title: Determining functionals for random partial differential equations

abstract:
  Determining functionals are tools to describe the finite dimensional
long-term dynamics of infinite dimensional dynamical systems. There also exist
several applications to infinite dimensional {\em random} dynamical systems. In
these applications the convergence condition of the trajectories of an infinite
dimensional random dynamical system with respect to a finite set of linear
functionals is assumed to be either in mean or exponential with respect to the
convergence almost surely. In contrast to these ideas we introduce a
convergence concept which is based on the convergence in probability. By this
ansatz we get rid of the assumption of exponential convergence. In addition,
setting the random terms to zero we obtain usual deterministic results. We
apply our results to the 2D Navier - Stokes equations forced by a white noise.


title: On the Violations of Local Equilibrium and Linear Response

abstract:
  We study how local equilibrium, and linear response predictions of transport
coefficients are violated as systems move far from equilibrium. This is done by
studying heat flow in classical lattice models with and without bulk transport
behavior, in 1--3 dimensions. We see that linear response and local equilibrium
assumptions break down at the same rate. The equation of state is also found to
develop non-local corrections in the steady state. We quantify the breakdown
through the analysis of both microscopic and macroscopic observables, which are
found to display non-trivial size dependence.


title: Topoi of Emergence. Foundations and Applications

abstract:
  We discuss aspects of emergence by introducing the concept of negator algebra
referring to topos-theoretic results obtained by Vladimir Trifonov some time
ago. As a possible application of this concept we present an example from
non-linear economic processes concerning the evolution of the global crude oil
market during the second half of the 20th century.


title: Basic Aspects of Negator Algebra in SOC

abstract:
  Recent developments in loop quantum gravity and topological quantum field
theory are being mirrored with a view to the emergent structure of self-
organized criticality (SOC). Referring back to an earlier paper [nlin.AO/
0105064], the relationship of SOC to negator algebra is discussed. It is shown
that introducing the categorial perspective leads to further holistic
conclusions of considerable universality. This present paper shall serve as a
preparation of a concrete research project under way designed to illustrate
this very universality. Hence, it belongs to a series of recent publications
discussing the modern and fruitful interaction between philosophy and the
sciences, beyond mere historical aspects and the traditional rephrasing of
well-known scientific results.


title:
The Zakharov-Shabat spectral problem on the semi-line: Hilbert
  formulation and applications

abstract:
  The inverse spectral transform for the Zakharov-Shabat equation on the
semi-line is reconsidered as a Hilbert problem. The boundary data induce an
essential singularity at large k to one of the basic solutions. Then solving
the inverse problem means solving a Hilbert problem with particular prescribed
behavior. It is demonstrated that the direct and inverse problems are solved in
a consistent way as soon as the spectral transform vanishes with 1/k at
infinity in the whole upper half plane (where it may possess single poles) and
is continuous and bounded on the real k-axis. The method is applied to
stimulated Raman scattering and sine-Gordon (light cone) for which it is
demonstrated that time evolution conserves the properties of the spectral
transform.


title: Tunneling Mechanism due to Chaos in a Complex Phase Space

abstract:
  We have revealed that the barrier-tunneling process in non-integrable systems
is strongly linked to chaos in complex phase space by investigating a simple
scattering map model. The semiclassical wavefunction reproduces complicated
features of tunneling perfectly and it enables us to solve all the reasons why
those features appear in spite of absence of chaos on the real plane.
Multi-generation structure of manifolds, which is the manifestation of
complex-domain homoclinic entanglement created by complexified classical
dynamics, allows a symbolic coding and it is used as a guiding principle to
extract dominant complex trajectories from all the semiclassical candidates.


title:
Equivalence between isospectrality and iso-length spectrality of a
  certain class of planar billiard domains

abstract:
  Isospectrality of planar domains which are obtained by successive unfolding
of a fundamental building block is studied in relation to iso-length
spectrality of the corresponding domains. Although an explicit and exact trace
formula such as Poisson's summation formula or Selberg's trace formula is not
known to exist for planar domains, equivalence between isospectrality and
iso-length spectrality in a certain setting can be proved by employing the
matrix representation of "transplantation of eigenfunctions". As an application
of the equivalence, transplantable pairs of domains, which are all isospectral
pairs of planar domains and therefore counter examples of Kac's question "can
one hear the shape of a drum?", are numerically enumerated and it is found at
least up to the domain composed of 13 building blocks transplantable pairs
coincide with those constructed by the method due to Sunada.


title:
Transverse Instability of Solitons Propagating on Current-Carrying Metal
  Thin Films

abstract:
  Small amplitude, long waves travelling over the surface of a current-carrying
metal thin film are studied. The equation of motion for the metal surface is
determined in the limit of high applied currents, when surface electromigration
is the predominant cause of adatom motion. If the surface height h is
independent of the transverse coordinate y, the equation of motion reduces to
the Korteweg-de Vries equation. One-dimensional solitons (i.e., those with h
independent of y) are shown to be unstable against perturbations to their shape
with small transverse wavevector.


title:
Dispersionless scalar integrable hierarchies, Whitham hierarchy and the
  quasi-classical dbar-dressing method

abstract:
  The quasi-classical limit of the scalar nonlocal dbar-problem is derived and
a quasi-classical version of the dbar-dressing method is presented.
Dispersionless KP, mKP and 2DTL hierarchies are discussed as illustrative
examples. It is shown that the universal Whitham hierarchy it is nothing but
the ring of symmetries for the quasi-classical dbar-problem. The reduction
problem is discussed and, in particular, the d2DTL equation of B type is
derived.


title: Retardation of the onset of yurbulence by minor viscosity contrasts

abstract:
  Motivated by the large effect of turbulent drag reduction by minute
concentrations of polymers we study the effects of minor viscosity contrasts on
the stability of hydrodynamic flows. The key player is a localized region where
the energy of fluctuations is produced by interactions with the mean flow (the
"critical layer"). We show that a layer of weakly space-dependent viscosity
placed near the critical layer can have very large stabilizing effect on
hydrodynamic fluctuations, retarding significantly the onset of turbulence. The
effect is not due to a modified dissipation (as is assumed in theories of drag
reduction), but due to reduced energy intake from the mean flow to the
fluctuations. We propose that similar physics act in turbulent drag reduction.


title: Nonlinear guided waves and spatial solitons in a periodic layered medium

abstract:
  We overview the properties of nonlinear guided waves and (bright and dark)
spatial optical solitons in a periodic medium created by a sequence of linear
and nonlinear layers. First, we consider a single layer with a cubic nonlinear
response (a nonlinear waveguide) embedded into a periodic layered linear
medium, and describe nonlinear localized modes (guided waves and Bragg-like
localized gap modes) and their stability. Then, we study modulational
instability as well as the existence and stability of discrete spatial solitons
in a periodic array of identical nonlinear layers, a one-dimensional nonlinear
photonic crystal. Both similarities and differences with the models described
by the discrete nonlinear Schrodinger equation (derived in the tight-binding
approximation) and coupled-mode theory (valid for the shallow periodic
modulations) are emphasized.


title:
Stable droplets and growth laws close to the modulational instability of
  a domain wall

abstract:
  We consider the curvature driven dynamics of a domain wall separating two
equivalent states in systems displaying a modulational instability of a flat
front. We derive an amplitude equation for the dynamics of the curvature close
to the bifurcation point from growing to schrinking circular droplets. We
predict the existence of stable droplets with a radius $R$ that diverges at the
bifurcation point, where a curvature driven growth law $R(t) \approx t^{1/4}$
is obtained. Our general analytical predictions, which are valid for a wide
variety of systems including models of nonlinear optical cavities and
reaction-diffusion systems, are illustrated in the parametrically driven
complex Ginzburg-Landau equation.


title: Microextensive Chaos of a Spatially Extended System

abstract:
  By analyzing chaotic states of the one-dimensional Kuramoto-Sivashinsky
equation for system sizes L in the range 79 <= L <= 93, we show that the
Lyapunov fractal dimension D scales microextensively, increasing linearly with
L even for increments Delta{L} that are small compared to the average cell size
of 9 and to various correlation lengths. This suggests that a spatially
homogeneous chaotic system does not have to increase its size by some
characteristic amount to increase its dynamical complexity, nor is the increase
in dimension related to the increase in the number of linearly unstable modes.


title:
Spectrum of the Frobenius-Perron operator for systems with stochastic
  perturbation

abstract:
  We analyze the problem of evolution in a system with stochastic perturbation
and point out that analytic properties of the noise present in the system might
determine spectral properties of the evolution operator (Frobenius-Perron
operator). We also propose a method for approximation of the spectrum and
eigenvectors of the FP-operator by applying a suitable noise. Moreover we
demonstrate that the eigenvalues of the FP-operator located outside the
essential spectrum are robust not only against local perturbation but also
against the non local perturbation and that these stable eigenvalues have a
direct physical meaning: they determine the rate of the exponential decay of
correlation in the system.


title: Extreme Wave Events in Directional, Random Oceanic Sea States

abstract:
  We discuss the effects of the directional spreading on the occurrence of
extreme wave events. We numerically integrate the envelope equation recently
proposed by Trulsen et al., Phys of Fluids 2000, as a weakly nonlinear model
for realistic oceanic gravity waves.Initial conditions for numerical
simulations are characterized by the spatial JONSWAP power spectrum for several
values of the significant wave height, steepness and directional spreading. We
show that by increasing the directionality of the initial spectrum the
appearance of extreme events is notably reduced.


title:
Fluctuation formula for nonreversible dynamics in the thermostated
  Lorentz gas

abstract:
  We investigate numerically the validity of the Gallavotti-Cohen fluctuation
formula in the two and three dimensional periodic Lorentz gas subjected to
constant electric and magnetic fields and thermostated by the Gaussian
isokinetic thermostat. The magnetic field breaks the time reversal symmetry,
and by choosing its orientation with respect to the lattice one can have either
a generalized reversing symmetry or no reversibility at all. Our results
indicate that the scaling property described by the fluctuation formula may be
approximately valid for large fluctuations even in the absence of
reversibility.


title: The Ising Model for Changes in Word Ordering Rule in Natural Language

abstract:
  The order of `noun and adposition' is the important parameter of word
ordering rules in the world's languages. The seven parameters, `adverb and
verb' and others, have a strong dependence on the `noun and adposition'.
Japanese as well as Korean, Tamil and several other languages seem to have a
stable structure of word ordering rules, as well as Thai and other languages
which have the opposite word ordering rules to Japanese. It seems that each
language in the world fluctuates between these two structures like the Ising
model for finite lattice.


title: Chaotic time-dependent billiards

abstract:
  A billiard in the form of a stadium with periodically perturbed boundary is
considered. Two types of such billiards are studied: stadium with strong
chaotic properties and a near-rectangle billiard. Phase portraits of such
billiards are investigated. In the phase plane areas corresponding to decrease
and increase of the velocity of billiard particles are found. Average
velocities of the particle ensemble as functions of the number of collisions
are obtained.


title: Hidden Chaos

abstract:
  When a medium composed of microscopic elements is subjected to a high
intensity field, the individual behaviors of microscopic elements can become
chaotic. In such cases it is important to consider the effects of this
irregularity at microscopical level onto the macroscopic behavior of the
medium. We show that the macroscopic field produced by a large group of chaotic
scatterers can remain regular, due to the partial or complete phase coherence
of the scattering elements and the incoherence of the chaotic components of
their responses. Thus when only macroscopic fields are observed, one may be
unaware of chaotic microscopical motion, as it appears to be hidden from the
observer. The coupling among the elements may lead to partial chaos
synchronization, which exposes the chaotic nature of the system making the
oscillations of macroscopic fields more irregular.


title:
Analysis of Hamiltonian Dynamics of Dispersion-Managed Coupled Breathers
  in Optical Transmission System

abstract:
  We theoretically derive the Hamiltonian dynamics of dispersion-managed
coupled breathers in optical transmission system as a result of phases-plane
dynamics of the system. We analytically show that the contribution of a
perturbation term caused by a noise sourced by amplifiers in optical fiber can
influence the amplitudes and chirps of the pulses dynamics. We also find that
the coupled breathers dynamics depent on a certain relationship of the chirps
and amplitudes. The results of the perturbed coupled NLS are then reduced to
that in the unperturbed system and the single NLS case.

categories: nlin.SI cond-mat.soft math-ph math.MP nlin.PS physics.optics


title:
Quantum Chaos of a particle in a square well : Competing Length Scales
  and Dynamical Localization

abstract:
  The classical and quantum dynamics of a particle trapped in a one-dimensional
infinite square well with a time periodic pulsed field is investigated. This is
a two-parameter non-KAM generalization of the kicked rotor, which can be seen
as the standard map of particles subjected to both smooth and hard potentials.
The virtue of the generalization lies in the introduction of an extra parameter
R which is the ratio of two length scales, namely the well width and the field
wavelength. If R is a non-integer the dynamics is discontinuous and non-KAM. We
have explored the role of R in controlling the localization properties of the
eigenstates. In particular the connection between classical diffusion and
localization is found to generalize reasonably well. In unbounded chaotic
systems such as these, while the nearest neighbour spacing distribution of the
eigenvalues is less sensitive to the nature of the classical dynamics, the
distribution of participation ratios of the eigenstates proves to be a
sensitive measure; in the chaotic regimes the latter being lognormal. We find
that the tails of the well converged localized states are exponentially
localized despite the discontinuous dynamics while the bulk part shows
fluctuations that tend to be closer to Random Matrix Theory predictions. Time
evolving states show considerable R dependence and tuning R to enhance
classical diffusion can lead to significantly larger quantum diffusion for the
same field strengths, an effect that is potentially observable in present day
experiments.


title: Fluctuation and Dissipation in Liquid Crystal Electroconvection

abstract:
  In this experiment a steady state current is maintained through a liquid
crystal thin film. When the applied voltage is increased through a threshold, a
phase transition is observed into a convective state characterized by the
chaotic motion of rolls. Above the threshold, an increase in power consumption
is observed that is manifested by an increase in the mean conductivity. A sharp
increase in the ratio of the power fluctuations to the mean power dissipated is
observed above the transition. This ratio is compared to the predictions of the
fluctuation theorem of Gallavotti and Cohen using an effective temperature
associated with the rolls' chaotic motion.


title:
Second moment of the Husimi distribution as a measure of complexity of
  quantum states

abstract:
  We propose the second moment of the Husimi distribution as a measure of
complexity of quantum states. The inverse of this quantity represents the
effective volume in phase space occupied by the Husimi distribution, and has a
good correspondence with chaoticity of classical system. Its properties are
similar to the classical entropy proposed by Wehrl, but it is much easier to
calculate numerically. We calculate this quantity in the quartic oscillator
model, and show that it works well as a measure of chaoticity of quantum
states.


title: Multi-valued mappings in generalized chaos synchronization

abstract:
  The onset of generalized synchronization of chaos in directionally-coupled
systems corresponds to the formation of a continuous mapping which enables one
to persistently define the state of the response system from the trajectory of
the drive system. The recently developed theory of generalized synchronization
of chaos deals only with the case where this synchronization mapping is a
single-valued function. In this paper, we explore generalized synchronization
in a regime where the synchronization mapping can become a multi-valued
function. Specifically, we study the properties of the multi-valued mapping
which occurs between the drive and response systems when the systems are
synchronized with a frequency ratio other than one-to-one, and address the
issues of the existence and continuity of such mappings. The basic theoretical
framework underlying the considered synchronization regimes is then developed.


title: Molecules in external fields: a semiclassical analysis

abstract:
  We undertake a semiclassical analysis of the spectral properties (modulations
of photoabsorption spectra, energy level statistics) of a simple Rydberg
molecule in static fields within the framework of Closed-Orbit/Periodic-Orbit
theories. We conclude that in addition to the usual classically allowed orbits
one must consider classically forbidden diffractive paths. Further, the
molecule brings in a new type of 'inelastic' diffractive trajectory, different
from the usual 'elastic' diffractive orbits encountered in previous studies of
atomic and analogous systems such as billiards with point-scatterers. The
relative importance of inelastic versus elastic diffraction is quantified by
merging the usual Closed Orbit theory framework with molecular quantum defect
theory.


title: Structures in BC_N Ruijsenaars-Schneider models

abstract:
  We construct the classical r-matrix structure for the Lax formulation of BC_N
Ruijsenaars-Schneider systems proposed in hep-th 0006004. The r-matrix
structure takes a quadratic form similar to the A_N Ruijsenaars-Schneider
Poisson bracket behavior, although the dynamical dependence is more
complicated. Commuting Hamiltonians stemming from the BC_N
Ruijsenaars-Schneider Lax matrix are shown to be linear combinations of
particular Koornwinder-van Diejen ``external fields'' Ruijsenaars-Schneider
models, for specific values of the exponential one-body couplings. Uniqueness
of such commuting Hamiltonians is established once the first of them and the
general analytic structure are given.


title: Power fluctuations in a driven damped chaotic pendulum

abstract:
  In this paper we investigate the power fluctuations in a driven, dampted
pendulum. When the motion of the pendulum is chaotic, the average power
supplied by the driving force is equal to the average dissipated power only for
an infinite long time period. We measure the fluctuations of the supplied power
during a time equal to the period of the driving force. Negative power
fluctuations occur and we estimate their probability. In a chaotic state the
histogram of the power distribution is broad and continuous although bounded.
For a value of the power not too close to the edge of the distribution the
Fluctuation Theorem of Gallavotti and Cohen is approximately satisfied.


title: Amplitude distribution of eigenfunctions in mixed systems

abstract:
  We study the amplitude distribution of irregular eigenfunctions in systems
with mixed classical phase space. For an appropriately restricted random wave
model a theoretical prediction for the amplitude distribution is derived and
good agreement with numerical computations for the family of limacon billiards
is found. The natural extension of our result to more general systems, e.g.
with a potential, is also discussed.


title: Autocorrelation function of eigenstates in chaotic and mixed systems

abstract:
  We study the autocorrelation function of different types of eigenfunctions in
quantum mechanical systems with either chaotic or mixed classical limits. We
obtain an expansion of the autocorrelation function in terms of the correlation
length. For localized states, like bouncing ball modes or states living on
tori, a simple model using only classical input gives good agreement with the
exact result. In particular, a prediction for irregular eigenfunctions in mixed
systems is derived and tested. For chaotic systems, the expansion of the
autocorrelation function can be used to test quantum ergodicity on different
length scales.


title:
Hexagon Formation as a Compromise Between Squares and Stripes in
  Vertically Vibrated Granular Layers

abstract:
  We propose a spatially continuous and temporally discrete model for pattern
formation in vertically vibrated granular layers. The grain transfer and the
grain mobility transitions are introduced qualitatively, but explicitly. This
model reproduces experimentally observed f/2 patterns, including squares in
high-mobility conditions and stripes in low-mobility conditions. Hexagons are
obtained when a high-mobility condition and a low-mobility condition are
alternately applied. We discuss this hexagon formation in connection with the
formation of squares and stripes. In addition to the experimentally observed
localized oscillon structures, this model also exhibits localized hexagonal
structures. These localized hexagonal structures grow by nucleating oscillons
of equal phase.


title: Special integrals of motion in quantum integrable systems

abstract:
  We investigate quantum integrals of motion in the sine-Gordon, Zhiber-Shabat
and similar systems. When the coupling constants in these models take special
values a new quantum symmetry appears. In those cases, correlation functions
can be obtained, and they have a power law behavior.


title: Do Chaotic Trajectories Care About Self-Similarity?

abstract:
  We investigate the relation between the chaotic dynamics and the hierarchical
phase-space structure of generic Hamiltonian systems. We demonstrate that even
in ideal situations when the phase space is dominated by an exactly
self-similar structure, the long-time dynamics is {\it not} dominated by this
structure. This has consequences for the power-law decay of correlations and
Poincar\'e recurrences.


title: Taming chaos by impurities in two-dimensional oscillator arrays

abstract:
  The effect of impurities in a two-dimensional lattice of coupled nonlinear
chaotic oscillators and their ability to control the dynamical behavior of the
system are studied. We show that a single impurity can produce synchronized
spatio-temporal patterns, even though all oscillators and the impurity are
chaotic when uncoupled. When a small number of impurities is arranged in a way,
that the lattice is divided into two disjoint parts, synchronization is
enforced even for small coupling. The synchronization is not affected as the
size of the lattice increases, although the impurity concentration tends to
zero.


title:
Calculation of the anomalous exponents in the rapid-change model of
  passive scalar advection to order $\varepsilon^{3}$

abstract:
  The field theoretic renormalization group and operator product expansion are
applied to the model of a passive scalar advected by the Gaussian velocity
field with zero mean and correlation function $\propto\delta(t-t')/k^{d+\eps}$.
Inertial-range anomalous exponents, identified with the critical dimensions of
various scalar and tensor composite operators constructed of the scalar
gradients, are calculated within the $\varepsilon$ expansion to order
$\varepsilon^{3}$ (three-loop approximation), including the exponents in
anisotropic sectors. The main goal of the paper is to give the complete
derivation of this third-order result, and to present and explain in detail the
corresponding calculational techniques. The character and convergence
properties of the $\varepsilon$ expansion are discussed; the improved
``inverse'' $\varepsilon$ expansion is proposed and the comparison with the
existing nonperturbative results is given.


title: On the Dubrovin Equations for the Finite-gap Potentials

abstract:
  The general technique of derivation of Dubrovin's equation for the arbitrary
operator pencils is suggested. The question of unique recovering of the
finite-gap potential by coordinates of zeroes of the Psi-function is discussed.
The crucial result of the paper is an autonomous form of Dubrovin's equations
and new trace-formulas for nontrivial spectral problems of the 3-rd order with
trigonal algebraic curve. We show only demonstrative examples, the method is
spread into arbitrary spectral problem including matrix ones.


title: Chaotic vibrations in a regenerative cutting process

abstract:
  We have analyzed vibrations generated in an orthogonal cutting process. Using
a simple one degree of freedom model of the regenerative cutting we have
observed the complex behaviour of the system. In presence of a shaped cutting
surface, the nonlinear interaction between the tool and a worpiece leads to
chatter vibrations of periodic, quasi-periodic or chaotic type depending on
system parameters. To describe the profile of the surface machined by the first
pass we used a harmonic function. We analysed the impact phenomenon between the
tool and a workpiece after their contact loss.


title: Nonlinear parametric instability in double-well lattices

abstract:
  A possibility of a nonlinear resonant instability of uniform oscillations in
dynamical lattices with harmonic intersite coupling and onsite nonlinearity is
predicted. Numerical simulations of a lattice with a double-well onsite
anharmonic potential confirm the existence of the nonlinear instability with an
anomalous value of the corresponding power index, 1.57, which is intermediate
between the values 1 and 2 characterizing the linear and nonlinear (quadratic)
instabilities. The anomalous power index may be a result of competition between
the resonant quadratic instability and nonresonant linear instabilities. The
observed instability triggers transition of the lattice into a chaotic
dynamical state.


title: Using Immunology Principles for Anomaly Detection in Electrical Systems

abstract:
  The immune system is a cognitive system of complexity comparable to the brain
and its computational algorithms suggest new solutions to engineering problems
or new ways of looking at these problems. Using immunological principles, a two
(or three-) module algorithm is developed which is capable of launching a
specific response to an anomalous situation. Applications are being developed
for electromechanical drives and network power transformers. Experimental
results illustrate an application to fault detection in squirrel-cage electric
motors.


title: Introduction to Spin and Lattice Models in the Social Sciences

abstract:
  In recent years, political economists, financial economists, and other social
scientists have introduced spin and lattice models into their theoretical tool
kit. To this end, the modeling skills of hard scientists may be of assistance.
This lecture introduces examples of how these models are used. A simple
dynamical model of how legal rules evolve and propagate in the courts is
described.

categories: nlin.AO cond-mat.stat-mech hep-lat nlin.CG nlin.PS physics.bio-ph physics.pop-ph physics.soc-ph


title:
Discrete dynamical systems with $W(A^{(1)}_{m-1} \times A^{(1)}_{n-1})$
  symmetry

abstract:
  We give a birational realization of affine Weyl group of type $A^{(1)}_{m-1}
\times A^{(1)}_{n-1}$. We apply this representation to construct some discrete
integrable systems and discrete Painlev\'e equations. Our construction has a
combinatorial counterpart through the ultra-discretization procedure.


title: Perron-Frobenius spectrum for random maps and its approximation

abstract:
  To study the convergence to equilibrium in random maps we developed the
spectral theory of the corresponding transfer (Perron-Frobenius) operators
acting in a certain Banach space of generalized functions. The random maps
under study in a sense fill the gap between expanding and hyperbolic systems
since among their (deterministic) components there are both expanding and
contracting ones. We prove stochastic stability of the Perron-Frobenius
spectrum and developed its finite rank operator approximations by means of a
``stochastically smoothed'' Ulam approximation scheme. A counterexample to the
original Ulam conjecture about the approximation of the SBR measure and the
discussion of the instability of spectral approximations by means of the
original Ulam scheme are presented as well.


title:
Inverse scattering theory for the perturbed 1-soliton potential of the
  heat equation

abstract:
  Inverse scattering transform method of the heat equation is developed for a
special subclass of potentials nondecaying at space infinity---perturbations of
the one-soliton potential by means of decaying two-dimensional functions.
Extended resolvent, Green's functions, and Jost solutions are introduced and
their properties are investigated in detail. The singularity structure of the
spectral data is given and then the Inverse problem is formulated in an exact
distributional sense.


title: Elliptic Solitons with Free Constants and their Isospectral Deformations

abstract:
  New exact solvable elliptic potentials with free constants for the spectral
problems of the third order are found. A time dependence of such potentials
gives their isospectral deformations and solutions of nonlinear integrable
equations.


title: Symplectic Integration of Hamiltonian Systems using Polynomial Maps

abstract:
  In order to perform numerical studies of long-term stability in nonlinear
Hamiltonian systems, one needs a numerical integration algorithm which is
symplectic. Further, this algorithm should be fast and accurate. In this paper,
we propose such a symplectic integration algorithm using polynomial map
refactorization of the symplectic map representing the Hamiltonian system. This
method should be particularly useful in long-term stability studies of particle
storage rings in accelerators.


title:
Pulse Bifurcations and Instabilities in an Excitable Medium:
  Computations in Finite Ring Domains

abstract:
  We investigate the instabilities and bifurcations of traveling pulses in a
model excitable medium; in particular we discuss three different scenarios for
the loss of stability resp. the disappearance of stable pulses. In numerical
simulations beyond the instabilities we observe replication of pulses
(backfiring) resulting in complex periodic or spatiotemporally chaotic dynamics
as well as modulated traveling pulses. We approximate the linear stability of
traveling pulses through computations in a finite albeit large domain with
periodic boundary conditions. The critical eigenmodes at the onset of the
instabilities are related to the resulting spatiotemporal dynamics and act upon
the back of the pulses. The first scenario has been analyzed earlier for high
excitability resp. low excitation threshold: it involves the collision of a
stable pulse branch with an unstable pulse branch in a so called T-point.


title:
Near action degeneracy of periodic orbits in systems with
  non-conventional time reversal

abstract:
  Recently, Sieber and Richter calculated semiclassically a first off-diagonal
contribution to the orthogonal form factor for a billiard on a surface of
constant negative curvature by considering orbit pairs having almost the same
action. For a generalization of this derivation to systems invariant under
non-conventional time reversal symmetry, which also belong to the orthogonal
symmetry class, we show in this paper that it is necessary to redefine the
configuration space in an appropriate way.


title: A semiquantal approach to finite systems of interacting particles

abstract:
  A novel approach is suggested for the statistical description of quantum
systems of interacting particles. The key point of this approach is that a
typical eigenstate in the energy representation (shape of eigenstates, SE) has
a well defined classical analog which can be easily obtained from the classical
equations of motion. Therefore, the occupation numbers for single-particle
states can be represented as a convolution of the classical SE with the quantum
occupation number operator for non-interacting particles. The latter takes into
account the wavefunctions symmetry and depends on the unperturbed energy
spectrum only. As a result, the distribution of occupation numbers $n_s$ can be
numerically found for a very large number of interacting particles. Using the
model of interacting spins we demonstrate that this approach gives a correct
description of $n_s$ even in a deep quantum region with few single-particle
orbitals.


title: A pseudointegrable Andreev billiard

abstract:
  A circular Andreev billiard in a uniform magnetic field is studied. It is
demonstrated that the classical dynamics is pseudointegrable in the same sense
as for rational polygonal billiards. The relation to a specific polygon, the
asymmetric barrier billiard, is discussed. Numerical evidence is presented
indicating that the Poincare map is typically weak mixing on the invariant
sets. This link between these different classes of dynamical systems throws
some light on the proximity effect in chaotic Andreev billiards.


title: Dimension-splitting for simplifying diffusion in lattice-gas models

abstract:
  We introduce a simplified technique for incorporating diffusive phenomena
into lattice-gas molecular dynamics models. In this method, spatial
interactions take place one dimension at a time, with a separate fractional
timestep devoted to each dimension, and with all dimensions treated
identically. We show that the model resulting from this technique is equivalent
to the macroscopic diffusion equation in the appropriate limit.
  This technique saves computational resources and reduces the complexity of
model design, programming, debugging, simulation and analysis. For example, a
reaction-diffusion simulation can be designed and tested as a one-dimensional
system, and then directly extended to two or more dimensions. We illustrate the
use of this approach in constructing a microscopically reversible model of
diffusion limited aggregation as well as in a model of growth of biological
films.


title: Resonances of the cusp family

abstract:
  We study a family of chaotic maps with limit cases the tent map and the cusp
map (the cusp family). We discuss the spectral properties of the corresponding
Frobenius--Perron operator in different function spaces including spaces of
analytic functions. A numerical study of the eigenvalues and eigenfunctions is
performed.


title:
Invariant description of solutions of hydrodynamic type systems in
  hodograph space: hydrodynamic surfaces

abstract:
  Hydrodynamic surfaces are solutions of hydrodynamic type systems viewed as
non-parametrized submanifolds of the hodograph space. We propose an invariant
differential-geometric characterization of hydrodynamic surfaces by expressing
the curvature form of the characteristic web in terms of the reciprocal
invariants.


title:
Penentration of dynamic localized states in DC-driven Josephson junction
  ladders by discrete jumps

abstract:
  We give a theoretical study of unusual resistive (dynamic) localized states
in anisotropic Josephson junction ladders, driven by a DC current at one edge.
These states comprise nonlinearly coupled rotating Josephson phases in adjacent
cells, and with increasing current they are found to expand into neighboring
cells by a sequence of sudden jumps. We argue that the jumps arise from
instabilities in the ladder's superconducting part, and our analytic
expressions for the peculiar voltage (rotational frequency) ratios and I-V
curves are in very good agreement with direct numerical simulations.


title:
Parabolic maps with spin: Generic spectral statistics with non-mixing
  classical limit

abstract:
  We investigate quantised maps of the torus whose classical analogues are
ergodic but not mixing. Their quantum spectral statistics shows non-generic
behaviour, i.e.it does not follow random matrix theory (RMT). By coupling the
map to a spin 1/2, which corresponds to changing the quantisation without
altering the classical limit of the dynamics on the torus, we numerically
observe a transition to RMT statistics. The results are interpreted in terms of
semiclassical trace formulae for the maps with and without spin respectively.
We thus have constructed quantum systems with non-mixing classical limit which
show generic (i.e. RMT) spectral statistics. We also discuss the analogous
situation for an almost integrable map, where we compare to Semi-Poissonian
statistics.


title: Stochastic magnetohydrodynamic turbulence in space dimensions $d\ge 2$

abstract:
  Interplay of kinematic and magnetic forcing in a model of a conducting fluid
with randomly driven magnetohydrodynamic equations has been studied in space
dimensions $d\ge 2$ by means of the renormalization group. A perturbative
expansion scheme, parameters of which are the deviation of the spatial
dimension from two and the deviation of the exponent of the powerlike
correlation function of random forcing from its critical value, has been used
in one-loop approximation. Additional divergences have been taken into account
which arise at two dimensions and have been inconsistently treated in earlier
investigations of the model. It is shown that in spite of the additional
divergences the kinetic fixed point associated with the Kolmogorov scaling
regime remains stable for all space dimensions $d\ge 2$ for rapidly enough
falling off correlations of the magnetic forcing. A scaling regime driven by
thermal fluctuations of the velocity field has been identified and analyzed.
The absence of a scaling regime near two dimensions driven by the fluctuations
of the magnetic field has been confirmed. A new renormalization scheme has been
put forward and numerically investigated to interpolate between the $\epsilon$
expansion and the double expansion.


title: On quantization of the KdV equation

abstract:
  Quantization procedure of the Gardner-Zakharov-Faddeev and Magri brackets by
means of the fermionic representation for the KdV field is considered. It is
shown that in both cases the corresponding Hamiltonians are given as sums of
two well defined operators. Each of them is bilinear and diagonal with respect
to either fermion, or boson (current) creation-annihilation operators. As a
result the quantization procedure needs no any space cut-off and can be
performed on the whole axis. Existence of the solitonic states in the Hilbert
space as well as quantization of soliton parameters are shown to result from
this approach. As a by-product it is also demonstrated that the dispersionless
KdV is uniquely and explicitly solvable in the quantum case.


title:
Time-delayed Spatial Patterns in a Two-dimensional Array of Coupled
  Oscillators

abstract:
  We investigated the effect of time delays on phase configurations in a set of
two-dimensional coupled phase oscillators. Each oscillator is allowed to
interact with its neighbors located within a finite radius, which serves as a
control parameter in this study. It is found that distance-dependent
time-delays induce various patterns including traveling rolls, square-like and
rhombus-like patterns, spirals, and targets. We analyzed the stability
boundaries of the emerging patterns and briefly pointed out the possible
empirical implications of such time-delayed patterns.


title:
On the semiclassical theory for universal transmission fluctuations in
  chaotic systems: the importance of unitarity

abstract:
  The standard semiclassical calculation of transmission correlation functions
for chaotic systems is severely influenced by unitarity problems. We show that
unitarity alone imposes a set of relationships between cross sections
correlation functions which go beyond the diagonal approximation. When these
relationships are properly used to supplement the semiclassical scheme we
obtain transmission correlation functions in full agreement with the exact
statistical theory and the experiment. Our approach also provides a novel
prediction for the transmission correlations in the case where time reversal
symmetry is present.


title:
Reciprocal figures, graphical statics and inversive geometry of the
  Schwarzian BKP hierarchy

abstract:
  A remarkable connection between soliton theory and an important and beautiful
branch of the theory of graphical statics developed by Maxwell and his
contemporaries is revealed. Thus, it is demonstrated that reciprocal triangles
which constitute the simplest pair of reciprocal figures representing both a
framework and a self-stress encapsulate the integrable discrete BKP equation
and its Schwarzian version. The inherent Moebius invariant nature of the
Schwarzian BKP equation is then exploited to define reciprocity in an inversive
geometric setting. Integrable pairs of lattices of non-trivial combinatorics
consisting of reciprocal triangles and their natural generalizations are
discussed. Particular reductions of these BKP lattices are related to the
integrable discrete versions of Darboux's 2+1-dimensional sine-Gordon equation
and the classical Tzitzeica equation of affine geometry. Furthermore, it is
shown that octahedral figures and their hexahedral reciprocals as considered by
Maxwell likewise give rise to discrete integrable systems and associated
integrable lattices.


title: Pattern formation and localization in the forced-damped FPU lattice

abstract:
  We study spatial pattern formation and energy localization in the dynamics of
an anharmonic chain with quadratic and quartic intersite potential subject to
an optical, sinusoidally oscillating field and a weak damping. The
zone-boundary mode is stable and locked to the driving field below a critical
forcing that we determine analytically using an approximate model which
describes mode interactions. Above such a forcing, a standing modulated wave
forms for driving frequencies below the band-edge, while a ``multibreather''
state develops at higher frequencies. Of the former, we give an explicit
approximate analytical expression which compares well with numerical data. At
higher forcing space-time chaotic patterns are observed.


title: Fluctuation-response relation in turbulent systems

abstract:
  We address the problem of measuring time-properties of Response Functions
(Green functions) in Gaussian models (Orszag-McLaughin) and strongly
non-Gaussian models (shell models for turbulence). We introduce the concept of
{\it halving time statistics} to have a statistically stable tool to quantify
the time decay of Response Functions and Generalized Response Functions of high
order. We show numerically that in shell models for three dimensional
turbulence Response Functions are inertial range quantities. This is a strong
indication that the invariant measure describing the shell-velocity
fluctuations is characterized by short range interactions between neighboring
shells.


title: $\hbar \to 0$ in Kicked Harper Model: Reassurances and Surprises

abstract:
  We investigate classical-quantum correspondence for kicked Harper model for
extremely small values of the Planck constant $\hbar$. In the asymmetric case a
pure quantum state shows clear signature of classical diffusive as well as
super diffusive transitions asymptotically independent of $\hbar$. However, for
the symmetric case, the $\hbar$ independent behavior occurs only for
renormalized parameter $\bar{K}=K/(2\hbar)$ with intriguing features such as a
sharp transition from integrable to non-integrable transport at
$\bar{K}=\pi/2$, a series of transitions at multiples of $\pi$ and periodicity
of the transmission probability. These results add new puzzles to the frontiers
of quantum chaos.


title: Nonlinear Aharonov-Bohm scattering by optical vortices

abstract:
  We study linear and nonlinear wave scattering by an optical vortex in a
self-defocusing nonlinear Kerr medium. In the linear case, we find a splitting
of a plane-wave front at the vortex proportional to its circulation, similar to
what occurs in the scattered wave of electrons for the Aharonov-Bohm effect.
For larger wave amplitudes, we study analytically and numerically the
scattering of a dark-soliton stripe (a nonlinear analog of a small-amplitude
wavepacket) by a vortex and observe a significant asymmetry of the scattered
wave. Subsequently, a wavefront splitting of the scattered wave develops into
transverse modulational instability, ``unzipping'' the stripe into trains of
vortices with opposite charges.


title: Scarring and the statistics of tunnelling

abstract:
  We show that the statistics of tunnelling can be dramatically affected by
scarring and derive distributions quantifying this effect. Strong deviations
from the prediction of random matrix theory can be explained quantitatively by
modifying the Gaussian distribution which describes wavefunction statistics.
The modified distribution depends on classical parameters which are determined
completely by linearised dynamics around a periodic orbit. This distribution
generalises the scarring theory of Kaplan [Phys. Rev. Lett. {\bf 80}, 2582
(1998)] to describe the statistics of the components of the wavefunction in a
complete basis, rather than overlaps with single Gaussian wavepackets. In
particular it is shown that correlations in the components of the wavefunction
are present, which can strongly influence tunnelling-rate statistics. The
resulting distribution for tunnelling rates is tested successfully on a
two-dimensional double-well potential.


title:
Crisis-induced intermittency due to attractor-widening in a
  buoyancy-driven solar dynamo

abstract:
  In a recent paper [M. A. J. H. Ossendrijver, A&A {\bf 359}, 364 (2000)]
numerical simulations of a 2D mean--field model where shown to produce grand
minima, typical of the long-term behavior of solar magnetic activity. The model
consisted of dynamo that features an $\alpha$ effect based on the buoyancy
instability of magnetic fluxtubes, which gives rise to the switching back and
forth from grand minima to ``regular'' solar behavior. In this Letter, we
report evidence from a time-series analysis of the model for the presence of
crisis--induced intermittency due to attractor--widening. We support this
finding by showing that the average duration of the minima, $<\tau>$, follows
the theoretically predicted scaling $<\tau> \sim
(C_{\delta\alpha}-C_{\delta\alpha}^*)^{-\gamma}$, where $C_{\delta\alpha}$ is
the bifurcation parameter of interest, together with other statistical
evidence. As far as we are aware, this is the first time concrete and detailed
evidence has been produced for the occurrence of this type of crisis--induced
intermittency -- due to attractor widening -- for such dynamo models.


title: Structure functions and energy dissipation dependence on Reynolds number

abstract:
  The dependence of the statistics of energy dissipation on the Reynolds number
is investigated in an experimental jet flow. In a range of about one decade of
$Re_{\lambda}$ (from about 200 to 2000) the adimensional mean energy
dissipation is found to be independent on $Re_{\lambda}$, while the higher
moments of dissipation show a power-law dependence. The scaling exponents are
found to be consistent with a simple prediction based on the multifractal model
for inertial range structure functions. This is an experimental confirmation of
the connection between inertial range quantities and dissipation statistics
predicted by the multifractal approach.


title:
Dynamical Systems approach to Saffman-Taylor fingering. A Dynamical
  Solvability Scenario

abstract:
  A dynamical systems approach to competition of Saffman-Taylor fingers in a
channel is developed. This is based on the global study of the phase space
structure of the low-dimensional ODE's defined by the classes of exact
solutions of the problem without surface tension. Some simple examples are
studied in detail, and general proofs concerning properties of fixed points and
existence of finite-time singularities for broad classes of solutions are
given. The existence of a continuum of multifinger fixed points and its
dynamical implications are discussed. The main conclusion is that exact
zero-surface tension solutions taken in a global sense as families of
trajectories in phase space spanning a sufficiently large set of initial
conditions, are unphysical because the multifinger fixed points are
nonhyperbolic, and an unfolding of them does not exist within the same class of
solutions. Hyperbolicity (saddle-point structure) of the multifinger fixed
points is argued to be essential to the physically correct qualitative
description of finger competition. The restoring of hyperbolicity by surface
tension is discussed as the key point for a generic Dynamical Solvability
Scenario which is proposed for a general context of interfacial pattern
selection.


title: Defect turbulence in inclined layer convection

abstract:
  We report experimental results on the defect turbulent state of undulation
chaos in inclined layer convection of a fluid withPrandtl number $\approx 1$.
By measuring defect density and undulation wavenumber, we find that the onset
of undulation chaos coincides with the theoretically predicted onset for
stable, stationary undulations. At stronger driving, we observe a competition
between ordered undulations and undulation chaos, suggesting bistability
between a fixed-point attractor and spatiotemporal chaos. In the defect
turbulent regime, we measured the defect creation, annihilation, entering,
leaving, and rates. We show that entering and leaving rates through boundaries
must be considered in order to describe the observed statistics. We derive a
universal probability distribution function which agrees with the experimental
findings.


title: Soliton ratchets

abstract:
  The mechanism underlying the soliton ratchet, both in absence and in presence
of noise, is investigated. We show the existence of an asymmetric internal mode
on the soliton profile which couples, trough the damping in the system, to the
soliton translational mode. Effective soliton transport is achieved when the
internal mode and the external force are phase locked. We use as working model
a generalized double sine-Gordon equation. The phenomenon is expected to be
valid for generic soliton systems.


title: Statistical predictability in two-dimensional turbulence

abstract:
  The predictability problem in the inverse energy cascade of two-dimensional
turbulence is addressed by means of high resolution direct numerical
simulations. The analysis is done in terms of the finite size Lyapunov exponent
(FSLE) which is a measure of the growth rate at fixed error level. The
numerical results are compared with classical closure predictions and good
quantitative agreement is found. Finally, it is shown that the inertial range
predictability properties are not affected by the presence of noise induced by
the small, not resolved, scales.


title: Ising Model Scaling Functions at Short Distance

abstract:
  A method for proving the Luther-Peschel formula for the short distance
asymptotics of the Ising model scaling functions is sketched.


title: Short Distance Asymptotics of Ising Correlation

abstract:
  The short distance asymptotics of the Ising Model scaling functions are
computed for the scaling functions that arise as continuum limits of lattice
correlations from below the critical temperature.


title: Integrability and Symmetry Algebra Associated with N=2 KP Flows

abstract:
  We show the complete integrability of N=2 nonstandard KP flows establishing
the biHamiltonian structures. One of Hamiltonian structures is shown to be
isomorphic to the nonlinear N=2 $\hat W_{\infty}$ algebra with the bosonic
sector having $\hat W_{1+\infty}\oplus \hat W_{\infty}$ structure. A consistent
free field representation of the super conformal algebra is obtained. The
bosonic generators are found to be an admixture of free fermions and free
complex bosons, unlike the linear one. The fermionic generators become
exponential in free fields, in general.


title:
Statistically Preserved Structures in Shell Models of Passive Scalar
  Advection

abstract:
  It was conjectured recently that Statiscally Preserved Structures underlie
the statistical physics of turbulent transport processes. We analyze here in
detail the time-dependent (non compact) linear operator that governs the
dynamics of correlation functions in the case of shell models of passive scalar
advection. The problem is generic in the sense that the driving velocity field
is neither Gaussian nor $\delta$-correlated in time. We show how to naturally
discuss the dynamics in terms of an effective compact operator that displays
"zero modes" which determine the anomalous scaling of the correlation
functions. Since shell models have neither Lagrangian structure nor "shape
dynamics" this example differs significantly from standard passive scalar
advection. Nevertheless with the necessary modifications the generality and
efficacy of the concept of Statistically Preserved Structures are further
exemplified. In passing we point out a bonus of the present approach, in
providing analytic predictions for the time-dependent correlation functions in
decaying turbulent transport.


title:
Application of Chaos Induced Near-Resonance Dynamics to Locate the
  Global Optimum of Functions

abstract:
  The problem of locating the global optimum of functions is studied in a
dynamic setting. The dynamics of simple multistable systems under the influence
of chaotic forcing is investigated. When the magnitude of the forcing signal
decays slowly, it is shown that the system attains an equilibrium state, which
corresponds to the global optimum of the corresponding multimodal potential
function. The role of bifurcations in facilitating the approach to the global
optimum is discussed.


title: Pattern formation at the bi-critical point of the Faraday instability

abstract:
  We present measurements on parametrically driven surface waves (Faraday
waves) performed in the vicinity of a bi-critical point in parameter space,
where modes with harmonic and subharmonic time dependence interact. The primary
patterns are squares in the subharmonic and hexagons in the harmonic regime. If
the primary instability is harmonic we observe a hysteretic secondary
transition from hexagons to squares without a perceptible variation of the
fundamental wavelength. The transition is understood in terms of a set of
coupled Landau equations and related to other canonical examples of phase
transitions in nonlinear dissipative systems. Moreover, the
subharmonic-harmonic mode competition gives rise to a variety of new
superlattice states. These structures are interpreted as mediator modes
involved in the transition between patterns of fourfold and sixfold rotational
symmetry.


title:
Bilinearization of Supersymmetric KP Hierarchies Associated with
  Non-Trivial Flows

abstract:
  We show that the supersymmetric KdV and KP equations, related to the
non-trivial flows, can be cast in the Hirota bilinear form. The existence of
one, two and subsequently $N$-soliton solutions is explicitly demonstrated.


title:
Classical versus Quantum Time Evolution of Densities at Limited
  Phase-Space Resolution

abstract:
  We study the interrelations between the classical (Frobenius-Perron) and the
quantum (Husimi) propagator for phase-space (quasi-)probability densities in a
Hamiltonian system displaying a mix of regular and chaotic behavior. We focus
on common resonances of these operators which we determine by blurring
phase-space resolution. We demonstrate that classical and quantum time
evolution look alike if observed with a resolution much coarser than a Planck
cell and explain how this similarity arises for the propagators as well as
their spectra. The indistinguishability of blurred quantum and classical
evolution implies that classical resonances can conveniently be determined from
quantum mechanics and in turn become effective for decay rates of quantum
correlations.


title:
Symmetry, singularities and integrability in complex dynamics V:
  Complete symmetry groups of certain relativistic spherically symmetric
  systems

abstract:
  We show that the concept of complete symmetry group introduced by Krause (J.
Math. Phys.35 (1994) 5734-5748) in the context of the Newtonian Kepler problem
has wider applicability, extending to the relativistic context of the Einstein
equations describing spherically symmetric bodies with certain conformal
Killing symmetries. We also provide a simple demonstration of the nonuniqueness
of the complete symmetry group.


title:
Perturbation theory for Lyapunov exponents of a toral map: extension of
  a result of Shub and Wilkinson

abstract:
  Starting from a hyperbolic toral automorphism, we obtain, for a small volume
preserving perturbation, an exact and rigorous second order perturbation
expansion of the Lyapunov exponents.


title: Pseudofractals and box counting algorithm

abstract:
  We show that for sets with the Hausdorff--Besicovitch dimension equal zero
the box counting algorithm commonly used to calculate Renyi exponents ($d_q$)
can exhibit perfect scaling suggesting non zero $d_q$'s. Properties of these
pathological sets ({\it pseudofractals}) are investigated. Numerical, as well
as analytical estimates for $d_q$'s are obtained. A simple indicator is given
to distinguish pseudofractals and fractals in practical applications of the box
counting method. Histograms made of pseudofractal sets are shown to have Pareto
tails.


title: The Arbitrary Trajectory Quantization Method

abstract:
  The arbitrary trajectory quantization method (ATQM) is a time dependent
approach to quasiclassical quantization based on the approximate dual
relationship that exists between the quantum energy spectra and classical
periodic orbits. It has recently been shown however, that, for polygonal
billiards, the periodicity criterion must be relaxed to include closed
almost-periodic (CAP) orbit families in this relationship. In light of this
result, we reinvestigate the ATQM and show that at finite energies, a
smoothened quasiclassical kernel corresponds to the modified formula that
includes CAP families while the delta function kernel corresponding to the
periodic orbit formula is recovered at high energies. Several clarifications
are also provided.


title: Semiclassical Inequivalence of Polygonalized Billiards

abstract:
  Polygonalization of any smooth billiard boundary can be carried out in
several ways. We show here that the semiclassical description depends on the
polygonalization process and the results can be inequivalent. We also establish
that generalized tangent-polygons are closest to the corresponding smooth
billiard and for de Broglie wavelengths larger than the average length of the
edges, the two are semiclassically equivalent.


title: Function Dynamics

abstract:
  We show mathematical structure of the function dynamics, i.e., the dynamics
of interval maps $f_{n+1} = (1-\e)f_n + \e f_n\circ f_n$ and clarify the types
of fixed points, the self-referential structure and the hierarchical structure.


title:
(1+1)-dimensional turbulence systems reduced from (2+1)-dimensional Lax
  integrable dispersive long wave equation

abstract:
  After extending the Clarkson-Kruskal's direct similarity reduction ansatz to
a more general form, one may obtain various new types of reduction equations.
Especially, some lower dimensional turbulence systems or chaotic systems may be
obtained from the general type of similarity reductions of a higher dimensional
Lax integrable model. Especially, the Kuramoto-Sivashinsky equation and an
arbitrary three order quasilinear equation which includes the Korteweg de-Vries
Burgers equation and the general Lorenz equation as two special cases are
obtained from the reductions of the (2+1)-dimensional dispersive long wave
equation system. Some types of periodic and chaotic solutions of the
(2+1)-dimensional dispersive long wave equation system are also discussed.


title:
Higher dimensional Lax pairs of lower dimensional chaos and turbulence
  systems

abstract:
  In this letter, a definition of the higher dimensional Lax pair for a lower
dimensional system which may be a chaotic system is given. A special concrete
(2+1)-dimensional Lax pair for a general (1+1)-dimensional three order
autonomous partial differential equation is studied. The result shows that any
(1+1)-dimensional three order semi-linear autonomous system (no matter it is
integrable or not) possesses infinitely many (2+1)-dimensional Lax pairs.
Especially, every solution of the KdV equation and the Harry-Dym equation with
their space variable being replaced by the field variable can be used to obtain
a (2+1)-dimensional Lax pair of any three order (1+1)-dimensional semi-linear
equation.


title: Chaos in soliton systems and special Lax pairs for chaos systems

abstract:
  In this letter, taking the well known (2+1)-dimensional soliton systems,
Davey-Stewartson (DS) model and the asymmetric Nizhnik-Novikov-Veselov (ANNV)
model, as two special examples, we show that some types of lower dimensional
chaotic behaviors may be found in higher dimensional soliton systems.
Especially, we derive the famous Lorenz system and its general form from the DS
equation and the ANNV equation. Some types of chaotic soliton solutions can be
obtained from analytic expression of higher dimensional soliton systems and the
numeric results of lower dimensional chaos systems. On the other hand, by means
of the Lax pairs of some soliton systems, a \em lower \rm dimensional chaos
system may have some types of \em higher \rm dimensional Lax pairs. An explicit
(2+1)-dimensional Lax pair for a (1+1)-dimensional chaotic equation is given.


title: Oscillating Fracture in Rubber

abstract:
  We have found an oscillating instability of fast-running cracks in thin
rubber sheets. A well-defined transition from straight to oscillating cracks
occurs as the amount of biaxial strain increases. Measurements of the amplitude
and wavelength of the oscillation near the onset of this instability indicate
that the instability is a Hopf bifurcation.


title:
$\hbar$ corrections in semi-classical formula for smooth chaotic
  dynamics

abstract:
  The validity of semiclassical expansions in the power of $\hbar$ for the
quantum Green's function have been extensively tested for billiards systems,
but in the case of chaotic dynamics with smooth potential, even if formula are
existing, a quantitative comparison is still missing. In this paper, extending
the theory developed by Gaspard et al., Adv. Chem. Phys. XC 105 (1995), based
on the classical Green's functions, we present an efficient method allowing the
calculation of $\hbar$ corrections for the propagator, the quantum Green's
function, and their traces. Especially, we show that the previously published
expressions for $\hbar$ corrections to the traces are incomplete.


title: Shell Models for Magnetohydrodynamic Turbulence

abstract:
  We review the main properties of shell models for magnetohydrodynamic (MHD)
turbulence. After a brief account on shell models with nearest neighbour
interactions, the paper focuses on the most recent results concerning dynamical
properties and intermittency of a model which is a generalization to MHD of the
Gledzer-Yamada-Okhitani (GOY) model for hydrodynamic. Applications to
astrophysical problems are also discussed.


title:
Scaling Behaviour of the Maximal Growth Rate in the Rosensweig
  Instability

abstract:
  The dependence of the maximal growth rate of the modes of the Rosensweig
instability on the properties of the magnetic fluid and the external magnetic
induction is studied. An expansion and a fit procedure are applied in the
appropriate ranges of the supercritical induction $\hat B$. With increasing
$\hat B$ the scaling of the maximal growth rate changes from linear to a
combination of linear and square-root-like scaling. The scaling of the
corresponding wave number alternates from quadratic to primarily linear. For
very small $\hat B$ the dependence of the maximal growth rate on the viscosity
is given. Suggestions are made for experiments to test the predicted scaling
behaviours.


title:
Solutions of the periodic Toda lattice via the projection procedure and
  by the algebra-geometric method

abstract:
  In this short review we compare different ways to construct solutions of the
periodic Toda lattice. We give two recipes that follow from the projection
method and compare them with the algebra-geometric construction of Krichever.


title: Averaging of an autoresonance model

abstract:
  Autoresonance is a phase locking phenomenon occurring in nonlinear
oscislatory system, which is forced by oscillating perturbation. Many physical
applicatcons of the autoresonance are known in nonlenear physics. The essence
of the phenomecon is that the nonlinear oscillator selfadjusts to the varying
external conditions so that it remains in resonance with the orivvr for a long
time. This long time resonance leads to a strong increase in the response
amplitude under weak drivinc perturbation. An analytic treatment of a simple
mathematical model is done here by means of asymptotic analysis using a small
driving parameter. The main result is finding threshold for entering the
autoresonance.


title: Dynamic Separation of Chaotic Signals in the Presence of Noise

abstract:
  The problem of separation of an observed sum of chaotic signals into the
individual components in the presence of noise on the path to the observer is
considered. A noise threshold is found above which high-quality separation is
impossible. Below the threshold, each signal is recovered with any prescribed
accuracy. This effect is shown to be associated with the information content of
the chaotic signals and a theoretical estimate is given for the threshold.


title: Crashes, Recoveries, and `Core-shifts' in a Model of Evolving Networks

abstract:
  A model of an evolving network of interacting molecular species is shown to
exhibit repeated rounds of crashes in which several species get rapidly
depopulated, followed by recoveries. The network inevitably self-organizes into
an autocatalytic structure, which consists of an irreducible `core' surrounded
by a parasitic `periphery'. Crashes typically occur when the existing
autocatalytic set becomes fragile and suffers a `core-shift', defined graph
theoretically. The nature of the recovery after a crash, in particular the time
of recovery, depends upon the organizational structure that survives the crash.
The largest eigenvalue of the adjacency matrix of the graph is an important
signal of network fragility or robustness.


title:
Large extinctions in an evolutionary model: The role of innovation and
  keystone species

abstract:
  The causes of major and rapid transitions observed in biological
macroevolution as well as in the evolution of social systems are a subject of
much debate. Here we identify the proximate causes of crashes and recoveries
that arise dynamically in a model system in which populations of (molecular)
species co-evolve with their network of chemical interactions. Crashes are
events that involve the rapid extinction of many species and recoveries the
assimilation of new ones. These are analyzed and classified in terms of the
structural properties of the network. We find that in the absence of large
external perturbation, `innovation' is a major cause of large extinctions and
the prime cause of recoveries. Another major cause of crashes is the extinction
of a `keystone species'. Different classes of causes produce crashes of
different characteristic sizes.


title: An Introduction to Quantum Chaos

abstract:
  Nonlinear dynamics (``chaos theory'') and quantum mechanics are two of the
scientific triumphs of the 20th century. The former lies at the heart of the
modern interdisciplinary approach to science, whereas the latter has
revolutionized physics. Both chaos theory and quantum mechanics have achieved a
fairly large level of glamour in the eyes of the general public. The study of
quantum chaos encompasses the application of dynamical systems theory in the
quantum regime. In the present article, we give a brief review of the origin
and fundamentals of both quantum mechanics and nonlinear dynamics. We recount
the birth of dynamical systems theory and contrast chaotic motion with
integrable motion. We similarly recall the transition from classical to quantum
mechanics and discuss the origin of the latter. We then consider the interplay
between nonlinear dynamics and quantum mechanics via a classification and
explanation of the three types of quantum chaos. We include several recent
results in this discussion.


title:
Unidirectional Propagation of an Ultra-Short Electromagnetic Pulse in a
  Resonant Medium with High Frequency Stark Shift

abstract:
  We consider in the unidirectional approximation the propagation of an ultra
short electromagnetic pulse in a resonant medium consisting of molecules
characterized by a transition operator with both diagonal and non-diagonal
matrix elements. We find the zero-curvature representation of the reduced
Maxwell-Bloch equations in the sharp line limit. This can be used to develop
the inverse scattering transform method to solve these equations. Finally we
obtain two types of exact traveling pulse solutions, one with the usual
exponential decay and another with an algebraic decay.


title: Semi-dispersing billiards with an infinite cusp

abstract:
  Let $f: [0, +\infty) \to (0, +\infty)$ be a sufficiently smooth convex
function, vanishing at infinity. Consider the planar domain $Q$ delimited by
the positive $x$-semiaxis, the positive $y$-semiaxis, and the graph of $f$.
  Under certain conditions on $f$, we prove that the billiard flow in $Q$ has a
hyperbolic structure and, for some examples, that it is also ergodic. This is
done using the cross section corresponding to collisions with the dispersing
part of the boundary. The relevant invariant measure for this Poincar\'e
section is infinite, whence the need to surpass the existing results, designed
for finite-measure dynamical systems.


title: Nonadiabatic dynamics in semiquantal physics

abstract:
  Every physical regime is some sort of approximation of reality. One
lesser-known realm that is the semiquantal regime, which may be used to
describe systems with both classical and quantum subcomponents. In the present
review, we discuss nonadiabatic dynamics in the semiquantal regime. Our primary
concern is electronic-nuclear coupling in polyatomic molecules, but we discuss
several other situations as well. We begin our presentation by formulating the
semiquantal approximation in quantum systems with degrees-of-freedom that
evolve at different speeds. We discuss nonadiabatic phenomena, focusing on
their relation to the Born-Oppenheimer approximation. We present several
examples--including Jahn-Teller distortion in molecules and crystals and the
dynamics of solvated electrons, buckyballs, nanotubes, atoms in a resonant
cavity, SQUIDs, quantum particle-spin systems, and micromasers. We also
highlight vibrating quantum billiards as a useful abstraction of semiquantal
dynamics.


title:
Effects of Transport Memory and Nonlinear Damping in a Generalized
  Fisher's Equation

abstract:
  Memory effects in transport require, for their incorporation into reaction
diffusion investigations, a generalization of traditional equations. The
well-known Fisher's equation, which combines diffusion with a logistic
nonlinearity, is generalized to include memory effects and traveling wave
solutions of the equation are found. Comparison is made with alternate
generalization procedures.


title: Golden rule decay versus Lyapunov decay of the quantum Loschmidt echo

abstract:
  The overlap of two wave functions evolving in time with slightly different
Hamiltonians decays exponentially, for perturbation strengths greater than the
level spacing. We present numerical evidence for a dynamical system that the
decay rate is given by the smallest of the Lyapunov exponent of the classical
chaotic dynamics and the level broadening that follows from the golden rule of
quantum mechanics. This limits the range of validity for the
perturbation-strength independent decay rate discovered by Jalabert and
Pastawski [Phys. Rev. Lett. 86, 2490 (2001)].


title: Vibrating quantum billiards on Riemannian manifolds

abstract:
  Quantum billiards provide an excellent forum for the analysis of quantum
chaos. Toward this end, we consider quantum billiards with time-varying
surfaces, which provide an important example of quantum chaos that does not
require the semiclassical ($\hbar \longrightarrow 0$) or high quantum-number
limits. We analyze vibrating quantum billiards using the framework of
Riemannian geometry. First, we derive a theorem detailing necessary conditions
for the existence of chaos in vibrating quantum billiards on Riemannian
manifolds. Numerical observations suggest that these conditions are also
sufficient. We prove the aforementioned theorem in full generality for one
degree-of-freedom boundary vibrations and briefly discuss a generalization to
billiards with two or more degrees-of-vibrations. The requisite conditions are
direct consequences of the separability of the Helmholtz equation in a given
orthogonal coordinate frame, and they arise from orthogonality relations
satisfied by solutions of the Helmholtz equation. We then state and prove a
second theorem that provides a general form for the coupled ordinary
differential equations that describe quantum billiards with one
degree-of-vibration boundaries. This set of equations may be used to illustrate
KAM theory and also provides a simple example of semiquantum chaos. Moreover,
vibrating quantum billiards may be used as models for quantum-well
nanostructures, so this study has both theoretical and practical applications.


title:
Dynamics of Coupling Functions in Globally Coupled Maps: Size,
  Periodicity and Stability of Clusters

abstract:
  It is shown how different globally coupled map systems can be analyzed under
a common framework by focusing on the dynamics of their respective global
coupling functions. We investigate how the functional form of the coupling
determines the formation of clusters in a globally coupled map system and the
resulting periodicity of the global interaction. The allowed distributions of
elements among periodic clusters is also found to depend on the functional form
of the coupling. Through the analogy between globally coupled maps and a single
driven map, the clustering behavior of the former systems can be characterized.
By using this analogy, the dynamics of periodic clusters in systems displaying
a constant global coupling are predicted; and for a particular family of
coupling functions, it is shown that the stability condition of these clustered
states can straightforwardly be derived.


title:
Relation between the solitons of Yang-Mills-Higgs fields in 2+1
  dimensional Minkowski space-time and anti-de Sitter space-time

abstract:
  The Yang-Mills-Higgs-Bogomolny equations in both 2+1 dimensional Minkowski
space-time and 2+1 dimensional anti-de Sitter space-time are known to be
integrable and their soliton solutions have already been obtained. In this
paper we show that there is a natural relation between the Lax pairs and
soliton solutions in these two space-times when the curvature changes from 0 to
-1. The change of the asymptotic behaviors of the solitons are also discussed.


title:
An Elementary Proof of Lyapunov Exponent Pairing for Hard-Sphere Systems
  at Constant Kinetic Energy

abstract:
  The conjugate pairing of Lyapunov exponents for a field-driven system with
smooth inter-particle interaction at constant total kinetic energy was first
proved by Dettmann and Morriss [Phys. Rev. E {\bf 53}, R5545 (1996)] using
simple methods of geometry. Their proof was extended to systems interacting via
hard-core inter-particle potentials by Wojtkowski and Liverani [Comm. Math.
Phys. {\bf 194}, 47 (1998)], using more sophisticated methods. Another, and
somewhat more direct version of the proof for hard-sphere systems has been
provided by Ruelle [J. Stat. Phys. {\bf 95}, 393 (1999)]. However, these
approaches for hard-sphere systems are somewhat difficult to follow. In this
paper, a proof of the pairing of Lyapunov exponents for hard-sphere systems at
constant kinetic energy is presented, based on a very simple explicit geometric
construction, similar to that of Ruelle. Generalizations of this construction
to higher dimensions and arbitrary shapes of scatterers or particles are
trivial. This construction also works for hard-sphere systems in an external
field with a Nos\'e-Hoover thermostat. However, there are situations of
physical interest, where these proofs of conjugate pairing rule for systems
interacting via hard-core inter-particle potentials break down.


title: n-th discrete KP hierarchy

abstract:
  We report an infinite class of discrete hierarchies which naturally
generalize familiar discrete KP one.


title: On the Rational Solutions of q-Painlev\'e V Equation

abstract:
  We give an explicit determinant formula for a class of rational solutions of
a q-analogue of the Painlev\'e V equation. The entries of the determinant are
given by the continuous q-Laguerre polynomials.


title: Quantum Expression of Classical Chaos

abstract:
  Even as we understand for long that the world is quantal and buried in it is
classical dynamics which is chaotic, finding eigenfunctions analytically from
the the Schroedinger equation has turned out to be a near-impossibility. Here,
we discover a class of chaotic quantum systems for which we obtain some
analytically exact eigenfunctions in closed form. This paves way to an exact
classical and quantum mechanical treatment of chaotic systems. Furthermore, we
bring out connections, underlying the discovery, between different areas of
physics and mathematics related to universality in fluctuations observed in a
wide variety of complex quantum systems.


title: Recursion Operator and Rational Lax Representation

abstract:
  We consider equations arising from rational Lax representations. A general
method to construct recursion operators for such equations is given. Several
examples are given, including a degenerate bi-Hamiltonian system with a
recursion operator.


title: Bifurcations in one degree-of-vibration quantum billiards

abstract:
  We classify the local bifurcations of one dov quantum billiards, showing that
only saddle-center bifurcations can occur. We analyze the resulting planar
system when there is no coupling in the superposition state. In so doing, we
also consider the global bifurcation structure. Using a double-well potential
as a representative example, we demonstrate how to locate bifurcations in
parameter space. We also discuss how to approximate the cuspidal loop using
AUTO as well as how to cross it via continuation by detuning the dynamical
system. Moreover, we show that when there is coupling, the resulting
five-dimensional system--though chaotic--has a similar underlying structure.
  We verify numerically that both homoclinic orbits and cusps occur and provide
an outline of an analytical argument for the existence of such homoclinic
orbits. Small perturbations of the system reveal homoclinic tangles that typify
chaotic behavior.


title: A class of integrable lattices and KP hierarchy

abstract:
  We introduce a class of integrable $l$-field first-order lattices together
with corresponding Lax equations. These lattices may be represented as
consistency condition for auxiliary linear systems defined on sequences of
formal dressing operators. This construction provides simple way to build
lattice Miura transformations between one-field lattice and $l$-field ($l\ge
2$) ones. We show that the lattices pertained to above class is in some sense
compatible with KP flows and define the chains of constrained KP Lax operators.


title: The Radially Vibrating Spherical Quantum Billiard

abstract:
  We consider the radially vibrating spherical quantum billiard as a
representative example of vibrating quantum billiards. We derive necessary
conditions for quantum chaos in $d$-term superposition states. These conditions
are symmetry relations corresponding to the relative quantum numbers of
eigenstates considered pairwise. In this discussion, we give special attention
to eigenstates with null angular momentum (for which the aforementioned
condition is automatically satisfied). Moreover, when these necessary
conditions are met, we observe numerically that there always exist parameter
values that correspond to chaotic configurations. We focus our numerical
studies on the ground and first excited states of the radially vibrating
spherical billiard with null angular momentum eigenstates. We observe chaotic
behavior in this configuration, which dispels the common belief that one is
required to pass to the semiclassical ($\hbar \longrightarrow 0$) or high
quantum number limits in order to meaningfully discuss quantum chaos. The
results in the present paper are also of practical import, as the radially
vibrating spherical quantum billiard may be used as a model for the quantum dot
nanostructure, the Fermi accelerating sphere, and intra-nuclear particle
behavior.


title: The Leading Off-Diagonal Correction to the Form Factor of Large Graphs

abstract:
  Using periodic-orbit theory beyond the diagonal approximation we investigate
the form factor, $K(\tau)$, of a generic quantum graph with mixing classical
dynamics and time-reversal symmetry. We calculate the contribution from pairs
of self-intersecting orbits that differ from each other only in the orientation
of a single loop. In the limit of large graphs, these pairs produce a
contribution $-2\tau^2$ to the form factor which agrees with random-matrix
theory.


title: On multifractality and fractional derivatives

abstract:
  It is shown phenomenologically that the fractional derivative $\xi=D^\alpha
u$ of order $\alpha$ of a multifractal function has a power-law tail $\propto
|\xi| ^{-p_\star}$ in its cumulative probability, for a suitable range of
$\alpha$'s. The exponent is determined by the condition $\zeta_{p_\star} =
\alpha p_\star$, where $\zeta_p$ is the exponent of the structure function of
order $p$. A detailed study is made for the case of random multiplicative
processes (Benzi {\it et al.} 1993 Physica D {\bf 65}: 352) which are amenable
to both theory and numerical simulations. Large deviations theory provides a
concrete criterion, which involves the departure from straightness of the
$\zeta_p$ graph, for the presence of power-law tails when there is only a
limited range over which the data possess scaling properties (e.g. because of
the presence of a viscous cutoff). The method is also applied to wind tunnel
data and financial data.

categories: nlin.CD cond-mat.stat-mech math.PR q-fin.ST


title:
Magnetization Reversal in Ferromagnetic Film Through Solitons by
  Electromagnetic Field

abstract:
  We study the reversal of magnetization in an isotropic ferromagnetic film
free from charges by exposing it to a circularly polarized electromagnetic (EM)
field. The magnetization excitations obtained in the form of line and lump
solitons of the completely integrable modified KP-II equation which is derived
using a reductive perturbation method from the set of coupled Landau-Lifshitz
and Maxwell equations. It is observed that when the polarization of the
EM-field is reversed followed by a rotation, for every $ {\pi \over 2}
$-degrees, the magnetization is reversed.


title: On a new approach to optical solitons in dielectric fibers

abstract:
  In this letter we describe a new and simple approach for modulating the
electromagnetic wave propagating in a dielectric medium in the form of solitons
by considering the torque developed between the induced dipoles in the medium
and external field without taking into account the nonlinear Kerr effect. A
reductive perturbation method deduces the Maxwell equation coupled with the
Larmor equation of the torque to the derivative nonlinear Schr\"odinger
equation that admits optical solitons in the medium.


title: Multiple Perron-Frobenius operators

abstract:
  A cycle expansion technique for discrete sums of several PF operators,
similar to the one used in standard classical dynamical zeta-function formalism
is constructed. It is shown that the corresponding expansion coefficients show
an interesting universal behavior, which illustrates the details of the
interference between the particlar mappings entering the sum.


title:
Relative dispersion in fully developed turbulence: The Richardson's Law
  and Intermittency Corrections

abstract:
  Relative dispersion in fully developed turbulence is investigated by means of
direct numerical simulations. Lagrangian statistics is found to be compatible
with Richardson description although small systematic deviations are found. The
value of the Richardson constant is estimated as $C_2 \simeq 0.55$, in a close
agreement with recent experimental findings [S. Ott and J. Mann J. Fluid Mech.
{\bf 422}, 207 (2000)]. By means of exit-time statistics it is shown that the
deviations from Richardson's law are a consequence of Eulerian intermittency.
The measured Lagrangian scaling exponents require a set of Eulerian structure
function exponents $\zeta_{p}$ which are remarkably close to standard ones
known for fully developed turbulence.


title: Relativistic Toda chain at root of unity II. Modified Q-operator

abstract:
  Matrix elements of quantum intertwiner as well as the modified Q-operator for
the quantum relativistic Toda chain at root of unity are constructed
explicitly. Modified Q-operators make isospectrality transformations of quantum
transfer matrices so that the classical counterparts of Q-operators correspond
to the Baecklund isospectrality transformations of the classical transfer
matrices. Separated vectors for the Functional Bethe Ansatz are constructed
with the help of modified Q-operators.


title:
Relativistic Toda chain at root of unity III. Relativistic Toda chain
  hierarchy

abstract:
  The hierarchy of the classical nonlinear integrable equations associated with
relativistic Toda chain model is considered. It is formulated for the N-th
powers of the quantum operators of the corresponding quantum integrable models.
Following the ideas of the paper hep-th/9606144 it is shown how one can obtain
such a system from 2D Toda lattice system. The reduction procedure is described
explicitly. The soliton solutions for the relativistic Toda chain are
constructed using results of solv-int/9304002 in terms of the rational
tau-functions. The vanishing properties of these tau-functions are
investigated.


title:
Differential constraints for the Kaup -- Broer system as a reduction of
  the 1D Toda lattice

abstract:
  It is shown that some special reduction of infinite 1D Toda lattice gives
differential constraints compatible with the Kaup -- Broer system. A family of
the travelling wave solutions of the Kaup -- Broer system and its higher
version is constructed.


title:
Hierarchy of (2+1)-dimensional nonlinear Schroedinger equation,
  self-dual Yang-Mills equation, and toroidal Lie algebras

abstract:
  The hierarchy structure associated with a (2+1)-dimensional Nonlinear
Schroedinger equation is discussed as an extension of the theory of the KP
hierarchy. Several methods to construct special solutions are given. The
relation between the hierarchy and a representation of toroidal Lie algebras
are established by using the language of free fermions. A relation to the
self-dual Yang-Mills equation is also discussed.

categories: nlin.SI hep-th math-ph math.DG math.MP math.QA math.RT


title: Finite Heat conduction in 2D Lattices

abstract:
  This paper gives a 2D hamonic lattices model with missing bond defects, when
the capacity ratio of defects is enough large, the temperature gradient can be
formed and the finite heat conduction is found in the model. The defects in the
2D harmonic lattices impede the energy carriers free propagation, by another
words, the mean free paths of the energy carrier are relatively short. The
microscopic dynamics leads to the finite conduction in the model.


title: Differential-difference system related to toroidal Lie algebra

abstract:
  We present a novel differential-difference system in (2+1)-dimensional
space-time (one discrete, two continuum), arisen from the Bogoyavlensky's
(2+1)-dimensional KdV hierarchy. Our method is based on the bilinear identity
of the hierarchy, which is related to the vertex operator representation of the
toroidal Lie algebra $\sl_2^{tor}$.

categories: nlin.SI hep-th math-ph math.MP math.QA math.RT


title: Periodicity Manifestations in the Non-Locally Coupled Maps

abstract:
  We study how periodicity manifestations recently found in the turbulent
globall coupled maps depend on the global feature of the couplings. We examine
three non-locally coupled map models. In the first two, the all-to-all
interaction is maintained 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 and in dimension $D=1,2,3$, periodicity
manifests universally from turbulence when the same suppression of the local
mean field fluctuation is achieved by the non-local averaging.


title: Energy fluxes in helical magnetohydrodynamics and dynamo action

abstract:
  Renormalized viscosity, renormalized resistivity, and various energy fluxes
are calculated for helical magnetohydrodynamics using perturbative field
theory. The calculation is to first-order in perturbation. Kinetic and magnetic
helicities do not affect the renormalized parameters, but they induce an
inverse cascade of magnetic energy. The sources for the the large-scale
magnetic field have been shown to be (1) energy flux from large-scale velocity
field to large-scale magnetic field arising due to nonhelical interactions, and
(2) inverse energy flux of magnetic energy caused by helical interactions.
Based on our flux results, a premitive model for galactic dynamo has been
constructed. Our calculations yields dynamo time-scale for a typical galaxy to
be of the order of $10^8$ years. Our field-theoretic calculations also reveal
that the flux of magnetic helicity is backward, consistent with the earlier
observations based on absolute equilibrium theory.


title: Spatiotemporal dynamics of pacing in models of anatomical reentry

abstract:
  Reentry around non-conducting ventricular scar tissue, which often causes
lethal arrhythmias, is typically treated by rapid stimulation from an
implantable defibrillator. However, the mechanisms of termination (success and
failure) are poorly understood. To elucidate such mechanisms, we studied pacing
of anatomical reentry in 1-D and 2-D excitable cardiac media models. Our
results suggest that the existence of inhomogeneity in the reentry circuit is
essential for pacing termination of tachycardia to be successful. Considering
the role of such inhomogeneities may lead to more effective pacing algorithms.


title: Backlund Transformations of Soliton Systems from Symmetry Constraints

abstract:
  Binary symmetry constraints are applied to constructing B\"acklund
transformations of soliton systems, both continuous and discrete. Construction
of solutions to soliton systems is split into finding solutions to
lower-dimensional Liouville integrable systems, which also paves a way for
separation of variables and exhibits integrability by quadratures for soliton
systems. Illustrative examples are provided for the KdV equation, the AKNS
system of nonlinear Schr\"odinger equations, the Toda lattice, and the Langmuir
lattice.


title: Pattern formation in 2-frequency forced parametric waves

abstract:
  We present an experimental investigation of superlattice patterns generated
on the surface of a fluid via parametric forcing with 2 commensurate
frequencies. The spatio-temporal behavior of 4 qualitatively different types of
superlattice patterns is described in detail. These states are generated via a
number of different 3--wave resonant interactions. They occur either as
symmetry--breaking bifurcations of hexagonal patterns composed of a single
unstable mode or via nonlinear interactions between the two primary unstable
modes generated by the two forcing frequencies. A coherent picture of these
states together with the phase space in which they appear is presented. In
addition, we describe a number of new superlattice states generated by 4--wave
interactions that arise when symmetry constraints rule out 3--wave resonances.


title: Dynamics of a simple evolutionary process

abstract:
  We study the simple evolutionary process in which we repeatedly find the
least fit agent in a population of agents and give it a new fitness which is
chosen independently at random from a specified distribution. We show that many
of the average properties of this process can be calculated exactly using
analytic methods. In particular we find the distribution of fitnesses at
arbitrary time, and the distribution of the lengths of runs of hits on the same
agent, the latter being found to follow a power law with exponent -1, similar
to the distribution of times between evolutionary events in the Bak-Sneppen
model and models based on the so-called record dynamics. We confirm our
analytic results with extensive numerical simulations.


title:
Transformations ${RS}_4^2(3)$ of the Ranks $\leq4$ and Algebraic
  Solutions of the Sixth Painlev\'e Equation

abstract:
  Compositions of rational transformations of independent variables of linear
matrix ordinary differential equations (ODEs) with the Schlesinger
transformations ($RS$-transformations) are used to construct algebraic
solutions of the sixth Painlev\'e equation. $RS$-Transformations of the ranks 3
and 4 of $2\times2$ matrix Fuchsian ODEs with 3 singular points into analogous
ODE with 4 singular points are classified.


title:
Volume preserving multidimensional integrable systems and Nambu--Poisson
  geometry

abstract:
  In this paper we study generalized classes of volume preserving
multidimensional integrable systems via Nambu--Poisson mechanics. These
integrable systems belong to the same class of dispersionless KP type equation.
Hence they bear a close resemblance to the self dual Einstein equation. All
these dispersionless KP and dToda type equations can be studied via twistor
geometry, by using the method of Gindikin's pencil of two forms. Following this
approach we study the twistor construction of our volume preserving systems.


title: Dark equations

abstract:
  Observing the Universe, astronomers have concluded that the motion of stars
can not be accounted for unless one assumes that most of the mass in the
Universe is carried on by a ``dark matter", so far impervious to all attempts
at being detected. There is now a similar concept of ``dark energy". I shall
discuss a different subject, ``dark equations". These have never indicated that
they influence anything or even exist, but if one supposes that they do exist,
one can systematically discover them and study their properties, some of which
turn out to be strange and others mysterious. These equations are similar in
spirit to what one gets when linearizing a given system, or studies how an
external linear wave interacts with a particular solution of a given system. We
define and study linear extensions of dynamical systems in general, and
integrable and Hamiltonian systems in particular. Systems discussed include the
KdV and mKdV equations and the associated Miura maps, the Burgers hierarchy and
the associated Hopf--Cole transformations, long wave equations, the Benney
hierarchy, and the KP hierarchy.


title:
Hyperelliptic Loop Solitons with Genus g: Investigations of a Quantized
  Elastica

abstract:
  In the previous work (J. Geom. Phys. {\bf{39}} (2001) 50-61), the closed loop
solitons in a plane, \it i.e., loops whose curvatures obey the modified
Korteweg-de Vries equations, were investigated for the case related to
algebraic curves with genera one and two. This article is a generalization of
the previous article to those of hyperelliptic curves with general genera. It
was proved that the tangential angle of loop soliton is expressed by the
Weierstrass hyperelliptic al function for a given hyperelliptic curve $y^2 =
f(x)$ with genus $g$.


title:
Hyperelliptic Solutions of Modified Korteweg-de Vries Equation of Genus
  g: Essentials of Miura Transformation

abstract:
  Explicit hyperelliptic solutions of the modified Korteweg-de Vries equations
without any ambiguous parameters were constructed in terms only of the
hyperelliptic $\al$-functions over non-degenerated hyperelliptic curve $y^2 =
f(x)$ of arbitrary genus $g$. In the derivation, any $\theta$-functions or
Baker-Akhiezer functions were not essentially used. Then the Miura
transformation naturally appears as the connections between the hyperelliptic
$\wp$-functions and hyperelliptic $\al$-functions.