDM soliton in one channel, and the crosstalk timing
jitter induced by collision between the solitons belonging to the different
channels are also estimated analytically. In particular, the JSF for the ASDM
soliton may be much larger than for its fundamental-soliton counterpart in the
same system.


title: On a class of inhomogeneous extensions for integrable evolution systems

abstract:
  In the present paper we prove the integrability (in the sense of existence of
formal symmetry of infinite rank) for a class of block-triangular inhomogeneous
extensions of (1+1)-dimensional integrable evolution systems. An important
consequence of this result is the existence of formal symmetry of infinite rank
for "almost integrable" systems, recently discovered by Sanders and van der
Kamp.


title: Associative memory on a small-world neural network

abstract:
  We study a model of associative memory based on a neural network with
small-world structure. The efficacy of the network to retrieve one of the
stored patterns exhibits a phase transition at a finite value of the disorder.
The more ordered networks are unable to recover the patterns, and are always
attracted to mixture states. Besides, for a range of the number of stored
patterns, the efficacy has a maximum at an intermediate value of the disorder.
We also give a statistical characterization of the attractors for all values of
the disorder of the network.


title: Universal level-spacing distribution in quantum systems

abstract:
  Classical counterparts of a great variety of quantum systems, from atomic
physics to quantum wells and quantum dots, to optical, microwave, and acoustic
resonators exhibit partially chaotic dynamics. Since it is often impossible to
measure the temporal dynamics in qunatum systems, the main and probably the
most dramatic manifestation of classical chaos in their phase space is seen in
the distribution of spacing between the neighboring energy levels. While the
mechanism leading to the onset of chaotic dynamics is different in every
system, the level spacing distribution obeys the universal law, changing from
Poissonian in the completely integrable systems to Wigner in completely chaotic
ones.


title: Symbolic analysis for some planar piecewise linear maps

abstract:
  In this paper a class of linear maps on the 2-torus and some planar piecewise
isometries are discussed. For these discontinuous maps, by introducing codings
underlying the map operations, symbolic descriptions of the dynamics and
admissibility conditions for itineraries are given, and explicit expressions in
terms of the codings for periodic points are presented.


title: Towards a theory of differential constraints of a hydrodynamic hierarchy

abstract:
  We present a theory of compatible differential constraints of a hydrodynamic
hierarchy of infinite-dimensional systems. It provides a convenient point of
view for studying and formulating integrability properties and it reveals some
hidden structures of the theory of integrable systems. Illustrative examples
and new integrable models are exhibited.


title: Vortex solitons in photonic crystal fibers

abstract:
  We demonstrate the existence of vortex soliton solutions in photonic crystal
fibers. We analyze the role played by the photonic crystal fiber defect in the
generation of optical vortices. An analytical prediction for the angular
dependence of the amplitude and phase of the vortex solution based on group
theory is also provided. Furthermore, all the analysis is performed in the
non-paraxial regime.


title:
Nonlinear Beltrami equation and tau-function for dispersionless
  hierarchies

abstract:
  It is proved that the action for nonlinear Beltrami equation (quasiclassical
dbar-problem) evaluated on its solution gives a tau-function for dispersionless
KP hierarchy. Infinitesimal transformations of tau-function corresponding to
variations of dbar-data are found. Determinant equations for the function
generating these transformations are derived. They represent a dispersionless
analogue of singular manifold (Schwarzian) KP equations. Dispersionless 2DTL
hierarchy is also considered.


title:
On bosonic limits of two recent supersymmetric extensions of the Harry
  Dym hierarchy

abstract:
  Two generalized Harry Dym equations, recently found by Brunelli, Das and
Popowicz in the bosonic limit of new supersymmetric extensions of the Harry Dym
hierarchy [J. Math. Phys. 44:4756--4767 (2003)], are transformed into
previously known integrable systems: one--into a pair of decoupled KdV
equations, the other one--into a pair of coupled mKdV equations from a
bi-Hamiltonian hierarchy of Kupershmidt.


title: Interaction of sine-Gordon kinks with defects: The two-bounce resonance

abstract:
  A model of soliton-defect interactions in the sine-Gordon equations is
studied using singular perturbation theory. Melnikov theory is used to derive a
critical velocity for strong interactions, which is shown to be exponentially
small for weak defects. Matched asymptotic expansions for nearly heteroclinic
orbits are constructed for the initial value problem, which are then used to
derive analytical formulas for the locations of the well known two- and
three-bounce resonance windows, as well as several other phenomena seen in
numerical simulations.


title: Adaptability and Diversity in Simulated Turn-taking Behaviour

abstract:
  Turn-taking behaviour is simulated in a coupled agents system. Each agent is
modelled as a mobile robot with two wheels. A recurrent neural network is used
to produce the motor outputs and to hold the internal dynamics. Agents are
developed to take turns on a two-dimensional arena by causing the network
structures to evolve.
  Turn-taking is established using either regular or chaotic behaviour of the
agents. It is found that chaotic turn-takers are more sensitive to the adaptive
inputs from the other agent. Conversely, regular turn-takers are comparatively
robust against noisy inputs, owing to their restricted dynamics. From many
observations, including turn-taking with virtual agents, we claim that there is
a complementary relationship between robustness and adaptability. Furthermore,
by investigating the recoupling of agents from different GA generations, we
report the emergence of a new turn-taking behaviour. Potential for synthesizing
a new form of motion is another characteristic of chaotic turn-takers.


title:
Measurement of synchronization properties in systems with different
  excitation levels

abstract:
  We devised a measure based on the distributions of relative event timings of
two coupled units. The measure dynamically evaluates temporal interdependencies
between the two coupled units. Using this we show that even in the event of
symmetrical coupling, non-identical units (having different control parameters)
have small but persistent shifts in their event timings that change
dramatically with the relative properties of their internal trajectories. We
link that finding with the activity dependent synaptic modification in the
brain showing that internally or externally driven excitation levels of the
neuron may determine direction of information processing in a network.


title: Quantum breathers in an attractive fermionic Hubbard model

abstract:
  In one-dimensional translationally invariant anharmonic lattices, an extended
Bloch state with two or more strongly correlated particles is usually called a
quantum breather. Here we study an attractive fermionic Hubbard model with two
kind of particles of opposite spin. We discuss the existence of breathers, and
several effects that break the translational symmetry of the system and
localize the breather in the lattice.


title: The analytic structure of 2D Euler flow at short times

abstract:
  Using a very high precision spectral calculation applied to the
incompressible and inviscid flow with initial condition $\psi_0(x_1, x_2) =
\cos x_1+\cos 2x_2$, we find that the width $\delta(t)$ of its analyticity
strip follows a $\ln(1/t)$ law at short times over eight decades. The
asymptotic equation governing the structure of spatial complex-space
singularities at short times (Frisch, Matsumoto and Bec 2003, J.Stat.Phys. 113,
761--781) is solved by a high-precision expansion method. Strong numerical
evidence is obtained that singularities have infinite vorticity and lie on a
complex manifold which is constructed explicitly as an envelope of analyticity
disks.


title:
An investigation into the vortex formation in a turbulent fluid with an
  application in tropical storm generation

abstract:
  The present work discusses about a possible physical interpretation of the
occurrence of turbulence in a dynamic fluid with mathematical modeling and
computer simulation. Here turbulence is defined to be a phenomenon of random
velocity field in the space-time continuum accompanied by chaotic occurrence of
vortices. This interpretation is independent of the Navier-Stokes equations. I
have reasoned how individual fluid percels are disintegrated with increasing
Reynolds number (Re) (or increasing velocity or decreasing viscosity or both)
leading to creation of smaller parcels with arbitrary speeds in arbitrary
directions, which destroys the laminar structure of the fluid flow. I have
modeled the occurrence of a vortex as a result of collision among fluid jets
under certain conditions. Chaotic occurrence of such vortices further
randomizes the velocity field. These together ultimately lead to turbulence. I
have also shown an application of vortex formation in a dynamic fluid in
atmospheric science., where it has been shown how an initial dusturbing
cyclonic vortex is created by collision between two linear wind jets under
certain conditions, which under favorable conditions, may mature into a severe
tropical storm. Then a three dimensional mathematical modeling of the vortex
(assuming that it is going to become a matured storm) has been proposed with
computer simulations. This helps us to understand the mystery of origin of
cyclonic and anticyclonic vortices in atmosphere and some of their observed
asymmetries.


title: Exact solutions for one of the extensive chaos model

abstract:
  Sampling equation method is presented to look for exact solutions of
nonlinear differential equations. Application of this approach to one of the
extensive chaos model is considered. Exact solutions of this model in
travelling wave are given. Nonlinear evolution equation for the considered
extensive chaos model is shown to have solitary and periodical waves.


title:
Periodic solutions of a many-rotator problem in the plane. II. Analysis
  of various motions

abstract:
  Various solutions are displayed and analyzed (both analytically and
numerically) of arecently-introduced many-body problem in the plane which
includes both integrable and nonintegrable cases (depending on the values of
the coupling constants); in particular the origin of certain periodic behaviors
is explained. The light thereby shone on the connection among
\textit{integrability} and \textit{analyticity} in (complex) time, as well as
on the emergence of a \textit{chaotic} behavior (in the guise of a sensitive
dependance on the initial data) not associated with any local exponential
divergence of trajectories in phase space, might illuminate interesting
phenomena of more general validity than for the particular model considered
herein.


title:
Information Flow through a Chaotic Channel: Prediction and Postdiction
  at Finite Resolution

abstract:
  We reconsider the persistence of information under the dynamics of the
logistic map in order to discuss communication through a nonlinear channel
where the sender can set the initial state of the system with finite
resolution, and the recipient measures it with the same accuracy. We separate
out the contributions of global phase space shrinkage and local phase space
contraction and expansion to the uncertainty in predicting and postdicting the
state of the system. Thus, we determine how the amplification parameter, the
time lag, and the resolution influence the possibility for communication. A
novel representation for real numbers is introduced that allows for a
visualization of the flow of information between scales.


title:
Integrable systems on $so(4)$ related with XXX spin chains with
  boundaries

abstract:
  We consider two-site XXX Heisenberg magnets with different boundary
conditions, which are integrable systems on so(4) possessing additional cubic
and quartic integrals of motion. The separated variables for these models are
constructed using the Sklyanin method.


title:
A unified description of the asymmetric q-P_{v} and d-P_{iv} equations
  and their Schlesinger transformations

abstract:
  We present a geometric description, based on the affine Weyl group
E_{6}^{(1)}, of two discrete analogues of the Painlev\'e VI equation, known as
the asymmetric q-P_{V} and asymmetric d-P_{IV}. This approach allows us to
describe in a unified way the evolution of the mapping along the independent
variable and along the various parameters (the latter evolution being the one
induced by the Schlesinger transformations). It turns out that both discrete
Painlev\'e equations exhibit the property of self-duality: the same equation
governs the evolution along any direction in the space of E_{6}^{(1)}.


title:
Thermal stability of charge transport in poly(dG)-poly(dC) and
  poly(dA)-poly(dT) DNA polymers

abstract:
  This paper has been withdrawn due to some inconsistency of the results
discovered after the submission. We apologize for it.


title: Interaction of moving discrete breathers with vacancies

abstract:
  In this paper a Frenkel--Kontorova model with a nonlinear interaction
potential is used to describe a vacancy defect in a crystal. According to
recent numerical results [Cuevas et al, Phys. Lett. A 315, 364 (2003)] the
vacancy can migrate when it interacts with a moving breather. We study more
thoroughly the phenomenology caused by the interaction of moving breathers with
a single vacancy and also with double vacancies. We show that vacancy mobility
is strongly correlated with the existence and stability properties of
stationary breathers centered at the particles adjacent to the vacancy, which
we will now call vacancy breathers.


title: The geometry of dual isomonodromic deformations

abstract:
  The JMMS equations are studied using the geometry of the spectral curve of a
pair of dual systems. It is shown that the equations can be represented as
time-independent Hamiltonian flows on a Jacobian bundle.


title:
Discrete canonical system and non-Abelian Toda lattice: Backlund-Darboux
  transformation and Weyl functions

abstract:
  A version of the iterated B\"acklund-Darboux transformation, where Darboux
matrix takes a form of the transfer matrix function from the system theory, is
constructed for the discrete canonical system and Non-Abelian Toda lattice.
Results on the transformations of the Weyl functions, insertion of the
eigenvalues, and construction of the bound states are obtained. A wide class of
the explicit solutions is given. An application to the semi-infinite block
Jacobi matrices is treated.


title: Laplacian Growth and Whitham Equations of Soliton Theory

abstract:
  The Laplacian growth (the Hele-Shaw problem) of multi-connected domains in
the case of zero surface tension is proven to be equivalent to an integrable
systems of Whitham equations known in soliton theory. The Whitham equations
describe slowly modulated periodic solutions of integrable hierarchies of
nonlinear differential equations. Through this connection the Laplacian growth
is understood as a flow in the moduli space of Riemann surfaces.


title: Complex dynamics in simple systems with seasonal parameter oscillations

abstract:
  We study systems with periodically oscillating parameters that can give way
to complex periodic or non periodic orbits. Performing the long time limit, we
can define ergodic averages such as Lyapunov exponents, where a negative
maximal Lyapunov exponent corresponds to a stable periodic orbit. By this,
extremely complicated periodic orbits composed of contracting and expanding
phases appear in a natural way. Employing the technique of $\epsilon$-uncertain
points, we find that values of the control parameters supporting such periodic
motion are densely embedded in a set of values for which the motion is chaotic.
When a tiny amount of noise is coupled to the system, dynamics with positive
and with negative non-trivial Lyapunov exponents are indistinguishable. We
discuss two physical systems, an oscillatory flow inside a duct and a dripping
faucet with variable water supply, where such a mechanism seems to be
responsible for a complicated alternation of laminar and turbulent phases.


title: Nonlinear polaritons in Bose-Einstein condensates in optical lattices

abstract:
  We study the interaction of a Bose-Einstein condensate in an optical lattice
with additional electromagnetic fields under Raman resonance condition. System
of evolution equations describing ultra-short optical pulse propagation and
photo-induced transport of cold atoms in optical lattice is derived. The steady
state solution of these equations was found. There are new kinds of polaritonic
solitary waves propagating.


title:
Phase Transitions in Random Boolean Networks with Different Updating
  Schemes

abstract:
  In this paper we study the phase transitions of different types of Random
Boolean networks. These differ in their updating scheme: synchronous,
semi-synchronous, or asynchronous, and deterministic or non-deterministic. It
has been shown that the statistical properties of Random Boolean networks
change considerable according to the updating scheme. We study with computer
simulations sensitivity to initial conditions as a measure of order/chaos. We
find that independently of their updating scheme, all network types have very
similar phase transitions, namely when the average number of connections of
nodes is between one and three. This critical value depends more on the size of
the network than on the updating scheme.

categories: nlin.AO cond-mat.stat-mech cs.CC nlin.CG q-bio.QM


title: Control of chaos in Hamiltonian systems

abstract:
  We present a technique to control chaos in Hamiltonian systems which are
close to integrable. By adding a small and simple control term to the
perturbation, the system becomes more regular than the original one. We apply
this technique to a forced pendulum model and show numerically that the control
is able to drastically reduced chaos.


title: Perturbation-induced radiation by the Ablowitz-Ladik soliton

abstract:
  An efficient formalism is elaborated to analytically describe dynamics of the
Ablowitz-Ladik soliton in the presence of perturbations. This formalism is
based on using the Riemann-Hilbert problem and provides the means of
calculating evolution of the discrete soliton parameters, as well as shape
distortion and perturbation-induced radiation effects. As an example, soliton
characteristics are calculated for linear damping and quintic perturbations.


title: Registering Seconds with a Conic Clock

abstract:
  The feasibility of registering seconds using the frictionless motion of a
point-like particle that slides under gravity on an inverted conical surface is
studied. Depending on the integer part of the relation between the angular and
radial frequencies of the particle trajectory, only an angular interval for the
cone is available for this purpose. For each one of these possible angles,
there exists a unique trajectory that has the capability of registering
seconds. The method to obtain the geometrical properties of these trajectories
and the necessary initial conditions to reach them are then established.


title:
Signatures of Quantum Chaos in Wave Functions Structure for Multi-well
  2D Potentials

abstract:
  We propose a new approach to investigation of quantum manifestations of
classical stochasticity (QMCS) in wave functions structure, which can be
realized in potentials with two and more local minima. The main advantage of
the proposed approach is the possibility to detect QMCS in comparison not
different wave functions, but different parts of the same wave function.
Efficiency of the approach is demonstrated for two potentials: surface
quadrupole oscillations (QO) and lower umbillic catastrophe (UC) $D_5$.


title: Multistep Parametric Processes in Nonlinear Optics

abstract:
  We present a comprehensive overview of different types of parametric
interactions in nonlinear optics which are associated with simultaneous
phase-matching of several optical processes in quadratic nonlinear media, the
so-called multistep parametric interactions. We discuss a number of
possibilities of double and multiple phase-matching in engineered structures
with the sign-varying second-order nonlinear susceptibility, including (i)
uniform and non-uniform quasi-phase-matched (QPM) periodic optical
superlattices, (ii) phase-reversed and periodically chirped QPM structures, and
(iii) uniform QPM structures in non-collinear geometry, including recently
fabricated two-dimensional nonlinear quadratic photonic crystals. We also
summarize the most important experimental results on the multi-frequency
generation due to multistep parametric processes, and overview the physics and
basic properties of multi-color optical parametric solitons generated by these
parametric interactions.


title: Singularity confinement and algebraic integrability

abstract:
  Two important notions of integrability for discrete mappings are algebraic
integrability and singularity confinement, have been used for discrete
mappings. Algebraic integrability is related to the existence of sufficiently
many conserved quantities whereas singularity confinement is associated with
the local analysis of singularities. In this paper, the relationship between
these two notions is explored for birational autonomous mappings. Two types of
results are obtained: first, algebraically integrable mappings are shown to
have the singularity confinement property. Second, a proof of the non-existence
of algebraic conserved quantities of discrete systems based on the lack of
confinement property is given.


title:
Nonequilibrium orientational patterns in two-component Langmuir
  monolayers

abstract:
  A model of a phase-separating two-component Langmuir monolayer in the
presence of a photo-induced reaction interconvering two components is
formulated. An interplay between phase separation, orientational ordering and
treaction is found to lead to a variety of nonequilibrium self-organized
patterns, both stationary and traveling. Examples of the patterns, observed in
numerical simulations, include flowing droplets, traveling stripes, wave
sources and vortex defects.


title: Dynamic Scaling of Bred Vectors in Chaotic Extended Systems

abstract:
  We argue that the spatiotemporal dynamics of bred vectors in chaotic extended
systems are related to a kinetic roughening process in the Kardar-Parisi-Zhang
universality class. This implies that there exists a characteristic length
scale corresponding to the typical extend over which the finite-size
perturbation is actually correlated in space. This can be used as a
quantitative parameter to characterize the degree of projection of the bred
vectors into the dynamical attractor.


title:
Analytic considerations in the study of spatial patterns arising from
  non-local interaction effects in population dynamics

abstract:
  Simple analytic considerations are applied to recently discovered patterns in
a generalized Fisher equation for population dynamics. The generalization
consists of the inclusion of non-local competition interactions among
individuals. We first show how stability arguments yield a condition for
pattern formation involving the ratio of the pattern wavelength and the
effective diffusion length of the individuals. We develop a mode-mode coupling
analysis which might be useful in shedding some light on the observed formation
of small-amplitude versus large-amplitude patterns.


title:
Nonlinear 2+1--Dimensional Field Equations from Incomplete Lie Algebra
  Structures

abstract:
  We show that the nonlinear 2+1--dimensional Three--Wave Resonant Interaction
equations, describing several important physical phenomena, can be generated
starting from incomplete Lie algebras in the framework of multidimensional
prolongation structures. We make use of an {\em ansatz} involving the structure
equations of a principal prolongation connection induced by an admissible
B\"acklund map.


title:
Pattern formation without heating in an evaporative convection
  experiment

abstract:
  We present an evaporation experiment in a single fluid layer. When latent
heat associated to the evaporation is large enough, the heat flow through the
free surface of the layer generates temperature gradients that can destabilize
the conductive motionless state giving rise to convective cellular structures
without any external heating. The sequence of convective patterns obtained here
without heating, is similar to that obtained in B\'enard-Marangoni convection.
This work present the sequence of spatial bifurcations as a function of the
layer depth. The transition between square to hexagonal pattern, known from
non-evaporative experiments, is obtained here with a similar change in
wavelength.


title:
Diffusion of Power in Randomly Perturbed Hamiltonian Partial
  Differential Equations

abstract:
  We study the evolution of the energy (mode-power) distribution for a class of
randomly perturbed Hamiltonian partial differential equations and derive {\it
master equations} for the dynamics of the expected power in the discrete modes.
In the case where the unperturbed dynamics has only discrete frequencies
(finitely or infinitely many) the mode-power distribution is governed by an
equation of discrete diffusion type for times of order $\cO(\ve^{-2})$. Here
$\ve$ denotes the size of the random perturbation. If the unperturbed system
has discrete and continuous spectrum the mode-power distribution is governed by
an equation of discrete diffusion-damping type for times of order
$\cO(\ve^{-2})$. The methods involve an extension of the authors' work on
deterministic periodic and almost periodic perturbations, and yield new results
which complement results of others, derived by probabilistic methods.


title:
Classical, semiclassical, and quantum investigations of the 4-sphere
  scattering system

abstract:
  A genuinely three-dimensional system, viz. the hyperbolic 4-sphere scattering
system, is investigated with classical, semiclassical, and quantum mechanical
methods at various center-to-center separations of the spheres. The efficiency
and scaling properties of the computations are discussed by comparisons to the
two-dimensional 3-disk system. While in systems with few degrees of freedom
modern quantum calculations are, in general, numerically more efficient than
semiclassical methods, this situation can be reversed with increasing dimension
of the problem. For the 4-sphere system with large separations between the
spheres, we demonstrate the superiority of semiclassical versus quantum
calculations, i.e., semiclassical resonances can easily be obtained even in
energy regions which are unattainable with the currently available quantum
techniques. The 4-sphere system with touching spheres is a challenging problem
for both quantum and semiclassical techniques. Here, semiclassical resonances
are obtained via harmonic inversion of a cross-correlated periodic orbit
signal.


title: A random matrix formulation of fidelity decay

abstract:
  We propose to study echo dynamics in a random matrix framework, where we
assume that the perturbation is time independent, random and orthogonally
invariant. This allows to use a basis in which the unperturbed Hamiltonian is
diagonal and its properties are thus largely determined by its spectral
statistics. We concentrate on the effect of spectral correlations usually
associated to chaos and disregard secular variations in spectral density. We
obtain analytic results for the fidelity decay in the linear response regime.
To extend the domain of validity, we heuristically exponentiate the linear
response result. The resulting expressions, exact in the perturbative limit,
are accurate approximations in the transition region between the ``Fermi golden
rule'' and the perturbative regimes, as examplarily verified for a
deterministic chaotic system. To sense the effect of spectral stiffness, we
apply our model also to the extreme cases of random spectra and equidistant
spectra. In our analytical approximations as well as in extensive Monte Carlo
calculations, we find that fidelity decay is fastest for random spectra and
slowest for equidistant ones, while the classical ensembles lie in between. We
conclude that spectral stiffness systematically enhances fidelity.


title: Ising-Bloch Transition in Degenerate Optical Parametric Oscillators

abstract:
  Domain walls in type I degenerate optical parametric oscillators are
numerically investigated. Both steady Ising and moving Bloch walls are found,
bifurcating one into another through a nonequilibrium Ising--Bloch transition.
Bloch walls are found that connect either homogeneous or roll planforms.
Secondary bifurcations affecting Bloch wall movement are characterized that
lead to a transition from a steady drift state to a temporal chaotic movement
as the system is moved far from the primary, Ising--Bloch bifurcation. Two
kinds of routes to chaos are found, both involving tori: a usual Ruelle-Takens
and an intermittent scenarios.


title:
A class of Einstein--Weyl spaces associated to an integrable system of
  hydrodynamic type

abstract:
  HyperCR Einstein--Weyl equations in 2+1 dimensions reduce to a pair of
quasi-linear PDEs of hydrodynamic type. All solutions to this hydrodynamic
system can be in principle constructed from a twistor correspondence, thus
establishing the integrability. Simple examples of solutions including the
hydrodynamic reductions yield new Einstein--Weyl structures.


title: Bifurcations due to small time-lag in coupled excitable systems

abstract:
  A system of ODE's is used to approximate the dynamics of two delayed coupled
  FitzHugh-Nagumo excitable units, and study the relevant bifurcations. It is
shown that the Bautin bifurcation acts as the organizing center for the
dynamics of ODE's for small time-lags.


title: Persistent holes in a fluid

abstract:
  We observe stable holes in a vertically oscillated 0.5 cm deep aqueous
suspension of cornstarch for accelerations a above 10g. Holes appear only if a
finite perturbation is applied to the layer. Holes are circular and
approximately 0.5 cm wide, and can persist for more than 10^5 cycles. Above a =
17g the rim of the hole becomes unstable producing finger-like protrusions or
hole division. At higher acceleration, the hole delocalizes, growing to cover
the entire surface with erratic undulations. We find similar behavior in an
aqueous suspension of glass microspheres.


title: Cyclic bases of zero-curvature representations: further examples

abstract:
  The paper continues nlin.SI/0212019 by giving three more examples of using
cyclic bases of zero-curvature representations in studies of relation between
strong Lax pairs and recursion operators.


title:
Symbolic algorithms for the Painleve test, special solutions, and
  recursion operators for nonlinear PDEs

abstract:
  This paper discusses the algorithms and implementations of three Mathematica
packages for the study of integrability and the computation of closed-form
solutions of nonlinear polynomial PDEs.
  The first package, PainleveTest.m, symbolically performs the Painleve
integrability test. The second package, PDESpecialSolutions.m, computes exact
solutions expressible in hyperbolic or elliptic functions. The third package,
PDERecursionOperator.m, generates and tests recursion operators.


title:
Symbolic Computation of Conserved Densities, Generalized Symmetries, and
  Recursion Operators for Nonlinear Differential-Difference Equations

abstract:
  Algorithms for the symbolic computation of conserved densities, fluxes,
generalized symmetries, and recursion operators for systems of nonlinear
differential-difference equations are presented. In the algorithms we use
discrete versions of the Frechet and variational derivatives, as well as
discrete Euler and homotopy operators.
  The algorithms are illustrated for prototypical nonlinear lattices, including
the Kac-van Moerbeke (Volterra) and Toda lattices. Results are shown for the
modified Volterra and Ablowitz-Ladik lattices.


title: Generalized fermionic discrete Toda hierarchy

abstract:
  Bi-Hamiltonian structure and Lax pair formulation with the spectral parameter
of the generalized fermionic Toda lattice hierarchy as well as its bosonic and
fermionic symmetries for different (including periodic) boundary conditions are
described. Its two reductions --N=4 and N=2 supersymmetric Toda lattice
hierarchies-- in different (including canonical) bases are investigated. Its
r-matrix description, monodromy matrix, and spectral curves are discussed.


title:
Extreme waves and modulational instability: wave flume experiments on
  irregular waves

abstract:
  We discuss the formation of large amplitude waves for sea states
characterized by JONSWAP spectra with random phases. In this context we discuss
experimental results performed in one of the largest wave tank facilities in
the world. We present experimental evidence that the tail of the cumulative
probability function of the wave heights for random waves strongly depends on
the ratio between the wave steepness and the spectral bandwidth. When this
ratio, called the Benjamin-Feir Index, is large the Rayleigh distribution
clearly underestimates the occurrence of large amplitude waves. Our
experimental results are also successfully compared with previously performed
numerical simulations of the Dysthe equation.


title: Mathematical Models of Bipolar Disorder

abstract:
  We use limit cycle oscillators to model Bipolar II disorder, which is
characterized by alternating hypomanic and depressive episodes and afflicts
about one percent of the United States adult population. We consider two
nonlinear oscillator models of a single bipolar patient. In both frameworks, we
begin with an untreated individual and examine the mathematical effects and
resulting biological consequences of treatment. We also briefly consider the
dynamics of interacting bipolar II individuals using weakly-coupled,
weakly-damped harmonic oscillators. We discuss how the proposed models can be
used as a framework for refined models that incorporate additional biological
data. We conclude with a discussion of possible generalizations of our work, as
there are several biologically-motivated extensions that can be readily
incorporated into the series of models presented here.


title: The PDEs of biorthogonal polynomials arising in the two-matrix model

abstract:
  The two-matrix model can be solved by introducing bi-orthogonal polynomials.
In the case the potentials in the measure are polynomials, finite sequences of
bi-orthogonal polynomials (called
  "windows") satisfy polynomial ODEs as well as deformation equations (PDEs)
and finite difference equations (Delta-E) which are all Frobenius compatible
and define discrete and continuous isomonodromic deformations for the irregular
ODE, as shown in previous works of ours.
  In the one matrix model an explicit and concise expression for the
coefficients of these systems is known and it allows to relate the partition
function with the isomonodromic tau-function of the overdetermined system.
Here, we provide the generalization of those expressions to the case of
bi-orthogonal polynomials, which enables us to compute the determinant of the
fundamental solution of the overdetermined system of ODE+PDEs+Delta-E.


title: A new two-dimensional lattice model that is "consistent around a cube"

abstract:
  For two-dimensional lattice equations one definition of integrability is that
the model can be naturally and consistently extended to three dimensions, i.e.,
that it is "consistent around a cube" (CAC). As a consequence of CAC one can
construct a Lax pair for the model. Recently Adler, Bobenko and Suris conducted
a search based on this principle and certain additional assumptions. One of
those assumptions was the "tetrahedron property", which is satisfied by most
known equations. We present here one lattice equation that satisfies the
consistency condition but does not have the tetrahedron property. Its Lax pair
is also presented and some basic properties discussed.


title:
Discrete zero curvature representations and infinitely many conservation
  laws for several 2+1 dimensional lattice hierarchies

abstract:
  In this article, several 2+1 dimensional lattice hierarchies proposed by
Blaszak and Szum [J. Math. Phys. {\bf 42}, 225(2001)] are further investigated.
We first describe their discrete zero curvature representations. Then, by means
of solving the corresponding discrete spectral equation, we demonstrate the
existence of infinitely many conservation laws for them and obtain the
corresponding conserved densities and associated fluxes formulaically. Thus,
their integrability is further confirmed.


title:
Phaselocked patterns and amplitude death in a ring of delay coupled
  limit cycle oscillators

abstract:
  We study the existence and stability of phaselocked patterns and amplitude
death states in a closed chain of delay coupled identical limit cycle
oscillators that are near a supercritical Hopf bifurcation. The coupling is
limited to nearest neighbors and is linear. We analyze a model set of discrete
dynamical equations using the method of plane waves. The resultant dispersion
relation, which is valid for any arbitrary number of oscillators, displays
important differences from similar relations obtained from continuum models. We
discuss the general characteristics of the equilibrium states including their
dependencies on various system parameters. We next carry out a detailed linear
stability investigation of these states in order to delineate their actual
existence regions and to determine their parametric dependence on time delay.
Time delay is found to expand the range of possible phaselocked patterns and to
contribute favorably toward their stability. The amplitude death state is
studied in the parameter space of time delay and coupling strength. It is shown
that death island regions can exist for any number of oscillators N in the
presence of finite time delay. A particularly interesting result is that the
size of an island is independent of N when N is even but is a decreasing
function of N when N is odd.


title:
Experimental test of the Gallavotti-Cohen fluctuation theorem in
  turbulent flows

abstract:
  We test the fluctuation theorem from measurements in turbulent flows. We
study the time fluctuations of the force acting on an obstacle, and we consider
two experimental situations: the case of a von K\'arm\'an swirling flow between
counter-rotating disks (VK) and the case of a wind tunnel jet. We first study
the symmetries implied by the Gallavotti-Cohen fluctuation theorem (FT) on the
probability density distributions of the force fluctuations; we then test the
Sinai scaling. We observe that in both experiments the symmetries implied by
the FT are well verified, whereas the Sinai scaling is established, as
expected, only for long times.


title: Chaos-induced breaking of the Franck-Condon approximation

abstract:
  We investigate the vibrational structure of electronic spectra for the
transition from the non-degenerate $A$ state to $E$ states in $E_g\otimes e_g$
Jahn-Teller systems with the trigonal field included. In connection with the
underlying chaotic behavior for vibronic energy levels in this model reported
by the present authors in Phys.Rev.{\bf E68}(2003)046201, we extend the
analysis by Longuet-Higgins et al. to the classically chaotic system. In
particular, the triple-humped structure is manifest with increasing the
anharmonicity. Such structure is completely inconsistent with the shape
obtained from Franck-Condon(FC) approximation, and is caused by the
chaos-induced beaking of the FC principle.


title: Cryptographic requirements for chaotic secure communications

abstract:
  In recent years, a great amount of secure communications systems based on
chaotic synchronization have been published. Most of the proposed schemes fail
to explain a number of features of fundamental importance to all cryptosystems,
such as key definition, characterization, and generation. As a consequence, the
proposed ciphers are difficult to realize in practice with a reasonable degree
of security. Likewise, they are seldom accompanied by a security analysis.
Thus, it is hard for the reader to have a hint about their security. In this
work we provide a set of guidelines that every new cryptosystems would benefit
from adhering to. The proposed guidelines address these two main gaps, i.e.,
correct key management and security analysis, to help new cryptosystems be
presented in a more rigorous cryptographic way. Also some recommendations are
offered regarding some practical aspects of communications, such as channel
noise, limited bandwith, and attenuation.


title:
Breaking a secure communication scheme based on the phase
  synchronization of chaotic systems

abstract:
  A security analysis of a recently proposed secure communication scheme based
on the phase synchronization of chaotic systems is presented. It is shown that
the system parameters directly determine the ciphertext waveform, hence it can
be readily broken by parameter estimation of the ciphertext signal.


title: Breaking parameter modulated chaotic secure communication system

abstract:
  This paper describes the security weakness of a recently proposed secure
communication method based on parameter modulation of a chaotic system and
adaptive observer-based synchronization scheme. We show that the security is
compromised even without precise knowledge of the chaotic system used.


title:
Cryptanalysis of a novel cryptosystem based on chaotic oscillators and
  feedback inversion

abstract:
  An analysis of a recently proposed cryptosystem based on chaotic oscillators
and feedback inversion is presented. It is shown how the cryptosystem can be
broken when Duffing's oscillator is considered. Some implementation problems of
the system are also discussed.


title: Cryptanalysis of dynamic look-up table based chaotic cryptosystems

abstract:
  In recent years many chaotic cryptosystems based on Baptista's seminal work
have been proposed. We analyze the security of two of the newest and most
interesting ones, which use a dynamically updated look-up table and also work
as stream ciphers. We provide different attack techniques to recover the
keystream used by the algorithms. The knowledge of this keystream provides the
attacker with the same information as the key and thus the security is broken.
We also show that the dependence on the plaintext, and not on the key, of the
look-up table updating mechanism facilitates cryptanalysis.


title: High Speed Chaos in Optical Feedback System with Flexible Timescales

abstract:
  We describe a new opto-electronic device with time-delayed feedback that uses
a Mach-Zehnder interferometer as passive nonlinearity and a semiconductor laser
as a current-to-optical-frequency converter. Bandlimited feedback allows tuning
of the characteristic time scales of both the periodic and high dimensional
chaotic oscillations that can be generated with the device. Our implementation
of the device produces oscillations in the frequency range of tens to hundreds
of MHz. We develop a model and use it to explore the experimentally observed
Andronov-Hopf bifurcation of the steady state and to estimate the dimension of
the chaotic attractor.


title: Instabilities induced by a weak breaking of a strong spatial resonance

abstract:
  Through multiple-scales and symmetry arguments we derive a model set of
amplitude equations describing the interaction of two steady-state
pattern-forming instabilities, in the case that the wavelengths of the
instabilities are nearly in the ratio 1:2. In the case of exact 1:2 resonance
the amplitude equations are ODEs; here they are PDEs.
  We discuss the stability of spatially-periodic solutions to long-wavelength
disturbances. By including these modulational effects we are able to explore
the relevance of the exact 1:2 results to spatially-extended physical systems
for parameter values near to this codimension-two bifurcation point. These new
instabilities can be described in terms of reduced `normal form' PDEs near
various secondary codimension-two points. The robust heteroclinic cycle in the
ODEs is destabilised by long-wavelength perturbations and a stable periodic
orbit is generated that lies close to the cycle. An analytic expression giving
the approximate period of this orbit is derived.


title:
Cryptanalyzing a discrete-time chaos synchronization secure
  communication system

abstract:
  This paper describes the security weakness of a recently proposed secure
communication method based on discrete-time chaos synchronization. We show that
the security is compromised even without precise knowledge of the chaotic
system used. We also make many suggestions to improve its security in future
versions.


title: Quaternion Dynamics of the Brain

abstract:
  A nonlinear dynamical modeling of interaction between automatic and conscious
processes in the brain is described. Effects of sensations, emotions and
reflections on the electromagnetic activity of the brain are represented in
terms of quanternion field.


title:
A Mean-Field Theory of Cellular Automata Model for Distributed Packet
  Networks

abstract:
  A Mean-Field theory is presented and applied to a Cellular Automata model of
distributed packet-switched networks. It is proved that, under a certain set of
assumptions, the critical input traffic is inversely proportional to the free
packet delay of the model. The applicability of Mean-Field theory in queue
length estimation is also investigated. Results of theoretical derivations are
compared with simulation samples to demonstrate the availability of the
Mean-Field approach.


title:
Spontaneous Channelization in Permeable Ground: Theory, Experiment, and
  Observation

abstract:
  Landscapes that are rhythmically dissected by natural drainage channels exist
in various geologic and climatic settings. Such landscapes are characterized by
a length-scale for the lateral spacing between channels. We observe a
small-scale version of this process in the form of beach rills and reproduce
channelization in a table-top seepage experiment. On the beach as well as in
the experiments, channels are spontaneously incised by surface flow, but once
initiated, they grow due to water emerging from underground. Field observation
and experiment suggest the process can be described in terms of flow through a
homogeneous porous medium with a freely shaped water table. According to this
theory, small deformations of the underground water table amplify the flux into
the channel and lead to further growth, a phenomenon we call ``Wentworth
instability''. Piracy of groundwater can occur over distances much larger than
the channel width. Channel spacing coarsens with time, until channels reach
their maximum length.


title: Nonlinear Noise Reduction Scheme Based on Information Flow

abstract:
  We present a measurement noise reduction scheme based on information flow of
a chaotic system. This scheme operates on conditions of chaoticity and
well-defined noise level, not depending on other detailed characteristics of
noise. Starting with a simple map and full knowledge of dynamics, we extend the
basic idea to general form applicable to higher dimensional systems. Reducing
noise in Lorenz system is demonstrated as an example. Inferring dynamics
without a priori knowledge is then discussed by proposing an indicator which
measures predictability.


title:
Synchronization of globally coupled nonidentical maps with inhomogeneous
  delayed interactions

abstract:
  We study the synchronization of a coupled map lattice consisting of a
one-dimensional chain of logistic maps. We consider global coupling with a
time-delay that takes into account the finite velocity of propagation of
interactions. We recently showed that clustering occurs for weak coupling,
while for strong coupling the array synchronizes into a global state where each
element sees all other elements in its current, present state [Physica A {\bf
325} (2003) 186, Phys. Rev. E {\bf 67} (2003) 056219]. In this paper we study
the effects of in-homogeneities, both in the individual maps, which are
non-identical maps evolving in period-2 orbits, and in the connection links,
which have non-uniform strengths. We find that the global synchronization
regime occurring for strong coupling is robust to heterogeneities: for strong
enough average coupling the inhomogeneous array still synchronizes in a global
state in which each element sees the other elements in positions close to its
current state. However, the clustering behaviour occurring for small coupling
is sensitive to inhomogeneities and differs from that occurring in the
homogeneous array.


title: Classical invariants and the quantization of chaotic systems

abstract:
  Long periodic orbits constitute a serious drawback in Gutzwiller's theory of
chaotic systems, and then it would be desirable that other classical
invariants, not suffering from the same problem, could be used in the
quantization of such systems. In this respect, we demonstrate how a suitable
dynamical analysis of chaotic quantum spectra unveils the fundamental role
played by classical invariant areas related to the stable and unstable
manifolds of short periodic orbits.


title:
Overcoming Obstacles to Integrability in Perturbed Nonlinear Evolution
  Equations

abstract:
  Obstacles to integrability in perturbed evolution equations are overcome by
allowing higher-order terms in the expansion of the solution to depend
explicitly on time and position. With a special expansion algorithm, obstacles
vanish explicitly for single-waves zero-order approximations, and
exponentially, outside the wave interaction-region, for multiple-waves. Simple
expressions for their asymptotic effects on the solution are obtained.


title: Sustained plankton blooms under open chaotic flows

abstract:
  We consider a predator-prey model of planktonic population dynamics, of
excitable character, living in an open and chaotic fluid flow, i.e., a state of
fluid motion in which fluid trajectories are unbounded but a chaotic region
exists that is restricted to a localized area. Despite that excitability is a
transient phenomenon and that fluid trajectories are continuously leaving the
system, there is a regime of parameters where the excitation remains
permanently in the system, given rise to a persistent plankton bloom. This
regime is reached when the time scales associated to fluid stirring become
slower than the ones associated to biological growth.


title: Induced Minority Dynamics in a Stock Market Model

abstract:
  In this paper, we present a simple stock market model (the market game) which
incorporates, as ab initio dynamics delayed majority dynamics, according to
which agents (with heterogeneous strategies and price expectations) are
rewarded if their actions at time t are the actions of the majority of agents
at time t+1. We observe that for a range of parameter settings, minority
dynamics are dynamically induced in this game, despite the fact that they are
not introduced ab initio. Central to the emergence of minority dynamics is the
introduction of the notion of price expectations for the agents. This leads to
the possibility of an agent not participating in the market for some time
steps. One consequence of the induced minority dynamics is an effective
reduction in market volatility. We also discuss the phase structure and
qualitative behavior of the market game for the entire parameter space.


title: Reactive dynamics of inertial particles in nonhyperbolic chaotic flows

abstract:
  Anomalous kinetics of infective (e.g., autocatalytic) reactions in open,
nonhyperbolic chaotic flows are important for many applications in biological,
chemical, and environmental sciences. We present a scaling theory for the
singular enhancement of the production caused by the universal, underlying
fractal patterns. The key dynamical invariant quantities are the effective
fractal dimension and effective escape rate, which are primarily determined by
the hyperbolic components of the underlying dynamical invariant sets. The
theory is general as it includes all previously studied hyperbolic reactive
dynamics as a special case. We introduce a class of dissipative embedding maps
for numerical verification.


title: Euler-Lagrange correspondence of generalized Burgers cellular automaton

abstract:
  Recently, we have proposed a {\em Euler-Lagrange transformation} for cellular
automata(CA) by developing new transformation formulas. Applying this method to
the Burgers CA(BCA), we have succeeded in obtaining the Lagrange representation
of the BCA. In this paper, we apply this method to multi-value generalized
Burgers CA(GBCA) which include the Fukui-Ishibashi model and the quick-start
model associated with traffic flow. As a result, we have succeeded in
clarifying the Euler-Lagrange correspondence of these models. It turns out,
moreover that the GBCA can naturally be considered as a simple model of a
multi-lane traffic flow.


title: Nonlinear differential equations with exact solutions

abstract:
  New problem is considered that is to find nonlinear differential equations
with special solutions. Method is presented to construct nonlinear ordinary
differential equations with exact solution. Crucial step to the method is the
assumption that nonlinear differential equations have exact solution which is
general solution of the simplest integrable equation. The Riccati equation is
shown to be a building block to find a lot of nonlinear differential equations
with exact solutions. Nonlinear differential equations of the second, third and
fourth order with special solutions are given. Most of these equations are used
at the description of processes in physics and in theory of nonlinear waves.


title: Controlled generation and steering of spatial gap solitons

abstract:
  We demonstrate the first fully controlled generation of immobile and slow
spatial gap solitons in nonlinear periodic systems with band-gap spectra, and
reveal the key features of gap solitons which distinguish them from
conventional counterparts, including a dynamical transformation of gap solitons
due to nonlinear inter-band coupling. We also predict theoretically and confirm
experimentally the effect of anomalous steering of gap solitons in
optically-induced photonic lattices.


title: Statistics of Poincar\' e recurrences for a class of smooth circle maps

abstract:
  Statistics of Poincar\' e recurrence for a class of circle maps, including
sub-critical, critical, and super-critical cases, are studied. It is shown how
the topological differences in the various types of the dynamics are manifested
in the statistics of the return times.


title: Landau-Lifshitz hierarchy and infinite dimensional Grassmann variety

abstract:
  The Landau-Lifshitz equation is an example of soliton equations with a
zero-curvature representation defined on an elliptic curve. This equation can
be embedded into an integrable hierarchy of evolution equations called the
Landau-Lifshitz hierarchy. This paper elucidates its status in Sato, Segal and
Wilson's universal description of soliton equations in the language of an
infinite dimensional Grassmann variety. To this end, a Grassmann variety is
constructed from a vector space of $2 \times 2$ matrices of Laurent series of
the spectral parameter $z$. A special base point $W_0$, called ``vacuum,'' of
this Grassmann variety is chosen. This vacuum is ``dressed'' by a Laurent
series $\phi(z)$ to become a point of the Grassmann variety that corresponds to
a general solution of the Landau-Lifshitz hierarchy. The Landau-Lifshitz
hierarchy is thereby mapped to a simple dynamical system on the set of these
dressed vacua. A higher dimensional analogue of this hierarchy (an elliptic
analogue of the Bogomolny hierarchy) is also presented.


title:
Towards a Noether-like conservation law theorem for one dimensional
  reversible cellular automata

abstract:
  Evidence and results suggesting that a Noether--like theorem for conservation
laws in 1D RCA can be obtained. Unlike Noether's theorem, the connection here
is to the maximal congruences rather than the automorphisms of the local
dynamics.
  We take the results of Takesue and Hattori (1992) on the space of additive
conservation laws in one dimensional cellular automata. In reversible automata,
we show that conservation laws correspond to the null spaces of certain
well-structured matrices.
  It is shown that a class of conservation laws exist that correspond to the
maximal congruences of index 2. In all examples investigated, this is all the
conservation laws. Thus we conjecture that there is an equality here,
corresponding to a Noether--like theorem.


title:
Symbolic computation of hyperbolic tangent solutions for nonlinear
  differential-difference equations

abstract:
  A new algorithm is presented to find exact traveling wave solutions of
differential-difference equations in terms of tanh functions. For systems with
parameters, the algorithm determines the conditions on the parameters so that
the equations might admit polynomial solutions in tanh.
  Examples illustrate the key steps of the algorithm. Parallels are drawn
through discussion and example to the tanh-method for partial differential
equations.
  The new algorithm is implemented in Mathematica. The package
DDESpecialSolutions.m can be used to automatically compute traveling wave
solutions of nonlinear polynomial differential-difference equations. Use of the
package, implementation issues, scope, and limitations of the software are
addressed.


title: Geometric discretization of the Bianchi system

abstract:
  We introduce the dual Koenigs lattices, which are the integrable discrete
analogues of conjugate nets with equal tangential invariants, and we find the
corresponding reduction of the fundamental transformation. We also introduce
the notion of discrete normal congruences. Finally, considering quadrilateral
lattices "with equal tangential invariants" which allow for harmonic normal
congruences we obtain, in complete analogy with the continuous case, the
integrable discrete analogue of the Bianchi system together with its geometric
meaning. To obtain this geometric meaning we also make use of the novel
characterization of the circular lattice as a quadrilateral lattice whose
coordinate lines intersect orthogonally in the mean.


title: Anisotropy in Homogeneous Rotating Turbulence

abstract:
  The effective stress tensor of a homogeneous turbulent rotating fluid is
anisotropic. This leads us to consider the most general axisymmetric four-rank
``viscosity tensor'' for a Newtonian fluid and the new terms in the turbulent
effective force on large scales that arise from it, in addition to the
microscopic viscous force. Some of these terms involve couplings to vorticity
and others are angular momentum non conserving (in the rotating frame).
Furthermore, we explore the constraints on the response function and the
two-point velocity correlation due to axisymmetry. Finally, we compare our
viscosity tensor with other four-rank tensors defined in current approaches to
non-rotating anisotropic turbulence.

categories: nlin.CD astro-ph math-ph math.MP physics.flu-dyn


title:
Self-similar solutions and collective coordinate methods for Nonlinear
  Schrodinger Equations

abstract:
  In this paper we study the phase of self-similar solutions to general
Nonlinear Schr\"odinger equations. From this analysis we gain insight on the
dynamics of nontrivial solutions and a deeper understanding of the way
collective coordinate methods work. We also find general evolution equations
for the most relevant dynamical parameter $w(t)$ corresponding to the width of
the solution. These equations are exact for self-similar solutions and provide
a shortcut to find approximate evolution equations for the width of
non-self-similar solutions similar to those of collective coordinate methods.


title: Multiple filamentation induced by input-beam ellipticity

abstract:
  The standard explanation for multiple filamentation (MF) of intense laser
beams has been that it is initiated by input beam noise (modulational
instability). In this study we provide the first experimental evidence that MF
can also be induced by input beam ellipticity. Unlike noise-induced beam
breakup, the MF pattern induced by ellipticity is reproducible shot to shot.
Moreover, our experiments show that ellipticity can dominate the effect of
noise, thus providing the first experimental methodology for controlling the MF
pattern of noisy beams. The results are explained using a theoretical model and
simulations.


title:
The Zakharov-Kuznetsov Equation as a Two-Dimensional Model for Nonlinear
  Rossby Waves

abstract:
  We study the dynamics of two-dimensional coherent structures in planetary
atmospheres and oceans. We derive the Zakharov-Kuznetsov equation for large
scale motion from the barotropic quasigeostrophic equation in a weakly
nonlinear, long wave approximation. We consider coherent structures emerging
out of an instability caused by a narrow jet-like meanflow. We use multiple
scale analysis combined with asymptotic matching.


title: Pattern forming pulled fronts: bounds and universal convergence

abstract:
  We analyze the dynamics of pattern forming fronts which propagate into an
unstable state, and whose dynamics is of the pulled type, so that their
asymptotic speed is equal to the linear spreading speed v^*. We discuss a
method that allows to derive bounds on the front velocity, and which hence can
be used to prove for, among others, the Swift-Hohenberg equation, the Extended
Fisher-Kolmogorov equation and the cubic Complex Ginzburg-Landau equation, that
the dynamically relevant fronts are of the pulled type. In addition, we
generalize the derivation of the universal power law convergence of the
dynamics of uniformly translating pulled fronts to both coherent and incoherent
pattern forming fronts. The analysis is based on a matching analysis of the
dynamics in the leading edge of the front, to the behavior imposed by the
nonlinear region behind it. Numerical simulations of fronts in the
Swift-Hohenberg equation are in full accord with our analytical predictions.


title: Optimal time delay embedding for nonlinear time series modeling

abstract:
  When building linear or nonlinear models one is faced with the problem of
selecting the best set of variable with which to predict the future dynamics.
In nonlinear time series analysis the problem is to select the correct time
delays in the time delay embedding. We propose a new technique which can
quantify the suitability of a particular set of variables and we suggests a
computationally efficient scheme to determine the best non-uniform time delay
embedding for modeling of time series. Our results are based on the assumption
that, in general, the variables which give the best local constant model will
also give the best nonlinear model. In a wide variety of experimental and
simulated systems we find that this method produces dynamics that are more
realistic and predictions that are more accurate than standard uniform
embeddings.


title: Rotating Concentric Circular Peakons

abstract:
  We study invariant manifolds of measure-valued solutions of the partial
differential equation for geodesic flow of a pressureless fluid. These
solutions describe interaction dynamics on lower-dimensional support sets; for
example, curves, or filaments, of momentum in the plane. The 2+1 solutions we
study are planar generalizations of the 1+1 peakon solutions of Camassa & Holm
[1993] for shallow water solitons. As an example, we study the canonical
Hamiltonian interaction dynamics of $N$ rotating concentric circles of peakons,
whose solution manifold is $2N-$dimensional. Thus, the problem is reduced from
infinite dimensions to a finite-dimensional, canonical, invariant manifold. The
existence of this reduced solution manifold and many of its properties may be
understood, by noticing that it is also the momentum map for the action of
diffeomorphisms on the space of curves in the plane. We show both analytical
and numerical results.


title:
Symmetry constraints for dispersionless integrable equations and systems
  of hydrodynamic type

abstract:
  Symmetry constraints for (2+1)-dimensional dispersionless integrable
equations are considered. It is demonstrated that they naturally lead to
systems of hydrodynamic type which arise within the reduction method. One also
easily obtaines an associated complex curve (Sato function) and corresponding
generating equations. Dispersionless KP and 2DTL hierarchy are considered as
illustrative examples.


title: Deterministic Chaos in Quantum Field Theory

abstract:
  We discuss the necessity and demonstrate the validity of introduction the
notion of deterministic chaos in quantum field theory. Brief review of the
existing approaches to this problem is given. We compare proposed chaos
criterion for quantum fields with existing ones. Its consequences in particle
physics are also discussed.


title:
Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the
  test for integrability

abstract:
  A (d+1)-dimensional dispersionless PDE is said to be integrable if its
n-component hydrodynamic reductions are locally parametrized by (d-1)n
arbitrary functions of one variable. Given a PDE which does not pass the
integrability test, the method of hydrodynamic reductions allows one to
effectively reconstruct additional differential constraints which, when added
to the equation, make it an integrable system in fewer dimensions (if
consistent).


title: Elliptic spectral parameter and infinite dimensional Grassmann variety

abstract:
  Recent results on the Grassmannian perspective of soliton equations with an
elliptic spectral parameter are presented along with a detailed review of the
classical case with a rational spectral parameter. The nonlinear Schr\"odinger
hierarchy is picked out for illustration of the classical case. This system is
formulated as a dynamical system on a Lie group of Laurent series with
factorization structure. The factorization structure induces a mapping to an
infinite dimensional Grassmann variety. The dynamical system on the Lie group
is thereby mapped to a simple dynamical system on a subset of the Grassmann
variety. Upon suitable modification, almost the same procedure turns out to
work for soliton equations with an elliptic spectral parameters. A clue is the
geometry of holomorphic vector bundles over the elliptic curve hidden (or
manifest) in the zero-curvature representation.


title: Spectral properties and pattern selection in fractal growth networks

abstract:
  A model for the generation of fractal growth networks in Euclidean spaces of
arbitrary dimension is presented. These networks are considered as the spatial
support of reaction-diffusion and pattern formation processes. The local
dynamics at the nodes of a fractal growth network is given by a nonlinear map,
giving raise to a coupled map system. The coupling is described by a matrix
whose eigenvectors constitute a basis on which spatial patterns on fractal
growth networks can be expressed by linear combination. The spectrum of
eigenvalues the coupling matrix exhibits a nonuniform distribution that is
reflected in the presence of gaps or niches in the boundaries of stability of
the synchronized states on the space of parameters of the system. These gaps
allow for the selection of specific spatial patterns by appropriately varying
the parameters of the system.


title: Loschmidt echo for a chaotic oscillator

abstract:
  Chaotic dynamics of a nonlinear oscillator is considered in the semiclassical
approximation. The Loschmidt echo is calculated for a time scale which is of
the power law in semiclassical parameter. It is shown that an exponential decay
of the Loschmidt echo is due to a Lyapunov exponent and it has a pure classical
nature.


title: Semiclassical approximation for a nonlinear oscillator with dissipation

abstract:
  An $S$--matrix approach is developed for the chaotic dynamics of a nonlinear
oscillator with dissipation. The quantum--classical crossover is studied in the
framework of the semiclassical expansion for the $S$--matrix. Analytical
expressions for the braking time and the $S$--matrix are obtained.


title: Numerical studies of stabilized Townes solitons

abstract:
  We study numerically stabilized solutions of the two-dimensional Schrodinger
equation with a cubic nonlinearity. We discuss in detail the numerical scheme
used and explain why choosing the right numerical strategy is very important to
avoid misleading results. We show that stabilized solutions are Townes
solitons, a fact which had only been conjectured previously. Also we make a
systematic study of the parameter regions in which these structures exist.


title: On Quantum Iterated Function Systems

abstract:
  Quantum Iterated Function System on a complex projective space is defined by
a family of linear operators on a complex Hilbert space. The operators define
both the maps and their probabilities by one algebraic formula. Examples with
conformal maps (relativistic boosts) on the Bloch sphere are discussed.


title:
Periodic orbit theory for the H\'enon-Heiles system in the continuum
  region

abstract:
  We investigate the resonance spectrum of the H\'enon-Heiles potential up to
twice the barrier energy. The quantum spectrum is obtained by the method of
complex coordinate rotation. We use periodic orbit theory to approximate the
oscillating part of the resonance spectrum semiclassically and Strutinsky
smoothing to obtain its smooth part. Although the system in this energy range
is almost chaotic, it still contains stable periodic orbits. Using Gutzwiller's
trace formula, complemented by a uniform approximation for a codimension-two
bifurcation scenario, we are able to reproduce the coarse-grained
quantum-mechanical density of states very accurately, including only a few
stable and unstable orbits.


title:
Geometric measures of redundance and irrelevance tradeoff exponent to
  choose suitable delay times for continuous systems

abstract:
  Using the concept of the geometric measures of redundance and irrelevance
tradeoff exponent (RITE)}, we present a new method to determine suitable delay
times for continuous systems. After applying the RITE algorithm to both
simulation and experimental observations, we find the results obtained are
close to those obtained from the criterion of the average mutual information
(AMI), while the RITE algorithm has the following advantages: simple
implementation, reasonable computational cost and robust performance against
observational noise.


title: Periodic orbit theory of two coupled Tchebyscheff maps

abstract:
  Coupled map lattices have been widely used as models in several fields of
physics, such as chaotic strings, turbulence, and phase transitions, as well as
in other disciplines, such as biology (ecology, evolution) and information
processing. This paper investigates properties of periodic orbits in two
coupled Tchebyscheff maps. The zeta function cycle expansions are used to
compute dynamical averages appearing in Beck's theory of chaotic strings. The
results show close agreement with direct simulation for most values of the
coupling parameter, and yield information about the system complementary to
that of direct simulation.


title:
Separable bi-Hamiltonian systems with quadratic in momenta first
  integrals

abstract:
  Geometric separability theory of Gel'fand-Zakharevich bi-Hamiltonian systems
on Riemannian manifolds is reviewed and developed. Particular attention is paid
to the separability of systems generated by the so-called special conformal
Killing tensors, i.e. Benenti systems. Then, infinitely many new classes of
separable systems are constructed by appropriate deformations of Benenti class
systems. All such systems can be lifted to the Gel'fand-Zakharevich
bi-Hamiltonian form.


title:
On the emergence of complex systems on the basis of the coordination of
  complex behaviors of their elements

abstract:
  We argue that the coordination of the activities of individual complex agents
enables a system to develop and sustain complexity at a higher level. We
exemplify relevant mechanisms through computer simulations of a toy system, a
coupled map lattice with transmission delays. The coordination here is achieved
through the synchronization of the chaotic operations of the individual
elements, and on the basis of this, regular behavior at a longer temporal scale
emerges that is inaccessible to the uncoupled individual dynamics.


title:
Hierarchical Structure in Healthy and Diseased Heart Rate Variability in
  Humans

abstract:
  It is shown that the heart rate variability (HRV) in healthy and diseased
humans possesses a hierarchical structure of the She-Leveque (SL) form. This
structure, first found in measurements in turbulent fluid flows, implies
further details in the HRV multifractal scaling. The potential of diagnosis is
also discussed based on the characteristics derived from the SL hierarchy.


title: Extraction of Plumes in Turbulent Thermal Convection

abstract:
  We present a scheme to extract information about plumes, a prominent coherent
structure in turbulent thermal convection, from simultaneous local velocity and
temperature measurements. Using this scheme, we study the temperature
dependence of the plume velocity and understand the results using the equations
of motion. We further obtain the average local heat flux in the vertical
direction at the cell center. Our result shows that heat is not mainly
transported through the central region but instead through the regions near the
sidewalls of the convection cell.


title: Transverse Patterns in Nonlinear Optical Resonators

abstract:
  The book is devoted to the formation and dynamics of localized structures
(vortices, solitons) and extended patterns (stripes, hexagons, tilted waves) in
nonlinear optical resonators such as lasers, optical parametric oscillators,
and photorefractive oscillators. The theoretical analysis is performed by
deriving order parameter equations, and also through numerical integration of
microscopic models of the systems under investigation. Experimental
observations, and possible technological implementations of transverse optical
patterns are also discussed. A comparison with patterns found in other
nonlinear systems, i.e. chemical, biological, and hydrodynamical systems, is
given. This article contains the table of contents and the introductory chapter
of the book.


title:
Conformal geometry of the (discrete) Schwarzian Davey-Stewartson II
  hierarchy

abstract:
  The conformal geometry of the Schwarzian Davey-Stewartson II hierarchy and
its discrete analogue is investigated. Connections with discrete and continuous
isothermic surfaces and generalised Clifford configurations are recorded. An
interpretation of the Schwarzian Davey-Stewartson II flows as integrable
deformations of conformally immersed surfaces is given.


title: Regular quantum graphs

abstract:
  We introduce the concept of regular quantum graphs and construct connected
quantum graphs with discrete symmetries. The method is based on a decomposition
of the quantum propagator in terms of permutation matrices which control the
way incoming and outgoing channels at vertex scattering processes are
connected. Symmetry properties of the quantum graph as well as its spectral
statistics depend on the particular choice of permutation matrices, also called
connectivity matrices, and can now be easily controlled. The method may find
applications in the study of quantum random walks networks and may also prove
to be useful in analysing universality in spectral statistics.


title: The normal dual congruences and the dual Bianchi lattice

abstract:
  The main goal of this paper is to find the discrete analogue of the Bianchi
system in spaces of arbitrary dimesion together with its geometric
interpretation. We show that the proper geometric framework of such
generalization is the language of dual quadrilateral lattices and of dual
congruences. After introducing the notion of the dual Koenigs lattice in a
projective space of arbitrary dimension we define the discrete dual congruences
and we present, as an important example, the normal dual discrete congruences.
Finally, we introduce the dual Bianchi lattice as a dual Koenigs lattice
allowing for a conjugate normal dual congruence, and we find its
characterization in terms of a system of integrable difference equations.


title: Entropy production away from the equilibrium

abstract:
  For a system moving away from equilibrium, we express the entropy production
via a two-point correlation function for any time and any distance from
equilibrium. The long-time limit gives the sum of the Lyapunov exponents for a
general dynamical system expressed via the formula of a Green-Kubo type.


title:
Cascade time-scales for energy and helicity in homogeneous isotropic
  turbulence

abstract:
  We extend the Kolmogorov phenomenology for the scaling of energy spectra in
high-Reynolds number turbulence, to explicitly include the effect of helicity.
There exists a time-scale $\tau_H$ for helicity transfer in homogeneous,
isotropic turbulence with helicity. We arrive at this timescale using the
phenomenological arguments used by Kraichnan to derive the timescale $\tau_E$
for energy transfer (J. Fluid Mech. {\bf 47}, 525--535 (1971)). We show that in
general $\tau_H$ may not be neglected compared to $\tau_E$, even for rather low
relative helicity. We then deduce an inertial range joint cascade of energy and
helicity in which the dynamics are dominated by $\tau_E$ in the low wavenumbers
with both energy and helicity spectra scaling as $k^{-5/3}$; and by $\tau_H$ at
larger wavenumbers with spectra scaling as $k^{-4/3}$. We demonstrate how,
within this phenomenology, the commonly observed ``bottleneck'' in the energy
spectrum might be explained. We derive a wavenumber $k_h$ which is less than
the Kolmogorov dissipation wavenumber, at which both energy and helicity
cascades terminate due to dissipation effects. Data from direct numerical
simulations are used to check our predictions.


title:
Nonlinear differential equations with exact solutions expressed via the
  Weierstrass function

abstract:
  New problem is studied that is to find nonlinear differential equations with
special solutions expressed via the Weierstrass function. Method is discussed
to construct nonlinear ordinary differential equations with exact solutions.
Main step of our method is the assumption that nonlinear differential equations
have exact solutions which are general solution of the simplest integrable
equation. We use the Weierstrass elliptic equation as a building block to find
a number of nonlinear differential equations with exact solutions. Nonlinear
differential equations of the second, third and fourth order with special
solutions expressed via the Weierstrass function are given. Most of these
equations are used at the description of nonlinear waves in physics.


title:
The Nonlinear Dynamic Conversion of Analog Signals into Excitation
  Patterns

abstract:
  Local periodic perturbations induce frequency-dependent propagation waves in
an excitable spatio-temporally chaotic system. We show how segments of
noise-contaminated and chaotic perturbations induce characteristic sequences of
excitations in the model system. Using a set of tuned excitable systems, it is
possible to characterize signals by their spectral composition of excitation
pattern. As an example we analyze an epileptic spike-and-wave time series.


title:
Control of Hamiltonian chaos as a possible tool to control anomalous
  transport in fusion plasmas

abstract:
  It is shown that a relevant control of Hamiltonian chaos is possible through
suitable small perturbations whose form can be explicitly computed. In
particular, it is possible to control (reduce) the chaotic diffusion in the
phase space of a Hamiltonian system with 1.5 degrees of freedom which models
the diffusion of charged test particles in a turbulent electric field across
the confining magnetic field in controlled thermonuclear fusion devices. Though
still far from practical applications, this result suggests that some strategy
to control turbulent transport in magnetized plasmas, in particular tokamaks,
is conceivable. The robustness of the control is investigated in terms of a
departure from the optimum magnitude, of a varying cut-off at large wave
vectors, and of random errors on the phases of the modes. In all three cases,
there is a significant region of maximum efficiency in the vicinity of the
optimum control term.


title:
Gap and out-gap breathers in a binary modulated discrete nonlinear
  Schr\"odinger model

abstract:
  We consider a modulated discrete nonlinear Schr\"odinger (DNLS) model with
alternating on-site potential, having a linear spectrum with two branches
separated by a 'forbidden' gap. Nonlinear localized time-periodic solutions
with frequencies in the gap and near the gap -- discrete gap and out-gap
breathers (DGBs and DOGBs) -- are investigated. Their linear stability is
studied varying the system parameters from the continuous to the
anti-continuous limit, and different types of oscillatory and real
instabilities are revealed. It is shown, that generally DGBs in infinite
modulated DNLS chains with hard (soft) nonlinearity do not possess any
oscillatory instabilities for breather frequencies in the lower (upper) half of
the gap. Regimes of 'exchange of stability' between symmetric and antisymmetric
DGBs are observed, where an increased breather mobility is expected. The
transformation from DGBs to DOGBs when the breather frequency enters the linear
spectrum is studied, and the general bifurcation picture for DOGBs with tails
of different wave numbers is described. Close to the anti-continuous limit, the
localized linear eigenmodes and their corresponding eigenfrequencies are
calculated analytically for several gap/out-gap breather configurations,
yielding explicit proof of their linear stability or instability close to this
limit.


title:
Frustrated quantum-spin system on a triangle coupled with $e_g$ lattice
  vibrations - Correspondence to Longuet-Higgins et al.'s Jahn-Teller model -

abstract:
  We investigate the quantum three spin model $({\bf S_1},{\bf S_2},{\bf S_3})$
of spin$=1/2$ on a triangle, in which spins are coupled with
lattice-vibrational modes through the exchange interaction depending on
distances between spin sites. The present model corresponds to the dynamic
Jahn-Teller system $E_g\otimes e_g$ proposed by Longuet-Higgins {\it et al.},
Proc.R.Soc.A.{\bf 244},1(1958). This correspondence is revealed by using the
transformation to Nakamura-Bishop's bases proposed in Phys.Rev.Lett.{\bf
54},861(1985). Furthermore, we elucidate the relationship between the behavior
of a chiral order parameter ${\hat \chi}={\bf S_1\cdot(S_2\times S_3)}$ and
that of the electronic orbital angular momentum ${\hat \ell_z}$ in $E_g\otimes
e_g$ vibronic model: The regular oscillatory behavior of the expectation value
$<{\hat \ell_z}>$ for vibronic structures with increasing energy can also be
found in that of $<{\hat \chi}>$. The increase of the additional
anharmonicity(chaoticity) is found to yield a rapidly decaying irregular
oscillation of $<{\hat \chi}>$.


title:
Speculative bubbles and fat tail phenomena in a heterogeneous agent
  model

abstract:
  The aim of this paper is to propose a heterogeneous agent model of stock
markets that develop complicated endogenous price fluctuations. We find
occurrences of non-stationary chaos, or speculative bubble, are caused by the
heterogeneity of traders' strategies. Furthermore, we show that the
distributions of returns generated from the heterogeneous agent model have fat
tails, a remarkable stylized fact observed in almost all financial markets.


title:
Scaling anomalies in the coarsening dynamics of fractal viscous
  fingering patterns

abstract:
  We analyze a recent experiment of Sharon \textit{et al.} (2003) on the
coarsening, due to surface tension, of fractal viscous fingering patterns
(FVFPs) grown in a radial Hele-Shaw cell. We argue that an unforced Hele-Shaw
model, a natural model for that experiment, belongs to the same universality
class as model B of phase ordering. Two series of numerical simulations with
model B are performed, with the FVFPs grown in the experiment, and with
Diffusion Limited Aggregates, as the initial conditions. We observed
Lifshitz-Slyozov scaling $t^{1/3}$ at intermediate distances and very slow
convergence to this scaling at small distances. Dynamic scale invariance breaks
down at large distances.


title: Existence of travelling waves in discrete sine-Gordon rings

abstract:
  We prove existence results for travelling waves in discrete, damped,
dc-driven sine-Gordon equations with periodic boundary conditions.


title: Hydrodynamic reductions and solutions of a universal hierarchy

abstract:
  The diagonal hydrodynamic reductions of a hierarchy of integrable
hydrodynamic chains are explicitly characterized. Their compatibility with
previously introduced reductions of differential type is analyzed and their
associated class of hodograph solutions is discussed.


title: Fractional Generalization of Liouville Equations

abstract:
  In this paper fractional generalization of Liouville equation is considered.
We derive fractional analog of normalization condition for distribution
function. Fractional generalization of the Liouvile equation for dissipative
and Hamiltonian systems was derived from the fractional normalization
condition. This condition is considered considered as a normalization condition
for systems in fractional phase space. The interpretation of the fractional
space is discussed.

categories: nlin.CD cond-mat.stat-mech hep-th math-ph math.MP physics.class-ph quant-ph


title: Diffractive orbits in isospectral billiards

abstract:
  Isospectral domains are non-isometric regions of space for which the spectra
of the Laplace-Beltrami operator coincide. In the two-dimensional Euclidean
space, instances of such domains have been given. It has been proved for these
examples that the length spectrum, that is the set of the lengths of all
periodic trajectories, coincides as well. However there is no one-to-one
correspondence between the diffractive trajectories. It will be shown here how
the diffractive contributions to the Green functions match nevertheless in a
''one-to-three'' correspondence.