title:
Current Statistics for Quantum Transport through Two-Dimensional Open
  Chaotic Billiards

abstract:
  The probability current statistics of two-dimensional open chaotic ballistic
billiards is studied both analytically and numerically. Assuming that the real
and imaginary parts of the scattering wave function are both random Gaussian
fields, we find a universal distribution function for the probability current.
In by-passing we recover previous analytic forms for wave function statistics.
The expressions bridge the entire region from GOE to GUE type statistics. Our
analytic expressions are verified numerically by explicit quantum-mechanical
calculations of transport through a Bunimovich billiard.


title: Anticipating the dynamics of chaotic maps

abstract:
  We study the regime of anticipated synchronization in unidirectionally
coupled chaotic maps such that the slave map has its own output reinjected
after a certain delay. For a class of simple maps, we give analytic conditions
for the stability of the synchronized solution, and present results of
numerical simulations of coupled 1D Bernoulli-like maps and 2D Baker maps, that
agree well with the analytic predictions.


title:
A note on the modified KP hierarchy and its (yet another) dispersionless
  limit

abstract:
  The modified KP hierarchies of Kashiwara and Miwa is formulated in Lax
formalism by Dickey. Their solutions are parametrised by flag varieties. Its
dispersionless limit is considered.


title: Integrable Systems and Metrics of Constant Curvature

abstract:
  In this article we present a Lagrangian representation for evolutionary
systems with a Hamiltonian structure determined by a differential-geometric
Poisson bracket of the first order associated with metrics of constant
curvature. Kaup-Boussinesq system has three local Hamiltonian structures and
one nonlocal Hamiltonian structure associated with metric of constant
curvature. Darboux theorem (reducing Hamiltonian structures to canonical form
''d/dx'' by differential substitutions and reciprocal transformations) for
these Hamiltonian structures is proved.


title: Stability of quantum motion and correlation decay

abstract:
  We derive a simple and general relation between the fidelity of quantum
motion, characterizing the stability of quantum dynamics with respect to
arbitrary static perturbation of the unitary evolution propagator, and the
integrated time auto-correlation function of the generator of perturbation.
Surprisingly, this relation predicts the slower decay of fidelity the faster
decay of correlations is. In particular, for non-ergodic and non-mixing
dynamics, where asymptotic decay of correlations is absent, a qualitatively
different and faster decay of fidelity is predicted on a time scale 1/delta as
opposed to mixing dynamics where the fidelity is found to decay exponentially
on a time-scale 1/delta^2, where delta is a strength of perturbation. A
detailed discussion of a semi-classical regime of small effective values of
Planck constant is given where classical correlation functions can be used to
predict quantum fidelity decay. Note that the correct and intuitively expected
classical stability behavior is recovered in the classical limit hbar->0, as
the two limits delta->0 and hbar->0 do not commute. In addition we also discuss
non-trivial dependence on the number of degrees of freedom. All the theoretical
results are clearly demonstrated numerically on a celebrated example of a
quantized kicked top.


title: On the compatible weakly-nonlocal Poisson brackets of Hydrodynamic Type

abstract:
  We consider the pairs of general weakly non-local Poisson brackets of
Hydrodynamic Type (Ferapontov brackets) and the corresponding integrable
hierarchies. We show that under the requirement of non-degeneracy of the
corresponding "first" pseudo-Riemannian metric $g_{(0)}^{\nu\mu}$ and also some
non-degeneracy requirement for the nonlocal part it is possible to introduce a
"canonical" set of "integrable hierarchies" based on the Casimirs, Momentum
functional and some "Canonical Hamiltonian functions". We prove also that all
the "Higher" "positive" Hamiltonian operators and the "negative" symplectic
forms have the weakly non-local form in this case. The same result is also true
for "negative" Hamiltonian operators and "positive" Symplectic structures in
the case when both pseudo-Riemannian metrics $g_{(0)}^{\nu\mu}$ and
$g_{(1)}^{\nu\mu}$ are non-degenerate.


title: Inverse Anticipating Synchronization

abstract:
  We report a new type of chaos synchronization:inverse anticipating
synchronization, where a time delay chaotic system can drive another system in
such a way that the driven system anticipates the driver by synchronizing with
its inverse future state. We extend the concept of inverse anticipating chaos
synchronization to cascaded systems. We propose means for the experimental
observation of inverse anticipating chaos synchronization in external cavity
lasers.


title: Engineering of spatial solitons in two-period QPM structures

abstract:
  We report on a scheme which might make it practically possible to engineer
the effective competing nonlinearities that on average govern the light
propagation in quasi-phase-matching (QPM) gratings. Modulation of the QPM
period with a second longer period, introduces an extra degree of freedom,
which can be used to engineer the effective quadratic and induced cubic
nonlinearity. However, in contrast to former work here we use a simple
phase-reversal grating for the modulation, which is practically realizable and
has already been fabricated. Furthermore, we develop the theory for arbitrary
relative lengths of the two periods and we consider the effect on solitons and
the bandwidth for their generation. We derive an expression for the bandwidth
of multicolor soliton generation in two-period QPM samples and we predict and
confirm numerically that the bandwidth is broader in the two-period QPM sample
than in homogeneous structures.


title: A model for predation pressure in colonial birds

abstract:
  Different explanations have been proposed for the existence of colonial
breeding behavior in birds, but field studies offer no conclusive results. We
analyze the interplay between learning habits and predation pressure by means
of numerical simulations. Our analysis suggests that extremely simple learning
processes and equally simplistic models of predation pressure lead to the
formation of stable colonies.


title:
Spectral curve and Hamiltonian structure of isomonodromic SU(2)
  Calogero-Gaudin system

abstract:
  This paper presents a new approach to the Hamiltonian structure of
isomonodromic deformations of a matrix system of ODE's on a torus. An
isomonodromic analogue of the $\rmSU(2)$ Calogero-Gaudin system is used for a
case study of this approach. A clue of this approach is a mapping to a finite
number of points on the spectral curve of the isomonodromic Lax equation. The
coordinates of these moving points give a new set of Darboux coordinates called
the spectral Darboux coordinates. The system of isomonodromic deformations is
thereby converted to a non-autonomous Hamiltonian system in the spectral
Darboux coordinates. The Hamiltonians turn out to resemble those of a
previously known isomonodromic system of a second order scalar ODE. The two
isomonodromic systems are shown to be linked by a simple relation.


title: Dual and Dual-Cross Synchronizations in Chaotic Systems

abstract:
  In this paper we investigate dual synchronization and dual-cross
synchronization between the transmitter and receiver, each one consisting of
two master and slave systems. These findings are considered to be of interest
in multichannel communication schemes.


title: Darboux transformation for the modified Veselov-Novikov equation

abstract:
  A Darboux transformation is constructed for the modified Veselov-Novikov
equation.


title: Baroclinic flow and the Lorenz-84 model

abstract:
  The bifurcation diagram of a truncation to six degrees of freedom of the
equations for quasi-geostrophic, baroclinic flow is investigated. Period
doubling cascades and Shil'nikov bifurcations lead to chaos in this model. The
low dimension of the chaotic attractor suggests the possibility to reduce the
model to three degrees of freedom. In a physically comprehensible limit of the
parameters this reduction is done explicitly. The bifurcation diagram of the
reduced model in this limit is compared to the diagram of the six degrees of
freedom model and agrees well. A numerical implementation of the graph
transform is used to approximate the three dimensional invariant manifold away
from the limit case. If the six dimensional model is reduced to a linearisation
of the invariant manifold about the Hadley state, the Lorenz-84 model is found.
Its parameters can then be calculated from the physical parameters of the
quasi-geostrophic model. Bifurcation diagrams at physical and traditional
parameter values are compared and routes to chaos in the Lorenz-84 model are
described.


title: Evolutionary reconstruction of networks

abstract:
  Can a graph specifying the pattern of connections of a dynamical network be
reconstructed from statistical properties of a signal generated by such a
system? In this model study, we present an evolutionary algorithm for
reconstruction of graphs from their Laplacian spectra. Through a stochastic
process of mutations and selection, evolving test networks converge to a
reference graph. Applying the method to several examples of random graphs,
clustered graphs, and small-world networks, we show that the proposed
stochastic evolution allows exact reconstruction of relatively small networks
and yields good approximations in the case of large sizes.


title:
Universality in the periodicity manifestations in the non-locally
  coupled map lattices in the turbulent regime

abstract:
  In the turbulent regime of coupled map lattice with non-local interaction the
maps systematically form periodic cluster attractors and their remnants by
synchronization due to the foliation of periodic windows of the element map. We
examine these periodicity manifestations in three types of coupled map lattices
in $D=1, 2, 3$. In the first two, the interaction is all-to-all but the
coupling decreases with distance in a power and an exponential law. In the
third, the interaction is uniform but cut off sharply. We find that in all
three models and in all dimensions periodicity manifests universally from
turbulence when the same suppression of the local mean field fluctuation is
achieved by the non-local averaging.


title:
Analytical Investigation of Anticipating Synchronization in time-delayed
  and Cascaded Systems

abstract:
  For the first time, using a modified Ikeda model it is demonstrated
analytically that anticipating synchronization can be obtained in chaotic
time-delay systems governed by two characteristic delay times. We derive
existence and stability conditions for the dual-time anticipating
synchronization manifold. We also show that increased anticipation times for
chaotic time-delay systems with two characteristic delay times can be obtained
by the use of cascaded systems.


title:
Pattern Formation on the Edge of Chaos: CO Oxidation on Pt(110) under
  Global Delayed Feedback

abstract:
  Experiments with catalytic CO oxidation on Pt(110) show that chemical
turbulence in this system can be suppressed by application of appropriate
global delayed feedbacks. Different spatiotemporal patterns, seen near a
transition from turbulence to uniform oscillations, are investigated. Using a
method based on the Hilbert transform, spatial distributions of local phase and
amplitude in such patterns are reconstructed from the experimental data. The
observed phenomena are reproduced in simulations using a theoretical model of
the reaction.


title: On N-wave type systems and their gauge equivalent

abstract:
  The class of nonlinear evolution equations - gauge equivalent to the N-wave
equations related to the simple Lie algebra g are derived and analyzed. They
are written in terms of the functions S(x,t) satisfying r= rank g nonlinear
constraints. The corresponding Lax pairs and the time evolution of the
scattering data are found. The Zakharov-Shabat dressing method is appropriately
modified to construct their soliton solutions.


title:
Avalanche dynamics in Bak-Sneppen evolution model observed with standard
  distribution width of fitness

abstract:
  We introduce the standard distribution width of fitness to characterize the
global and individual features of a ecosystem in the Bak-Sneppen evolution
model. Through tracking this quantity in evolution, a different hierarchy of
avalanche dynamics, $w_{0}$ avalanche is observed. The corresponding gap
equation and the self-organized threshold $w_{c}$ are obtained. The critical
exponents $\tau ,$ $\gamma $and $\rho $, which describe the behavior of the
avalanche size distribution, the average avalanche size and the relaxation to
attractor, respectively, are calculated with numerical simulation. The exact
master equation and $\gamma $ equation are derived. And the scaling relations
are established among the critical exponents of this new avalanche.


title: Spectral properties of quantized barrier billiards

abstract:
  The properties of energy levels in a family of classically pseudointegrable
systems, the barrier billiards, are investigated. An extensive numerical study
of nearest-neighbor spacing distributions, next-to-nearest spacing
distributions, number variances, spectral form factors, and the level dynamics
is carried out. For a special member of the billiard family, the form factor is
calculated analytically for small arguments in the diagonal approximation. All
results together are consistent with the so-called semi-Poisson statistics.


title:
Statistically Preserved Structures and Anomalous Scaling in Turbulent
  Active Scalar Advection

abstract:
  The anomalous scaling of correlation functions in the turbulent statistics of
active scalars (like temperature in turbulent convection) is understood in
terms of an auxiliary passive scalar which is advected by the same turbulent
velocity field. While the odd-order correlation functions of the active and
passive fields differ, we propose that the even-order correlation functions are
the same to leading order (up to a trivial multiplicative factor). The leading
correlation functions are statistically preserved structures of the passive
scalar decaying problem, and therefore universality of the scaling exponents of
the even-order correlations of the active scalar is demonstrated.


title:
Experimental investigation of the initial regime in fingering
  electrodeposition: dispersion relation and velocity measurements

abstract:
  Recently a fingering morphology, resembling the hydrodynamic Saffman-Taylor
instability, was identified in the quasi-two-dimensional electrodeposition of
copper. We present here measurements of the dispersion relation of the growing
front. The instability is accompanied by gravity-driven convection rolls at the
electrodes, which are examined using particle image velocimetry. While at the
anode the theory presented by Chazalviel et al. describes the convection roll,
the flow field at the cathode is more complicated because of the growing
deposit. In particular, the analysis of the orientation of the velocity vectors
reveals some lag of the development of the convection roll compared to the
finger envelope.


title: Universal R operator with deformed conformal symmetry

abstract:
  We study the general solution of the Yang-Baxter equation with deformed
$sl(2)$ symmetry. The universal R operator acting on tensor products of
arbitrary representations is obtained in spectral decomposition and in integral
forms. The results for eigenvalues, eigenfunctions and integral kernel appear
as deformations of the ones in the rational case. They provide a basis for the
construction of integrable quantum systems generalizing the XXZ spin models to
the case of arbitrary not necessarily finite-dimensional representations on the
sites.


title:
Small-scale structure of nonlinearly interacting species advected by
  chaotic flows

abstract:
  We study the spatial patterns formed by interacting biological populations or
reacting chemicals under the influence of chaotic flows. Multiple species and
nonlinear interactions are explicitly considered, as well as cases of smooth
and nonsmooth forcing sources. The small-scale structure can be obtained in
terms of characteristic Lyapunov exponents of the flow and of the chemical
dynamics. Different kinds of morphological transitions are identified.
Numerical results from a three-component plankton dynamics model support the
theory, and they serve also to illustrate the influence of asymmetric
couplings.


title:
Soliton-radiation coupling in the parametrically driven, damped
  nonlinear Schr\"odinger equation

abstract:
  We use the Riemann-Hilbert problem to study the interaction of the soliton
with radiation in the parametrically driven, damped nonlinear Schr\"odinger
equation. The analysis is reduced to the study of a finite-dimensional
dynamical system for the amplitude and phase of the soliton and the complex
amplitude of the long-wavelength radiation. In contrast to previously utilised
Inverse Scattering-based perturbation techniques, our approach is valid for
arbitrarily large driving strengths and damping coefficients. We show that,
contrary to suggestions made in literature, the complexity observed in the
soliton's dynamics cannot be accounted for just by its coupling to the
long-wavelength radiation.


title:
On the Lax pairs for the generalized Kowalewski and Goryachev-Chaplygin
  tops

abstract:
  A polynomial deformation of the Kowalewski top is considered. This
deformation includes as a degeneration a new integrable case for the Kirchhoff
equations found recently by one of the authors. A $5\times 5$ matrix Lax pair
for the deformed Kowalewski top is proposed. Also deformations of the two-field
Kowalewski gyrostat and the $so(p,q)$ Kowalewski top are found. All our Lax
pairs are deformations of the corresponding Lax representations found by Reyman
and Semenov-Tian {S}hansky. In addition, a similar deformation of the
Goryachev-Chaplygin top and its $3\times 3$ matrix Lax representation is
constructed.


title: Dynamical Localization in Quasi-Periodic Driven Systems

abstract:
  We investigate how the time dependence of the Hamiltonian determines the
occurrence of Dynamical Localization (DL) in driven quantum systems with two
incommensurate frequencies. If both frequencies are associated to impulsive
terms, DL is permanently destroyed. In this case, we show that the evolution is
similar to a decoherent case. On the other hand, if both frequencies are
associated to smooth driving functions, DL persists although on a time scale
longer than in the periodic case. When the driving function consists of a
series of pulses of duration $\sigma$, we show that the localization time
increases as $\sigma^{-2}$ as the impulsive limit, $\sigma\to 0$, is
approached. In the intermediate case, in which only one of the frequencies is
associated to an impulsive term in the Hamiltonian, a transition from a
localized to a delocalized dynamics takes place at a certain critical value of
the strength parameter. We provide an estimate for this critical value, based
on analytical considerations. We show how, in all cases, the frequency spectrum
of the dynamical response can be used to understand the global features of the
motion. All results are numerically checked.


title:
Speed Selection Mechanism for Propagating Fronts in Reaction-Diffusion
  Systems with Multiple Fields

abstract:
  We introduce a speed selection mechanism for front propagation in
reaction-diffusion systems with multiple fields. This mechanism applies to
pulled and pushed fronts alike, and operates by restricting the fields to large
"finite" intervals in the comoving frames of reference. The unique velocity for
which the center of a monotonic solution for a particular field is insensitive
to the location of the ends of the finite interval is the velocity that is
physically selected for that field, making thus the solution approximately
translation invariant. The fronts for the various fields may propagate at
different speeds, all of them being determined through this mechanism. We
present analytic results for the case of piecewise parabolic potentials, and
numerical results for other cases.


title:
Coarse Stability and Bifurcation Analysis Using Stochastic Simulators:
  Kinetic Monte Carlo Examples

abstract:
  We implement a computer-assisted approach that, under appropriate conditions,
allows the bifurcation analysis of the coarse dynamic behavior of microscopic
simulators without requiring the explicit derivation of closed macroscopic
equations for this behavior. The approach is inspired by the so-called
time-step per based numerical bifurcation theory. We illustrate the approach
through the computation of both stable and unstable coarsely invariant states
for Kinetic Monte Carlo models of three simple surface reaction schemes. We
quantify the linearized stability of these coarsely invariant states, perform
pseudo-arclength continuation, detect coarse limit point and coarse Hopf
bifurcations and construct two-parameter bifurcation diagrams.


title:
Resonances and superlattice pattern stabilization in two-frequency
  forced Faraday waves

abstract:
  We investigate the role weakly damped modes play in the selection of Faraday
wave patterns forced with rationally-related frequency components m*omega and
n*omega. We use symmetry considerations to argue for the special importance of
the weakly damped modes oscillating with twice the frequency of the critical
mode, and those oscillating primarily with the "difference frequency"
|n-m|*omega and the "sum frequency" (n+m)*omega. We then perform a weakly
nonlinear analysis using equations of Zhang and Vinals (1997, J. Fluid Mech.
336) which apply to small-amplitude waves on weakly inviscid, semi-infinite
fluid layers. For weak damping and forcing and one-dimensional waves, we
perform a perturbation expansion through fourth order which yields analytical
expressions for onset parameters and the cubic bifurcation coefficient that
determines wave amplitude as a function of forcing near onset. For stronger
damping and forcing we numerically compute these same parameters, as well as
the cubic cross-coupling coefficient for competing waves travelling at an angle
theta relative to each other. The resonance effects predicted by symmetry are
borne out in the perturbation results for one spatial dimension, and are
supported by the numerical results in two dimensions. The difference frequency
resonance plays a key role in stabilizing superlattice patterns of the SL-I
type observed by Kudrolli, Pier and Gollub (1998, Physica D 123).


title: Coarse Bifurcation Studies of Bubble Flow Microscopic Simulations

abstract:
  The parametric behavior of regular periodic arrays of rising bubbles is
investigated with the aid of 2-dimensional BGK Lattice-Boltzmann (LB)
simulators. The Recursive Projection Method is implemented and coupled to the
LB simulators, accelerating their convergence towards what we term coarse
steady states. Efficient stability/bifurcation analysis is performed by
computing the leading eigenvalues/eigenvectors of the coarse time stepper. Our
approach constitutes the basis for system-level analysis of processes modeled
through microscopic simulations.


title: "Dispersion management" for solitons in a Korteweg-de Vries system

abstract:
  The existence of ``dispersion-managed solitons'', i.e., stable pulsating
solitary-wave solutions to the nonlinear Schr\"{o}dinger equation with
periodically modulated and sign-variable dispersion is now well known in
nonlinear optics. Our purpose here is to investigate whether similar structures
exist for other well-known nonlinear wave models. Hence, here we consider as a
basic model the variable-coefficient Korteweg-de Vries equation; this has the
form of a Korteweg-de Vries equation with a periodically varying third-order
dispersion coefficient, that can take both positive and negative values. More
generally, this model may be extended to include fifth-order dispersion. Such
models may describe, for instance, periodically modulated waveguides for long
gravity-capillary waves. We develop an analytical approximation for solitary
waves in the weakly nonlinear case, from which it is possible to obtain a
reduction to a relatively simple integral equation, which is readily solved
numerically. Then, we describe some systematic direct simulations of the full
equation, which use the soliton shape produced by the integral equation as an
initial condition. These simulations reveal regions of stable and unstable
pulsating solitary waves in the corresponding parametric space. Finally, we
consider the effects of fifth-order dispersion.


title: Ehrenfest times for classically chaotic systems

abstract:
  We describe the quantum mechanical spreading of a Gaussian wave packet by
means of the semiclassical WKB approximation of Berry and Balazs. We find that
the time scale $\tau$ on which this approximation breaks down in a chaotic
system is larger than the Ehrenfest times considered previously. In one
dimension $\tau=\fr{7}{6}\lambda^{-1}\ln(A/\hbar)$, with $\lambda$ the Lyapunov
exponent and $A$ a typical classical action.


title:
Spontaneous Branching of Anode-Directed Streamers between Planar
  Electrodes

abstract:
  Non-ionized media subject to strong fields can become locally ionized by
penetration of finger-shaped streamers. We study negative streamers between
planar electrodes in a simple deterministic continuum approximation. We observe
that for sufficiently large fields, the streamer tip can split. This happens
close to Firsov's limit of `ideal conductivity'. Qualitatively the tip
splitting is due to a Laplacian instability quite like in viscous fingering.
For future quantitative analytical progress, our stability analysis of planar
fronts identifies the screening length as a regularization mechanism.


title: Evidences of Bolgiano scaling in 3D Rayleigh-Benard convection

abstract:
  We present new results from high-resolution high-statistics direct numerical
simulations of a tri-dimensional convective cell. We test the fundamental
physical picture of the presence of both a Bolgiano-like and a Kolmogorov-like
regime. We find that the dimensional predictions for these two distinct regimes
(characterized respectively by an active and passive role of the temperature
field) are consistent with our measurements.


title:
A host of traveling waves in a model of three-dimensional water-wave
  dynamics

abstract:
  We describe traveling waves in a basic model for three-dimensional water-wave
dynamics in the weakly nonlinear long-wave regime. Small solutions that are
periodic in the direction of translation (or orthogonal to it) form an
infinite-dimensional family. We characterize these solutions through spatial
dynamics, by reducing a linearly ill-posed mixed-type initial-value problem to
a center manifold of infinite dimension and codimension. A unique global
solution exists for arbitrary small initial data for the two-component bottom
velocity, specified along a single line in the direction of translation (or
orthogonal to it). A dispersive, nonlocal, nonlinear wave equation governs the
spatial evolution of bottom velocity.


title:
Canonical explicit B\"{a}cklund transformations with spectrality for
  constrained flows of soliton hierarchies

abstract:
  It is shown that explicit B\"{a}cklund transformations (BTs) for the
high-order constrained flows of soliton hierarchy can be constructed via their
Darboux transformations and Lax representation, and these BTs are canonical
transformations including B\"{a}cklund parameter $\eta$ and possess a
spectrality property with respect to $\eta$ and the 'conjugated' variable $\mu$
for which the pair $(\eta, \mu)$ lies on the spectral curve. As model we
present the canonical explicit BTs with the spectrality for high-order
constrained flows of the Kaup-Newell hierarchy and the KdV hierarchy.


title:
Constructing N-soliton solution for the mKdV equation through
  constrained flows

abstract:
  Based on the factorization of soliton equations into two commuting integrable
x- and t-constrained flows, we derive N-soliton solutions for mKdV equation via
its x- and t-constrained flows. It shows that soliton solution for soliton
equations can be constructed directly from the constrained flows.


title:
Kinematics of the Three Moving Space Curves Associated with the
  Nonlinear Schr\"{o}dinger Equation

abstract:
  Starting with the general description of a moving curve, we have recently
presented a unified formalism to show that three distinct space curve
evolutions can be identified with a given integrable equation. Applying this to
the nonlinear Schr\"{o}dinger equation (NLS), we find three sets of coupled
equations for the evolution of the curvature and the torsion, one set for each
moving curve. The first set is given by the usual Da Rios-Betchov equations.
The velocity at each point of the curve that corresponds to this set is well
known to be a local expression in the curve variables. In contrast, the
velocities of the other two curves are shown to be nonlocal expressions. Each
of the three curves is endowed with a corresponding infinite set of geometric
constraints. These moving space curves are found by using their connection with
the integrable Landau- Lifshitz equation. The three evolving curves
corresponding to the envelope soliton solution of the NLS are presented and
their behaviors compared.


title:
The role of diffusion in the chaotic advection of a passive scalar with
  finite lifetime

abstract:
  We study the influence of diffusion on the scaling properties of the first
order structure function, S_1, of a two-dimensional chaotically advected
passive scalar with finite lifetime, i.e., with a decaying term in its
evolution equation. We obtain an analytical expression for S_1 where the
dependence on the diffusivity, the decaying coefficient and the stirring due to
the chaotic flow is explicitly stated.
  We show that the presence of diffusion introduces a crossover length-scale,
the diffusion scale (L_d), such that the scaling behaviour for the structure
function is analytical for length-scales shorter than L_d, and shows a scaling
exponent that depends on the decaying term and the mixing of the flow for
larger scales. Therefore, the scaling exponents for scales larger than L_d are
not modified with respect to those calculated in the zero diffusion limit.
Moreover, L_d turns out to be independent of the decaying coeficient, being its
value the same as for the passive scalar with infinite lifetime.
  Numerical results support our theoretical findings.


title: Turbulent boundary layer equations

abstract:
  We study a boundary layer problem for the Navier-Stokes-alpha model obtaining
a generalization of the Prandtl equations conjectured to represent the averaged
flow in a turbulent boundary layer. We solve the equations for the
semi-infinite plate, both theoretically and numerically. The latter solutions
agree with some experimental data in the turbulent boundary layer.


title: Sensitivity to perturbations in a quantum chaotic billiard

abstract:
  The Loschmidt echo (LE) measures the ability of a system to return to the
initial state after a forward quantum evolution followed by a backward
perturbed one. It has been conjectured that the echo of a classically chaotic
system decays exponentially, with a decay rate given by the minimum between the
width $\Gamma$ of the local density of states and the Lyapunov exponent. As the
perturbation strength is increased one obtains a cross-over between both
regimes. These predictions are based on situations where the Fermi Golden Rule
(FGR) is valid. By considering a paradigmatic fully chaotic system, the
Bunimovich stadium billiard, with a perturbation in a regime for which the FGR
manifestly does not work, we find a cross over from $\Gamma$ to Lyapunov decay.
We find that, challenging the analytic interpretation, these conjetures are
valid even beyond the expected range.


title: New geometries associated with the nonlinear Schr\"{o}dinger equation

abstract:
  We apply our recent formalism establishing new connections between the
geometry of moving space curves and soliton equations, to the nonlinear
  Schr\"{o}dinger equation (NLS).
  We show that any given solution of the NLS gets associated with three
distinct space curve evolutions. The tangent vector of the first of these
curves, the binormal vector of the second and the normal vector of the third,
are shown to satisfy the integrable Landau-Lifshitz (LL) equation
  ${\bf S}_u = {\bf S} \times {\bf S}_{ss}$, (${\bf S}^2=1$). These connections
enable us to find the three surfaces swept out by the moving curves associated
with the NLS. As an example, surfaces corresponding to a stationary envelope
soliton solution of the NLS are obtained.


title:
Analysis of cancellation in two-dimensional magnetohydrodynamic
  turbulence

abstract:
  A signed measure analysis of two-dimensional intermittent magnetohydrodynamic
turbulence is presented. This kind of analysis is performed to characterize the
scaling behavior of the sign-oscillating flow structures, and their geometrical
properties. In particular, it is observed that cancellations between positive
and negative contributions of the field inside structures, are inhibited for
scales smaller than the Taylor microscale, and stop near the dissipative scale.
Moreover, from a simple geometrical argument, the relationship between the
cancellation exponent and the typical fractal dimension of the structures in
the flow is obtained.

categories: nlin.CD physics.data-an physics.flu-dyn physics.plasm-ph


title: Shape transformations in rotating ferrofluid drops

abstract:
  Floating drops of magnetic fluid can be brought into rotation by applying a
rotating magnetic field. We report theoretical and experimental results on the
transition from a spheroid equilibrium shape to non-axissymmetrical three-axes
ellipsoids at certain values of the external field strength. The transitions
are continuous for small values of the magnetic susceptibility and show
hysteresis for larger ones. In the non-axissymmetric shape the rotational
motion of the drop consists of a vortical flow inside the drop combined with a
slow rotation of the shape. Nonlinear magnetization laws are crucial to obtain
quantitative agreement between theory and experiment.


title:
Going with the Flow: a Lagrangian approach to self-similar dynamics and
  its consequences

abstract:
  We present a systematic computational approach to the study of self-similar
dynamics.
  The approach, through the use of what can be thought of as a ``dynamic
pinning condition" factors out self-similarity, and yields a transformed,
non-local evolution equation.
  The approach, which is capable of treating both first and second kind
self-similar solutions, yields as a byproduct the self-similarity exponents of
the original dynamics. We illustrate the approach through the porous medium
equation, showing how both the Barenblatt (first kind) and the Graveleau
(second kind) self-similar solutions arise in this framework.
  We also discuss certain implications of the dynamics of the transformed
equation (which we will name "MN-dynamics"); in particular we discuss the
discrete-time implementation of the approach, and connections with time-stepper
based methods for the "coarse" integration/bifurcation analysis of microscopic
simulators.


title: Nonlinearity Management in Dispersion Managed System

abstract:
  We propose using a nonlinear phase shift interferometric converter (NPSIC) (a
new device) for lumped compensation of the nonlinearity in optical fibers. The
NPSIC is a nonlinear analog of the Mach-Zehnder interferometer and provides a
way to control the sign of the nonlinear phase shift. We investigate a
potential use of NPSIC for compensation of the nonlinearity to make a
dispersion-managed system closer to an ideal linear system. More importantly
NPSIC can be used to essentially improve single channel capacity in the
nonlinear regime.


title:
Deterministic diffusion in one-dimensional maps - Calculation of
  diffusion constants by harmonic inversion of periodic orbit sums

abstract:
  A method is proposed for the calculation of diffusion constants for
one-dimensional maps exhibiting deterministic diffusion. The procedure is based
on harmonic inversion and uses a known relation between the diffusion constant
and the periodic orbits of a map. The method is tested on an example map for
which results calculated by different other techniques are available for
comparison.


title: Multivortex Solutions of the Weierstrass Representation

abstract:
  The connection between the complex Sine and Sinh-Gordon equations on the
complex plane associated with a Weierstrass type system and the possibility of
construction of several classes of multivortex solutions is discussed in
detail. We perform the Painlev\'e test and analyse the possibility of deriving
the B\"acklund transformation from the singularity analysis of the complex
Sine-Gordon equation. We make use of the analysis using the known relations for
the Painlev\'{e} equations to construct explicit formulae in terms of the
Umemura polynomials which are $\tau$-functions for rational solutions of the
third Painlev\'{e} equation. New classes of multivortex solutions of a
Weierstrass system are obtained through the use of this proposed procedure.
  Some physical applications are mentioned in the area of the vortex Higgs
model when the complex Sine-Gordon equation is reduced to coupled Riccati
equations.


title:
Hopf-Turing mixed mode and pattern selection in reaction diffusion
  systems

abstract:
  The amplitude equation of Gierer-Mainhardt model has been actually derived
near the boundary abuot which Turing and Hopf modes exist. In a parameter
region where Hopf-Turing mixed mode solution is stable, a chaotic state that
generally results from interaction between mixed modes, is observed. This
chaotic region follows a strong selection of a spatially periodic order
followed by a local, resonant, very large frequency temporal oscillation. A
spatio-temporal forcing, responsible for what obseved, has been identified.


title: Singularity analysis of a new discrete nonlinear Schrodinger equation

abstract:
  We apply the Painleve test for integrability to a new discrete
(differential-difference) nonlinear Schrodinger equation introduced by Leon and
Manna. Since the singular expansions of solutions of this equation turn out to
contain nondominant logarithmic terms, we conclude that the studied equation is
nonintegrable. This result supports the observation of Levi and Yamilov that
the Leon-Manna equation does not admit high-order generalized symmetries. As a
byproduct of the singularity analysis carried out, we obtain a new discrete
equation which should be integrable according to a conjecture of Weiss.


title:
Spectral Statistics of Rectangular Billiards with Localized
  Perturbations

abstract:
  The form factor $K(\tau)$ is calculated analytically to the order $\tau^3$ as
well as numerically for a rectangular billiard perturbed by a $\delta$-like
scatterer with an angle independent diffraction constant, $D$. The cases where
the scatterer is at the center and at a typical position in the billiard are
studied. The analytical calculations are performed in the semiclassical
approximation combined with the geometrical theory of diffraction. Non diagonal
contributions are crucial and are therefore taken into account.
  The numerical calculations are performed for a self adjoint extension of a
$\delta$ function potential.
  We calculate the angle dependent diffraction constant for an arbitrary
perturbing potential $U({\bf r})$, that is large in a finite but small region
(compared to the wavelength of the particles that in turn is small compared to
the size of the billiard). The relation to the idealized model of the
$\delta$-like scatterer is formulated. The angle dependent diffraction constant
is used for the analytic calculation of the form factor to the order $\tau^2$.
If the scatterer is at a typical position, the form factor is found to reduce
(in this order) to the one found for angle independent diffraction. If the
scatterer is at the center, the large degeneracy in the lengths of the orbits
involved leads to an additional small contribution to the form factor,
resulting of the angle dependence of the diffraction constant. The robustness
of the results is discussed.


title:
Quasi-classical $\dbar$-method: Generating equations for dispersionless
  integrable hierarchies

abstract:
  The quasi-classical $\bar{\partial}$-dressing method is used to derive
compact generating equations for dispersionless hierarchies. Dispersionless
Kadomtsev-Petviashvili (KP) and two-dimensional Toda lattice (2DTL) hierarchies
are considered as illustrative examples.


title: Localized Perturbations of Integrable Systems

abstract:
  The statistics of energy levels of a rectangular billiard, that is perturbed
by a strong localized potential, are studied analytically and numerically, when
this perturbation is at the center or at a typical position. Different results
are found for these two types of positions. If the scatterer is at the center,
the symmetry leads to additional contributions, some of them are related to the
angular dependence of the potential. The limit of the $\delta$-like scatterer
is obtained explicitly. The form factor, that is the Fourier transform of the
energy-energy correlation function, is calculated analytically, in the
framework of the semiclassical geometrical theory of diffraction, and
numerically. Contributions of classical orbits that are non diagonal are
calculated and are found to be essential.


title:
Dependence of heat transport on the strength and shear rate of
  prescribed circulating flows

abstract:
  We study numerically the dependence of heat transport on the maximum velocity
and shear rate of physical circulating flows, which are prescribed to have the
key characteristics of the large-scale mean flow observed in turbulent
convection. When the side-boundary thermal layer is thinner than the viscous
boundary layer, the Nusselt number (Nu), which measures the heat transport,
scales with the normalized shear rate to an exponent 1/3. On the other hand,
when the side-boundary thermal layer is thicker, the dependence of Nu on the
Peclet number, which measures the maximum velocity, or the normalized shear
rate when the viscous boundary layer thickness is fixed, is generally not a
power law. Scaling behavior is obtained only in an asymptotic regime. The
relevance of our results to the problem of heat transport in turbulent
convection is also discussed.


title: Measuring billiard eigenfunctions with arbitrary trajectories

abstract:
  We propose a method of measuring approximate quantum eigenfunctions in
polygonalized billiard geometries, based on a quasiclassical evolution operator
having a (smoothened) Perron-Frobenius kernel modulated by a phase arising from
quantum considerations. We show that quasiclassical evolution differs from
semiclassical (or quantum) evolution but the operators have the same set of
eigenfunctions. We demonstrate this by determining the quasiclassical
eigenfunctions of the polygonalized stadium billiard using arbitrary
trajectories and comparing this with the exact quantum stadium eigenfunctions.


title: Intermittency in two-dimensional Ekman-Navier-Stokes turbulence

abstract:
  We study the statistics of the vorticity field in two-dimensional
Navier-Stokes turbulence with a linear Ekman friction. We show that the
small-scale vorticity fluctuations are intermittent, as conjectured by Nam et
al. [Phys. Rev. Lett. vol.84 (2000) 5134]. The small-scale statistics of
vorticity fluctuations coincides with the one of a passive scalar with finite
lifetime transported by the velocity field itself.


title: How to solve Yang-Baxter equation using the Taylor expansion of R-matrix

abstract:
  A new method for solving the Yang-Baxter equation is presented. It is based
on the Taylor expansion of R-matrix which is developed up to the power
lambda^6. Using this method the R-matrix for integrable spin ladder is
calculated.


title: On the construction of evolution equations admitting a master symmetry

abstract:
  A method for constructing evolution equations admitting a master symmetry is
proposed. Several examples illustrating the method are presented. It is also
noted that for certain evolution equations master symmetries can be useful for
obtaining new conservation laws from a given one.


title: Periodic orbits for three particles with finite angular momentum

abstract:
  We present a novel numerical method to calculate periodic orbits for
dynamical systems by an iterative process which is based directly on the action
integral in classical mechanics. New solutions are obtained for the planar
motion of three equal mass particles on a common periodic orbit with finite
total angular momentum, under the action of attractive pairwise forces of the
form $1/r^{p+1}$. It is shown that for $-2 < p \le 0$, Lagrange's 1772 circular
solution is the limiting case of a complex symmetric orbit. The evolution of
this orbit and another recently discovered one in the shape of a figure eight
is investigated for a range of angular momenta. Extensions to n equal mass
particles and to three particles of different masses are also discussed
briefly.


title:
A Lax Pair for the Dynamics of DNA Modeled as a Shearable and Extensible
  Elastic Rod: II. Discretization of the Arc Length

abstract:
  In previous work, the dynamics of the elastic rod was recast in a Lax pair
formulation, with fiducial arc length s and time t as continuous independent
variables. However, the solution of these equations cannot apply directly to a
system where the fiducial arc length s is a discrete variable. In this paper,
we show how to discretize the continuous s variable in a way that preserves the
integrability of the original system. The t parameter is not discretized, so
this algorithm will be especially useful for solutions of the s-discrete and
t-continuous elastic rod problem, as may occur in problems where the polymeric
structure of the DNA is made explicit.


title:
Polarization coupling and pattern selection in a type-II optical
  parametric oscillator

abstract:
  We study the role of a direct intracavity polarization coupling in the
dynamics of transverse pattern formation in type-II optical parametric
oscillators. Transverse intensity patterns are predicted from a stability
analysis, numerically observed, and described in terms of amplitude equations.
Standing wave intensity patterns for the two polarization components of the
field arise from the nonlinear competition between two concentric rings of
unstable modes in the far field. Close to threshold a wavelength is selected
leading to standing waves with the same wavelength for the two polarization
components. Far from threshold the competition stabilizes patterns in which two
different wavelengths coexist.


title: Duality of spectral curves arising in two-matrix models

abstract:
  The two matrix model is considered, with measure given by the exponential of
a sum of polynomials in two different variables. It is shown how to derive a
sequence of pairs of ``dual'' finite size systems of ODEs for the corresponding
biorthonormal polynomials. An inverse theorem is proved showing how to
reconstruct such measures from pairs of semi-infinite finite band matrices
defining the recursion relations and satisfying the string equation. A proof is
given in the $N\to \infty$ limit that the dual systems obtained share the same
spectral curve.

categories: nlin.SI cond-mat.stat-mech hep-th math-ph math.MP


title:
New contiguity relation of the sixth Painlev\'e equation from a
  truncation

abstract:
  For the master Painlev\'e equation P6(u), we define a consistent method,
adapted from the Weiss truncation for partial differential equations, which
allows us to obtain the first degree birational transformation of Okamoto. Two
new features are implemented to achieve this result. The first one is the
homography between the derivative of the solution $u$ and a Riccati
pseudopotential. The second one is an improvement of a conjecture by Fokas and
Ablowitz on the structure of this birational transformation. We then build the
contiguity relation of P6, which yields one new second order nonautonomous
discrete equation.


title: A PDE-Based Approach to Classical Phase-Space Deformations

abstract:
  This paper presents a PDE-based approach to finding an optimal canonical
basis with which to represent a nearly integrable Hamiltonian. The idea behind
the method is to continuously deform the initial canonical basis in such a way
that the dependence of the Hamiltonian on the canonical position of the final
basis is minimized. The final basis incorporates as much of the classical
dynamics as possible into an integrable Hamiltonian, leaving a much smaller
non-integrable component than in the initial representation. With this approach
it is also possible to construct the semiclassical wavefunctions corresponding
to the final canonical basis. This optimized basis is potentially useful in
quantum calculations, both as a way to minimize the required size of basis
sets, and as a way to provide physical insight by isolating those effects
resulting from integrable dynamics.


title: A Bi-Hamiltonian Formulation for Triangular Systems by Perturbations

abstract:
  A bi-Hamiltonian formulation is proposed for triangular systems resulted by
perturbations around solutions, from which infinitely many symmetries and
conserved functionals of triangular systems can be explicitly constructed,
provided that one operator of the Hamiltonian pair is invertible. Through our
formulation, four examples of triangular systems are exhibited, which also show
that bi-Hamiltonian systems in both lower dimensions and higher dimensions are
many and varied. Two of four examples give local 2+1 dimensional bi-Hamiltonian
systems and illustrate that multi-scale perturbations can lead to
higher-dimensional bi-Hamiltonian systems.


title: Non-linear Fractal Interpolating Functions

abstract:
  We consider two non-linear generalizations of fractal interpolating functions
generated from iterated function systems. The first corresponds to fitting data
using a Kth-order polynomial, while the second relates to the freedom of adding
certain arbitrary functions. An escape-time algorithm that can be used for such
systems to generate fractal images like those associated with Julia or
Mandelbrot sets is also described.


title:
Commutative Poisson subalgebras for the Sklyanin bracket and
  deformations of known integrable models

abstract:
  A hierarchy of commutative Poisson subalgebras for the Sklyanin bracket is
proposed. Each of the subalgebras provides a complete set of integrals in
involution with respect to the Sklyanin bracket. Using different
representations of the bracket, we find some integrable models and a separation
of variables for them. The models obtained are deformations of known integrable
systems like the Goryachev-Chaplygin top, the Toda lattice and the Heisenberg
model.


title:
Stable and chaotic solutions of the complex Ginzburg-Landau equation
  with periodic boundary conditions

abstract:
  We study, analytically and numerically, the dynamical behavior of the
solutions of the complex Ginzburg-Landau equation with diffraction but without
diffusion, which governs the spatial evolution of the field in an active
nonlinear laser cavity. Accordingly, the solutions are subject to periodic
boundary conditions. The analysis reveals regions of stable stationary
solutions in the model’s parameter space, and a wide range of oscillatory
and chaotic behaviors. Close to the first bifurcation destabilizing the
spatially uniform solution, a stationary single-humped solution is found in an
asymptotic analytical form, which turns out to be in very good agreement with
the numerical results. Simulations reveal a series of stable stationary
multi-humped solutions


title: On Generation of magnetic field in astrophysical bodies

abstract:
  In this letter we compute energy transfer rates from velocity field to
magnetic field in MHD turbulence using field-theoretic method. The striking
result of our field theoretic calculation is that there is a large energy
transfer rate from the large-scale velocity field to the large-scale magnetic
field. We claim that the growth of large-scale magnetic energy is primarily due
to this transfer. We reached the above conclusion without any linear
approximation like that in $\alpha$-dynamo.


title: Misleading signatures of quantum chaos

abstract:
  The main signature of chaos in a quantum system is provided by spectral
statistical analysis of the nearest neighbor spacing distribution and the
spectral rigidity given by $\Delta_3(L)$. It is shown that some standard
unfolding procedures, like local unfolding and Gaussian broadening, lead to a
spurious increase of the spectral rigidity that spoils the $\Delta_3(L)$
relationship with the regular or chaotic motion of the system. This effect can
also be misinterpreted as Berry's saturation.


title: Measuring the Lyapunov exponent using quantum mechanics

abstract:
  We study the time evolution of two wave packets prepared at the same initial
state, but evolving under slightly different Hamiltonians. For chaotic systems,
we determine the circumstances that lead to an exponential decay with time of
the wave packet overlap function. We show that for sufficiently weak
perturbations, the exponential decay follows a Fermi golden rule, while by
making the difference between the two Hamiltonians larger, the characteristic
exponential decay time becomes the Lyapunov exponent of the classical system.
We illustrate our theoretical findings by investigating numerically the overlap
decay function of a two-dimensional dynamical system.


title: Schlesinger transformations and quantum R-matrices

abstract:
  Schlesinger transformations are discrete monodromy preserving symmetry
transformations of a meromorphic connection which shift by integers the
eigenvalues of its residues. We study Schlesinger transformations for twisted
sl_N-valued connections on the torus. A universal construction is presented
which gives the elementary two-point transformations in terms of Belavin's
elliptic quantum R-matrix. In particular, the role of the quantum deformation
parameter is taken by the difference of the two poles whose residue eigenvalues
are shifted. Elementary one-point transformations (acting on the residue
eigenvalues at a single pole) are constructed in terms of the classical
elliptic r-matrix.
  The action of these transformations on the tau-function of the system may
completely be integrated and we obtain explicit expressions in terms of the
parameters of the connection. In the limit of a rational R-matrix, our
construction and the tau-quotients reduce to the classical results of Jimbo and
Miwa in the complex plane.


title:
B\"{a}cklund transformations for high-order constrained flows of the
  AKNS hierarchy: canonicity and spectrality property

abstract:
  New infinite number of one- and two-point B\"{a}cklund transformations (BTs)
with explicit expressions are constructed for the high-order constrained flows
of the AKNS hierarchy. It is shown that these BTs are canonical transformations
including B\"{a}cklund parameter $\eta$ and a spectrality property holds with
respect to $\eta$ and the 'conjugated' variable $\mu$ for which the point
$(\eta, \mu)$ belongs to the spectral curve. Also the formulas of m-times
repeated Darboux transformations for the high-order constrained flows of the
AKNS hierarchy are presented.


title: Extended Self-Similarity and Hierarchical Structure in Turbulence

abstract:
  It is shown that the two remarkable properties of turbulence, the Extended
Self-Similarity (ESS) [R. Benzi {\it et al.}, Phy. Rev. E {\bf 48}, R29,
(1993)] and the She-Leveque Hierarchical Structure (SLHS) [Z.S. She and E.
Leveque, Phy. Rev. Lett. {\bf 72}, 336, (1994)] are related to each other. In
particular, we have shown that a generalized hierarchical structure together
with the most intense structures being shock-like give rise to ESS. Our
analysis thus suggests that the ESS measured in turbulent flows is an
indication of the shock-like intense structures. Results of analysis of
velocity measurements in a pipe-flow turbulence support our conjecture.


title: When do tracer particles dominate the Lyapunov spectrum?

abstract:
  Dynamical instability is studied in a deterministic dynamical system of
Hamiltonian type composed of a tracer particle in a fluid of many particles.
The tracer and fluid particles are hard balls (disks, in two dimensions, or
spheres, in three dimensions) undergoing elastic collisions. The dynamical
instability is characterized by the spectrum of Lyapunov exponents. The tracer
particle is shown to dominate the Lyapunov spectrum in the neighborhoods of two
limiting cases: the Lorentz-gas limit in which the tracer particle is much
lighter than the fluid particles and the Rayleigh-flight limit in which the
fluid particles have a vanishing radius and form an ideal gas. In both limits,
a gap appears in the Lyapunov spectrum between the few largest Lyapunov
exponents associated with the tracer and the rest of the Lyapunov spectrum.


title:
A fast method to simulate travelling waves in nonhomogeneous chemical or
  biological media

abstract:
  Waves in excitable media can be treated by a simple geometric theory. The
propagation velocity is assumed known and evolution of wave fronts is
determined by elementary physical principles (Fermat's principle, Huygens'
principle). Based on this geometric theory a fast computational method is
developed. By this method the distorting effect of the spatial grid is avoided.
The method is applied to the cases when a circular obstacle is surrounded by a
homogeneous and heterogeneous medium, respectively. The numerical simulations
show that the method is convenient, fast and reliable.


title:
Escape-rate formalism, decay to steady states, and divergences in the
  entropy-production rate

abstract:
  In summer 1997 we were sitting with Bob Dorfman and a few other friends
interested in chaotic systems and transport theory on a terrace close to
Oktogon in Budapest. While taking our (decaf) coffee after a very nice Italian
meal, we discussed about logarithmic divergences in the entropy production of
systems with absorbing boundary conditions and their consequences for the
escape-rate formalism. It was guessed at that time that the problem could be
resolved by a careful discussion of the physical content of the absorbing
boundary conditions. To our knowledge a thorough analysis of this long-standing
question is still missing. We dedicate it hereby to Bob on occasion of his 65th
birthday.


title: Computing A Glimpse of Randomness

abstract:
  A Chaitin Omega number is the halting probability of a universal Chaitin
(self-delimiting Turing) machine. Every Omega number is both computably
enumerable (the limit of a computable, increasing, converging sequence of
rationals) and random (its binary expansion is an algorithmic random sequence).
In particular, every Omega number is strongly non-computable. The aim of this
paper is to describe a procedure, which combines Java programming and
mathematical proofs, for computing the exact values of the first 64 bits of a
Chaitin Omega:
0000001000000100000110001000011010001111110010111011101000010000. Full
description of programs and proofs will be given elsewhere.


title: Synchronization in small-world systems

abstract:
  We quantify the dynamical implications of the small-world phenomenon. We
consider the generic synchronization of oscillator networks of arbitrary
topology, and link the linear stability of the synchronous state to an
algebraic condition of the Laplacian of the graph. We show numerically that the
addition of random shortcuts produces improved network synchronizability.
Further, we use a perturbation analysis to place the synchronization threshold
in relation to the boundaries of the small-world region. Our results also show
that small-worlds synchronize as efficiently as random graphs and hypercubes,
and more so than standard constructive graphs.


title:
Stepwise structure of Lyapunov spectra for many particle systems by a
  random matrix dynamics

abstract:
  The structure of Lyapunov spectra for many particle systems with a random
interaction between the particles is discussed. The dynamics of the tangent
space is expressed as a master equation, which leads to a formula that connects
the positive Lyapunov exponents and the time correlations of the particle
interaction matrix. Applying this formula to one and two dimensional models we
investigate the stepwise structure of the Lyapunov spectra, which appear in the
region of small positive Lyapunov exponents. Long range interactions lead to a
clear separation of the Lyapunov spectra into a part exhibiting stepwise
structure and a part changing smoothly. The part of the Lyapunov spectrum
containing the stepwise structure is clearly distinguished by a wave like
structure in the eigenstates of the particle interaction matrix. The two
dimensional model has the same step widths as found numerically in a
deterministic chaotic system of many hard disks.


title: Solutions of Fisher and Burger's equations with finite transport memory

abstract:
  The Fisher and Burgers' equations with finite memory transport, describing
reaction-diffusion and convection-diffusion processes, respectively have
recently attracted a lot of attention in the context of chemical kinetics,
mathematical biology and turbulence. We show here that they admit exact
solutions. While the speed of the traveling wavefront is dependent on the
relaxation time in Fisher equation, memory effects significantly smoothen out
the shock wave nature of Burgers' solution, without making any influence on the
corresponding wave speed. We numerically analyze the ansatz for the exact
solution and show that for the reaction-diffusion system the strength of the
reaction term must be moderate enough not to exceed a critical limit to allow
travelling wave solution to exist for appreciable finite memory effect.


title:
The Theory of Binary Nonlinearization and Its Applications to Soliton
  Equations

abstract:
  Binary symmetry constraints are applied to the nonlinearization of spectral
problems and adjoint spectral problems into so-called binary constrained flows,
which provide candidates for finite-dimensional Liouville integrable
Hamiltonian systems. The resulting constraints on the potentials of spectral
problems give rise to a kind of involutive solutions to zero curvature
equations, and thus the integrability by quadratures can be shown for zero
curvature equations once the corresponding binary constrained flows are proved
to be integrable. The whole process to carry out binary symmetry constraints is
called binary nonlinearization. The principal task of binary nonlinearization
is to expose the Liouville integrability for the resulting binary constrained
flows, which can usually be achieved as a consequence of the existence of
hereditary recursion operators. The theory of binary nonlinearization is
applied to the multi-wave interaction equations associated with a 4x4 matrix
spectral problem as an illustrative example. The Backlund transformations
resulted from symmetry constraints are given for the multi-wave interaction
equations, and thus a kind of involutive solutions is presented and the
integrability by quadratures is shown for the multi-wave interaction equations.


title: True and fake Lax pairs: how to distinguish them

abstract:
  The gauge-invariant description of zero-curvature representations of
evolution equations is applied to the problem of how to distinguish the fake
Lax pairs from the true Lax pairs. The main difference between the true Lax
pairs and the fake ones is found in the structure of their cyclic bases.


title: Nearest-neighbor distribution for singular billiards

abstract:
  The exact computation of the nearest-neighbor spacing distribution P(s) is
performed for a rectangular billiard with point-like scatterer inside for
periodic and Dirichlet boundary conditions and it is demonstrated that for
large s this function decreases exponentially. Together with the results of
[Bogomolny et al., Phys. Rev. E 63, 036206 (2001)] it proves that spectral
statistics of such systems is of intermediate type characterized by level
repulsion at small distances and exponential fall-off of the nearest-neighbor
distribution at large distances. The calculation of the n-th nearest-neighbor
spacing distribution and its asymptotics is performed as well for any boundary
conditions.


title:
Parameter estimation in spatially extended systems: The Karhunen-Loeve
  and Galerkin multiple shooting approach

abstract:
  Parameter estimation for spatiotemporal dynamics for coupled map lattices and
continuous time domain systems is shown using a combination of multiple
shooting, Karhunen-Loeve decomposition and Galerkin's projection methodologies.
The resulting advantages in estimating parameters have been studied and
discussed for chaotic and turbulent dynamics using small amounts of data from
subsystems, availability of only scalar and noisy time series data, effects of
space-time parameter variations, and in the presence of multiple time-scales.


title: Integration of a generalized H\'enon-Heiles Hamiltonian

abstract:
  The generalized H\'enon-Heiles Hamiltonian
$H=1/2(P_X^2+P_Y^2+c_1X^2+c_2Y^2)+aXY^2-bX^3/3$ with an additional
nonpolynomial term $\mu Y^{-2}$ is known to be Liouville integrable for three
sets of values of $(b/a,c_1,c_2)$. It has been previously integrated by genus
two theta functions only in one of these cases. Defining the separating
variables of the Hamilton-Jacobi equations, we succeed here, in the two other
cases, to integrate the equations of motion with hyperelliptic functions.


title: A Note on the Ruelle Pressure for a Dilute Disordered Sinai Billiard

abstract:
  The topological pressure is evaluated for a dilute random Lorentz gas, in the
approximation that takes into account only uncorrelated collisions between the
moving particle and fixed, hard sphere scatterers. The pressure is obtained
analytically as a function of a temperature-like parameter, beta, and of the
density of scatterers. The effects of correlated collisions on the topological
pressure can be described qualitatively, at least, and they significantly
modify the results obtained by considering only uncorrelated collision
sequences. As a consequence, for large systems, the range of beta-values over
which our expressions for the topological pressure are valid becomes very
small, approaching zero, in most cases, as the inverse of the logarithm of
system size.


title: Torus Fractalization and Intermittency

abstract:
  The bifurcation transition is studied for the onset of intermittency
analogous to the Pomeau-Manneville mechanism of type-I, but generalized for the
presence of a quasiperiodic external force. The analysis is concentrated on the
torus-fractalization (TF) critical point that occurs at some critical amplitude
of driving. (At smaller amplitudes the bifurcation corresponds to a collision
and subsequent disappearance of two smooth invariant curves, and at larger
amplitudes it is a touch of fractal attractor and repeller at some set of
exceptional points, without coincidence.) For the TF critical point,
renormalization group (RG) analysis is developed. For the golden mean rotation
number a nontrivial fixed-point solution of the RG equation is found in a class
of fractional-linear functions with coefficients depending on the phase
variable. Universal constants are computed responsible for scaling in phase
space ($\alpha=2.890053...$ and $\beta=-1.618034...$) and in parameter space
($\delta_1=3.134272...$ and $\delta_2=1.618034...$). An analogy with the Harper
equation is outlined, which reveals important peculiarities of the transition.
For amplitudes of driving less than the critical value the transition leads (in
the presence of an appropriate re-injection mechanism) to intermittent chaotic
regimes; in the supercritical case it gives rise to a strange nonchaotic
attractor.


title: Pseudochaos

abstract:
  A family of the billiard-type systems with zero Lyapunov exponent is
considered as an example of dynamics which is between the regular one and
chaotic mixing. This type of dynamics is called ``pseudochaos''. We demonstrate
how the fractional kinetic equation can be introduced for the pseudochaos and
how the main critical exponents of the fractional kinetics can be evaluated
from the dynamics. Problems related to pseudochaos are discussed: Poincare
recurrences, continued fractions, log-periodicity, rhombic billiards, and
others. Pseudochaotic dynamics and fractional kinetics can be applied to
streamlines or magnetic field lines behavior.


title:
New renormalization procedure for perturbed ordinally differential
  equations with given initial condition and prescribed secular terms

abstract:
  For perturbed ordinally differential equations, a procedure of
renormalization group method is proposed. To uniquely obtain renormalized
solutions for given initial conditions, the procedure assumes that the extra
integral constants yielded by perturbative expansion are determined as they
cancel out all the prescribed secular terms. A sufficient condition for the
renormalized solutions coinciding with exact solutions is also stated.


title:
Darboux Transformations, Infinitesimal Symmetries and Conservation Laws
  for Nonlocal Two-Dimensional Toda Lattice

abstract:
  The technique of Darboux transformation is applied to nonlocal partner of
two-dimensional periodic A_{n-1} Toda lattice. This system is shown to admit a
representation as the compatibility conditions of direct and dual
overdetermined linear systems with quantized spectral parameter. The
generalization of the Darboux transformation technique on linear equations of
such a kind is given. The connections between the solutions of overdetermined
linear systems and their expansions in series at singular points neighborhood
are presented. The solutions of the nonlocal Toda lattice and infinite
hierarchies of the infinitesimal symmetries and conservation laws are obtained.


title: Random unistochastic matrices

abstract:
  An ensemble of random unistochastic (orthostochastic) matrices is defined by
taking squared moduli of elements of random unitary (orthogonal) matrices
distributed according to the Haar measure on U(N) (or O(N), respectively). An
ensemble of symmetric unistochastic matrices is obtained with use of unitary
symmetric matrices pertaining to the circular orthogonal ensemble. We study the
distribution of complex eigenvalues of bistochastic, unistochastic and
ortostochastic matrices in the complex plane. We compute averages (entropy,
traces) over the ensembles of unistochastic matrices and present inequalities
concerning the entropies of products of bistochastic matrices.


title: Turbulence in small-world networks

abstract:
  The transition to turbulence via spatiotemporal intermittency is investigated
in the context of coupled maps defined on small-world networks. The local
dynamics is given by the Chat\'e-Manneville minimal map previously used in
studies of spatiotemporal intermittency in ordered lattices. The critical
boundary separating laminar and turbulent regimes is calculated on the
parameter space of the system, given by the coupling strength and the rewiring
probability of the network. Windows of relaminarization are present in some
regions of the parameter space. New features arise in small-world networks; for
instance, the character of the transition to turbulence changes from second
order to a first order phase transition at some critical value of the rewiring
probability. A linear relation characterizing the change in the order of the
phase transition is found. The global quantity used as order parameter for the
transition also exhibits nontrivial collective behavior for some values of the
parameters. These models may describe several processes occurring in nonuniform
media where the degree of disorder can be continuously varied through a
parameter.


title:
Explicit centre manifold reduction and full bifurcation analysis for an
  astigmatic Maxwell-Bloch laser

abstract:
  We set up a general framework to study the bifurcations of nearly degenerate
modes of a Maxwell-Bloch laser model, and we apply it specifically to the study
of interactions of two modes of a laser with broken circular symmetry. We use
an explicit centre manifold reduction to analyse the behaviour of this laser.
Complex bifurcation sequences involving single mode solutions, mode locking and
mode beating regimes are predicted. These sequences are organized by a Hopf
bifurcation with 1:1 resonance and a Z_2-symmetric Takens-Bogdanov bifurcation.
Numerical simulations of the original system show good agreement with theory.


title: Rotating optical soliton clusters

abstract:
  We introduce the concept of soliton clusters -- multi-soliton bound states in
a homogeneous bulk optical medium, and reveal a key physical mechanism for
their stabilization associated with a staircase-like phase distribution that
induces a net angular momentum and leads to cluster rotation. The ringlike
soliton clusters provide a nontrivial generalization of the concepts of
two-soliton spiraling, optical vortex solitons, and necklace-type optical
beams.


title: DNA Torsional Solitons in Presence of localized Inhomogeneities

abstract:
  In the present paper we investigate the influence of inhomogeneities in the
dynamics and stability of DNA open states, modeled as propagating solitons in
the spirit of a Generalized Yakushevish Model. It is a direct consecuence of
our model that there exists a critical distance between the soliton's center of
mass and the inhomogeneity at which the interaction between them can change the
stability of the open state.Furtherly from this results was derived a
renormalized potential funtion.


title: On a Periodic Soliton Cellular Automaton

abstract:
  We propose a box and ball system with a periodic boundary condition (pBBS).
The time evolution rule of the pBBS is represented as a Boolean recurrence
formula, an inverse ultradiscretization of which is shown to be equivalent with
the algorithm of the calculus for the 2Nth root. The relations to the pBBS of
the combinatorial R matrix of ${U'}_q(A_N^{(1)})$ are also discussed.


title:
Moments of generalized Husimi distributions and complexity of many-body
  quantum states

abstract:
  We consider generalized Husimi distributions for many-body systems, and show
that their moments are good measures of complexity of many-body quantum states.
Our construction of the Husimi distribution is based on the coherent state of
the single-particle transformation group. Then the coherent states are
independent-particle states, and, at the same time, the most localized states
in the Husimi representation. Therefore delocalization of the Husimi
distribution, which can be measured by the moments, is a sign of many-body
correlation (entanglement). Since the delocalization of the Husimi distribution
is also related to chaoticity of the dynamics, it suggests a relation between
entanglement and chaos. Our definition of the Husimi distribution can be
applied not only to the systems of distinguishable particles, but also to those
of identical particles, i.e., fermions and bosons. We derive an algebraic
formula to evaluate the moments of the Husimi distribution.

categories: nlin.CD cond-mat.str-el nucl-th quant-ph


title:
Generating Function Associated with the Rational Solutions of the
  Painlev\'e II Equation

abstract:
  We consider the Hankel determinant representation for the rational solutions
of the Painlev\'e II equation. We give an explicit formula for the generating
function of the entries in terms of logarithmic derivative of the Airy
function, which by itself is a particular solution of the Painlev\'e II
equation.


title:
Surface Quality of a Work Material Influence on Vibrations in a Cutting
  Process

abstract:
  The problem of stability in the machining processes is an important task. It
is strictly connected with the final quality of a product. In this paper we
consider vibrations of a tool-workpiece system in a straight turning process
induced by random disturbances and their effect on a product surface. Basing on
experimentally obtained system parameters we have done the simulations using
one degree of freedom model. The noise has been introduced to the model by the
Langevin equation. We have also analyzed the product surface shape and its
dependence on the level of noise.


title: q-Painlev\'e systems arising from q-KP hierarchy

abstract:
  A system of q-Painlev\'e type equations with multi-time variables t_1,...,t_M
is obtained as a similarity reduction of the N-reduced q-KP hierarchy. This
system has affine Weyl group symmetry of type A^{(1)}_{M-1} \times
A^{(1)}_{N-1}. Its rational solutions are constructed in terms of q-Schur
functions.


title: Multi-channel pulse dynamics in a stabilized Ginzburg-Landau system

abstract:
  We study the stability and interactions of chirped solitary pulses in a
system of nonlinearly coupled cubic Ginzburg-Landau (CGL) equations with a
group-velocity mismatch between them, where each CGL equation is stabilized by
linearly coupling it to an additional linear dissipative equation. In the
context of nonlinear fiber optics, the model describes transmission and
collisions of pulses at different wavelengths in a dual-core fiber, in which
the active core is furnished with bandwidth-limited gain, while the other,
passive (lossy) one is necessary for stabilization of the solitary pulses.
Complete and incomplete collisions of pulses in two channels in the cases of
anomalous and normal dispersion in the active core are analyzed by means of
perturbation theory and direct numerical simulations. It is demonstrated that
the model may readily support fully stable pulses whose collisions are
quasi-elastic, provided that the group-velocity difference between the two
channels exceeds a critical value. In the case of quasi-elastic collisions, the
temporal shift of pulses, predicted by the analytical approach, is in
semi-quantitative agrement with direct numerical results in the case of
anomalous dispersion (in the opposite case, the perturbation theory does not
apply). We also consider a simultaneous collision between pulses in three
channels, concluding that this collision remains quasi-elastic, and the pulses
remain completely stable. Thus, the model may be a starting point for the
design of a stabilized wavelength-division-multiplexed (WDM) transmission
system.


title: Two and three-dimensional oscillons in nonlinear Faraday resonance

abstract:
  We study 2D and 3D localised oscillating patterns in a simple model system
exhibiting nonlinear Faraday resonance. The corresponding amplitude equation is
shown to have exact soliton solutions which are found to be always unstable in
3D. On the contrary, the 2D solitons are shown to be stable in a certain
parameter range; hence the damping and parametric driving are capable of
suppressing the nonlinear blowup and dispersive decay of solitons in two
dimensions. The negative feedback loop occurs via the enslaving of the
soliton's phase, coupled to the driver, to its amplitude and width.

categories: nlin.PS math-ph math.MP nlin.AO physics.flu-dyn physics.optics


title:
Random Wandering Around Homoclinic-like Manifolds in Symplectic Map
  Chain

abstract:
  We present a method to construct a symplecticity preserving renormalization
group map of a chain of weakly nonlinear symplectic maps and obtain a general
reduced symplectic map describing its long-time behaviour. It is found that the
modulational instability in the reduced map triggers random wandering of orbits
around some homoclinic-like manifolds, which is understood as the Bernoulli
shifts.


title: How To Attain Maximum Profit In Minority Game?

abstract:
  What is the physical origin of player cooperation in minority game? And how
to obtain maximum global wealth in minority game? We answer the above questions
by studying a variant of minority game from which players choose among $N_c$
alternatives according to strategies picked from a restricted set of strategy
space. Our numerical experiment concludes that player cooperation is the result
of a suitable size of sampling in the available strategy space. Hence, the
overall performance of the game can be improved by suitably adjusting the
strategy space size.


title: Topological Shocks in Burgers Turbulence

abstract:
  The dynamics of the multi-dimensional randomly forced Burgers equation is
studied in the limit of vanishing viscosity. It is shown both theoretically and
numerically that the shocks have a universal global structure which is
determined by the topology of the configuration space. This structure is shown
to be particularly rigid for the case of periodic boundary conditions.


title: Topological Compactons

abstract:
  One dimensional topological kink which has strictly finite size without any
exponential or power-like tail is presented. It can be observed in a simple
mechanical system akin to the one used in order to demonstrate sinus-Gordon
solitons.


title:
Constraints on the spectral distribution of energy and enstrophy
  dissipation in forced two-dimensional turbulence

abstract:
  We study two-dimensional turbulence in a doubly periodic domain driven by a
monoscale-like forcing and damped by various dissipation mechanisms of the form
$\nu_{\mu}(-\Delta)^{\mu}$. By ``monoscale-like'' we mean that the forcing is
applied over a finite range of wavenumbers $k_\min \leq k \leq k_\max$, and
that the ratio of enstrophy injection $\eta \geq 0$ to energy injection
$\epsilon \geq 0$ is bounded by $k_\min^2 \epsilon \leq \eta \leq k_\max^2
\epsilon$. It is shown that for $\mu\geq 0$ the asymptotic behaviour satisfies
(eqnarray) \norm u_1^2&\leq&k_\max^2\norm u^2,(eqnarray) where $\norm u^2$ and
$\norm u_1^2$ are the energy and enstrophy, respectively. It is also shown that
for Navier-Stokes turbulence ($\mu = 1$), the time-mean enstrophy dissipation
rate is bounded from above by $2\nu_1 k_\max^2$. These results place strong
constraints on the spectral distribution of energy and enstrophy and of their
dissipation, and thereby on the existence of energy and enstrophy cascades, in
such systems. In particular, the classical dual cascade picture is shown to be
invalid for forced two-dimensional Navier--Stokes turbulence ($\mu=1$) when it
is forced in this manner. Inclusion of Ekman drag ($\mu=0$) along with
molecular viscosity permits a dual cascade, but is incompatible with the
log-modified -3 power law for the energy spectrum in the enstrophy-cascading
inertial range. In order to achieve the latter, it is necessary to invoke an
inverse viscosity ($\mu<0$).


title: Particle transport in a random velocity field with Lagrangian statistics

abstract:
  The transport properties of a random velocity field with Kolmogorov spectrum
and time correlations defined along Lagrangian trajectories are analyzed. The
analysis is carried on in the limit of short correlation times, as a
perturbation theory in the ratio, scale by scale, of the eddy decay and
turn-over time. Various quantities such as the Batchelor constant and the
dimensionless constants entering the expression for particle relative and
self-diffusion are given in terms of this ratio and of the Kolmogorov constant.
Particular attention is paid to particles with finite inertia. The
self-diffusion properties of a particle with Stokes time longer than the
Kolmogorov time are determined, verifying on an analytical example the
dimensional results of [nlin.CD/0103018]. Expressions for the fluid velocity
Lagrangian correlations and correlation times along a solid particle
trajectory, are provided in several parameter regimes, including the infinite
Stokes time limit corresponding to Eulerian correlations. The concentration
fluctuation spectrum and the non-ergodic properties of a suspension of heavy
particles in a turbulent flow, in the same regime, are analyzed. The
concentration spectrum is predicted to obey, above the scale of eddies with
lifetime equal to the Stokes time, a power law with universal -4/3 exponent,
and to be otherwise independent of the nature of the turbulent flow. A
preference of the solid particle to lie in less energetic regions of the flow
is observed.


title:
Simultaneous separation for the Kowalevski and Goryachev-Chaplygin
  gyrostats

abstract:
  In the special case of zero square integral the Kowalevski gyrostat and
Goryachev-Chaplygin gyrostat share a simple separation of variables originated
from the 4x4 Lax matrix.


title: Semiclassical analysis of Wigner functions

abstract:
  In this work we study the Wigner functions, which are the quantum analogues
of the classical phase space density, and show how a full rigorous
semiclassical scheme for all orders of \hbar can be constructed for them
without referring to the actual coordinate space wavefunctions from which the
Wigner functions are typically calculated. We find such a picture by a careful
analysis around the stationary points of the main quantization equation, and
apply this approach to the harmonic oscillator solving it for all orders of
\hbar.


title: Modeling of Spiking-Bursting Neural Behavior Using Two-Dimensional Map

abstract:
  A simple model that replicates the dynamics of spiking and spiking-bursting
activity of real biological neurons is proposed. The model is a two-dimensional
map which contains one fast and one slow variable. The mechanisms behind
generation of spikes, bursts of spikes, and restructuring of the map behavior
are explained using phase portrait analysis. The dynamics of two coupled maps
which model the behavior of two electrically coupled neurons is discussed.
Synchronization regimes for spiking and bursting activity of these maps are
studied as a function of coupling strength. It is demonstrated that the results
of this model are in agreement with the synchronization of chaotic
spiking-bursting behavior experimentally found in real biological neurons.


title:
Coupling the Sine-Gordon Field Theory to a Mechanical System at the
  Boundary

abstract:
  We describe an integrable system consisting of the sine-Gordon field,
restricted to the half line, and coupled to a non-linear oscillator at the
boundary. By extension of the coupling constant to imaginary values we also
outline the equivalent system for the sinh-Gordon field. We show how Sklyanin's
formalism can be applied to situations with dynamic boundary conditions, and
illustrate the method with the derivation of our example system.


title:
Symbolic computation of exact solutions expressible in hyperbolic and
  elliptic functions for nonlinear PDEs

abstract:
  Algorithms are presented for the tanh- and sech-methods, which lead to
closed-form solutions of nonlinear ordinary and partial differential equations
(ODEs and PDEs). New algorithms are given to find exact polynomial solutions of
ODEs and PDEs in terms of Jacobi's elliptic functions.
  For systems with parameters, the algorithms determine the conditions on the
parameters so that the differential equations admit polynomial solutions in
tanh, sech, combinations thereof, Jacobi's sn or cn functions. Examples
illustrate key steps of the algorithms.
  The new algorithms are implemented in Mathematica. The package
DDESpecialSolutions.m can be used to automatically compute new special
solutions of nonlinear PDEs. Use of the package, implementation issues, scope,
limitations, and future extensions of the software are addressed.
  A survey is given of related algorithms and symbolic software to compute
exact solutions of nonlinear differential equations.


title:
Topology of the real part of hyperelliptic Jacobian associated with the
  periodic Toda lattice

abstract:
  This paper concerns the topology of the isospectral {\it real} manifold of
the ${\mathfrak sl}(N)$ periodic Toda lattice consisting of $2^{N-1}$ different
systems. The solutions of those systems contain blow-ups, and the set of those
singular points defines a devisor of the manifold. Then adding the divisor, the
manifold is compactified as the real part of the $(N-1)$-dimensional Jacobi
variety associated with a hyperelliptic Riemann surface of genus $g=N-1$. We
also study the real structure of the divisor, and then provide conjectures on
the topology of the affine part of the real Jacobian and on the gluing rule
over the divisor to compactify the manifold based upon the sign-representation
of the Weyl group of ${\mathfrak sl}(N)$.


title: Spectral cross correlations of magnetic edge states

abstract:
  We observe strong, non-trivial cross-correlations between the edge states
found in the interior and the exterior of magnetic quantum billiards. Our
analysis is based on a novel definition of the edge state spectral density
which is rigorous, practical and semiclassically accessible.


title: Time Reversal for Waves in Random Media

abstract:
  In time reversal acoustics experiments, a signal is emitted from a localized
source, recorded at an array of receivers-transducers, time reversed, and
finally re-emitted into the medium. A celebrated feature of time reversal
experiments is that the refocusing of the re-emitted signals at the location of
the initial source is improved when the medium is heterogeneous. Contrary to
intuition, multiple scattering enhances the spatial resolution of the refocused
signal and allows one to beat the diffraction limit obtained in homogeneous
media. This paper presents a quantitative explanation of time reversal and
other more general refocusing phenomena for general classical waves in
heterogeneous media. The theory is based on the asymptotic analysis of the
Wigner transform of wave fields in the high frequency limit. Numerical
experiments complement the theory.


title: Magnetic edge states

abstract:
  Magnetic edge states are responsible for various phenomena of
magneto-transport. Their importance is due to the fact that, unlike the bulk of
the eigenstates in a magnetic system, they carry electric current along the
boundary of a confined domain. Edge states can exist both as interior (quantum
dot) and exterior (anti-dot) states. In the present report we develop a
consistent and practical spectral theory for the edge states encountered in
magnetic billiards. It provides an objective definition for the notion of edge
states, is applicable for interior and exterior problems, facilitates efficient
quantization schemes, and forms a convenient starting point for both the
semiclassical description and the statistical analysis. After elaborating these
topics we use the semiclassical spectral theory to uncover nontrivial spectral
correlations between the interior and the exterior edge states. We show that
they are the quantum manifestation of a classical duality between the
trajectories in an interior and an exterior magnetic billiard.


title: Coherent switching of semiconductor resonator solitons

abstract:
  We demonstrate switching on and off of spatial solitons in a semiconductor
microresonator by injection of light coherent with the background illumination.
Evidence results that the formation of the solitons and their switching does
not involve thermal processes.


title: On a Functional Equation of Ruijsenaars

abstract:
  We obtain the general solution of the functional equation introduced by
Ruijsenaars guaranteeing the commutativity of n operators associated with the
quantum Ruijsenaars-Schneider models.


title: Thin front propagation in steady and unsteady cellular flows

abstract:
  Front propagation in two dimensional steady and unsteady cellular flows is
investigated in the limit of very fast reaction and sharp front, i.e., in the
geometrical optics limit. In the steady case, by means of a simplified model,
we provide an analytical approximation for the front speed,
$v_{{\scriptsize{f}}}$, as a function of the stirring intensity, $U$, in good
agreement with the numerical results and, for large $U$, the behavior
$v_{{\scriptsize{f}}}\sim U/\log(U)$ is predicted. The large scale of the
velocity field mainly rules the front speed behavior even in the presence of
smaller scales. In the unsteady (time-periodic) case, the front speed displays
a phase-locking on the flow frequency and, albeit the Lagrangian dynamics is
chaotic, chaos in front dynamics only survives for a transient. Asymptotically
the front evolves periodically and chaos manifests only in the spatially
wrinkled structure of the front.


title:
Conservation laws of generalized higher Burgers and linear evolution
  equations

abstract:
  By the Cole-Hopf transformation, with any linear evolution equation in 1+1
dimensions a generalized Burgers equation is associated. We describe local
conservation laws of these equations. It turns out that any generalized Burgers
equation has only one conservation law, while a linear evolution equation with
constant coefficients has an infinite number of (x,t)-independent conservation
laws iff the equation involves only odd order terms and, therefore, is
bi-Hamiltonian.


title: Freely decaying weak turbulence for sea surface gravity waves

abstract:
  We study numerically the generation of power laws in the framework of weak
turbulence theory for surface gravity waves in deep water. Starting from a
random wave field, we let the system evolve numerically according to the
nonlinear Euler equations for gravity waves in infinitely deep water. In
agreement with the theory of Zakharov and Filonenko, we find the formation of a
power spectrum characterized by a power law of the form of $|{\bf k}|^{-2.5}$.


title: Symmetric Periodic Solutions of the Anisotropic Manev Problem

abstract:
  We consider the Manev Potential in an anisotropic space, i.e., such that the
force acts differently in each direction. Using a generalization of the
  Poincare' continuation method we study the existence of periodic solutions
for weak anisotropy. In particular we find that the symmetric periodic orbits
of the Manev system are perturbed to periodic orbits in the anisotropic
problem.


title: On Relations of Hyperelliptic Weierstrass al Functions

abstract:
  We study relations of the Weierstrass's hyperelliptic al-functions over a
non-degenerated hyperelliptic curve $y^2 = f(x)$ of arbitrary genus $g$ as
solutions of sine-Gordon equation using Weierstrass's local parameters, which
are characterized by two ramified points. Though the hyperelliptic solutions of
the sine-Gordon equation had already obtained, our derivations of them are
simple; they need only residual computations over the curve and primitive
matrix computations.


title: Inverse Cascade Regime in Shell Models of 2-Dimensional Turbulence

abstract:
  We consider shell models that display an inverse energy cascade similar to
2-dimensional turbulence (together with a direct cascade of an enstrophy-like
invariant). Previous attempts to construct such models ended negatively,
stating that shell models give rise to a "quasi-equilibrium" situation with
equipartition of the energy among the shells. We show analytically that the
quasi-equilibrium state predicts its own disappearance upon changing the model
parameters in favor of the establishment of an inverse cascade regime with K41
scaling. The latter regime is found where predicted, offering a useful model to
study inverse cascades.


title:
Nonlinear Optical Vector Ampliude Equations. Polarization and Vortex
  Solutions

abstract:
  We investigate two kind of polarization of localized optical waves in
nonlinear Kerr type media, linear and combination of linear and circular. In
the first case of linear polarized components we obtained the vector version of
3D+1 Nonlinear Schr\"{o}dinger Equation (VNSE). It is shown that these
equations admit exact vortex solutions with spin l=1. We have determined the
dispersion region and the medium parameters necessary for experimental
observation of these vortices. In the second case we represent the electric and
magnetic fields as sum of circular and linear components. We supouse also that
our nonlinear media admit linear magnetic polarization. This allows us to
reduce the Maxwell's equations to a set of amplitude Nonlinear Dirac Equations
(NDE). WE find two representation of the NDE- spherical and spinor. In the
spherical representation we obtain exact vortex solutions with spin l-1 and in
the spinor representation, exact vortex solutions with spin j=1/2.


title: Dissipative chaotic scattering

abstract:
  We show that weak dissipation, typical in realistic situations, can have a
metamorphic consequence on nonhyperbolic chaotic scattering in the sense that
the physically important particle-decay law is altered, no matter how small the
amount of dissipation. As a result, the previous conclusion about the unity of
the fractal dimension of the set of singularities in scattering functions, a
major claim about nonhyperbolic chaotic scattering, may not be observable.


title: Lie point symmetries and first integrals: the Kowalevsky top

abstract:
  We show how the Lie group analysis method can be used in order to obtain
first integrals of any system of ordinary differential equations.
  The method of reduction/increase of order developed by Nucci (J. Math. Phys.
37, 1772-1775 (1996)) is essential. Noether's theorem is neither necessary nor
considered. The most striking example we present is the relationship between
Lie group analysis and the famous first integral of the Kowalevski top.


title:
Statistical Analysis of a Semilinear Hyperbolic System Advected by a
  White in Time Random Velocity Field

abstract:
  We study a system of semilinear hyperbolic equations passively advected by
smooth white noise in time random velocity fields. Such a system arises in
modeling non-premixed isothermal turbulent flames under single-step kinetics of
fuel and oxidizer. We derive closed equations for one-point and multi-point
probability distribution functions (PDFs) and closed form analytical formulas
for the one point PDF function, as well as the two-point PDF function under
homogeneity and isotropy. Exact solution formulas allows us to analyze the
ensemble averaged fuel/oxidizer concentrations and the motion of their level
curves. We recover the empirical formulas of combustion in the thin reaction
zone limit and show that these approximate formulas can either underestimate or
overestimate average concentrations when reaction zone is not tending to zero.
We show that the averaged reaction rate slows down locally in space due to
random advection induced diffusion; and that the level curves of ensemble
averaged concentration undergo diffusion about mean locations.