title:
One abstract method for the study of network transportation and its
  application in the traffic problem in the process of city expansion

abstract:
  One abstract method for the study of network transportation is proposed in
this paper. By interpolating the properties of the edges that constitute
network into the two leading parameters of the nodes, this method can abstract
the configuration of real-life transportation networks without losing the
influence of the distance and the capability of the physical facilities, which
are essential parts of the transportation networks. With this method we can
study network transportation from the local point of view and describe network
congestion in detail, especially the dispersing of congestion. After applying
this method, as an example, to the traffic problem in the process of city
expansion, simulation results show that: the intuitions that with the same rate
of particle generation and the same capacity and power, the smaller the network
is the harder for the network to become congested; and the network will never
be congested if the power of each node equals or is bigger than its capacity
are incorrect. And the distance from one crossroad to another and the width of
the roads can be designed properly, when designing a city, so as to enlarge the
tolerance size of the city in the process of city expansion. This method may
provide more appropriate guide for real-life traffic design of network
transportation.


title: Self-organising mechanism of neuronal avalanche activity

abstract:
  A self-organising model is proposed to explain the criticality in cortical
networks deduced from recent observations of neuronal avalanches. Prevailing
understanding of self-organised criticality (SOC) dictates that conservation of
energy is essential to its emergence. Neuronal networks however are inherently
non-conservative as demonstrated by microelectrode recordings. The model
presented here shows that SOC can arise in non-conservative systems as well, if
driven internally. Evidence suggests that synaptic background activity provides
the internal drive for non-conservative cortical networks to achieve and
maintain a critical state. SOC is robust to any degree $\eta \in (0,1]$ of
background activity when the network size $N$ is large enough such that $\eta
N\sim 10^3$. For small networks, a strong background leads to epileptiform
activity, consistent with neurophysiological knowledge about epilepsy.


title: Noise-induced perturbations of dispersion-managed solitons

abstract:
  We study noise-induced perturbations of dispersion-managed solitons by
developing soliton perturbation theory for the dispersion-managed nonlinear
Schroedinger (DMNLS) equation, which governs the long-term behavior of optical
fiber transmission systems and certain kinds of femtosecond lasers. We show
that the eigenmodes and generalized eigenmodes of the linearized DMNLS equation
around traveling-wave solutions can be generated from the invariances of the
DMNLS equations, we quantify the perturbation-induced parameter changes of the
solution in terms of the eigenmodes and the adjoint eigenmodes, and we obtain
evolution equations for the solution parameters. We then apply these results to
guide importance-sampled Monte-Carlo simulations and reconstruct the
probability density functions of the solution parameters under the effect of
noise.


title:
Amplification in the auditory periphery: the effect of coupling tuning
  mechanisms

abstract:
  A mathematical model describing the coupling between two independent
amplification mechanisms in auditory hair cells is proposed and analyzed. Hair
cells are cells in the inner ear responsible for translating sound-induced
mechanical stimuli into an electrical signal that can then be recorded by the
auditory nerve. In nonmammals, two separate mechanisms have been postulated to
contribute to the amplification and tuning properties of the hair cells. Models
of each of these mechanisms have been shown to be poised near a Hopf
bifurcation. Through a weakly nonlinear analysis that assumes weak periodic
forcing, weak damping, and weak coupling, the physiologically-based models of
the two mechanisms are reduced to a system of two coupled amplitude equations
describing the resonant response. The predictions that follow from an analysis
of the reduced equations, as well as performance benefits due to the coupling
of the two mechanisms, are discussed and compared with published experimental
auditory nerve data.


title: Soliton Propagation in Chains with Simple Nonlocal Defects

abstract:
  We study the propagation of solitons on complex chains built by inserting
finite graphs at two sites of an unbranched chain. We compare numerical
findings with the results of an analytical linear approximation scheme
describing the interaction of large-fast solitons with non-local topological
defects on a chain. We show that the transmission properties of the solitons
strongly depend on the structure of the inserted graph, giving a tool to
control the soliton propagation through the choice of pertinent graphs to be
attached to the chain.


title: Non-equilibrium Lorentz gas on a curved space

abstract:
  The periodic Lorentz gas with external field and iso-kinetic thermostat is
equivalent, by conformal transformation, to a billiard with expanding
phase-space and slightly distorted scatterers, for which the trajectories are
straight lines. A further time rescaling allows to keep the speed constant in
that new geometry. In the hyperbolic regime, the stationary state of this
billiard is characterized by a phase-space contraction rate, equal to that of
the iso-kinetic Lorentz gas. In contrast to the iso-kinetic Lorentz gas where
phase-space contraction occurs in the bulk, the phase-space contraction rate
here takes place at the periodic boundaries.


title: Periodic orbit bifurcations and scattering time delay fluctuations

abstract:
  We study fluctuations of the Wigner time delay for open (scattering) systems
which exhibit mixed dynamics in the classical limit. It is shown that in the
semiclassical limit the time delay fluctuations have a distribution that
differs markedly from those which describe fully chaotic (or strongly
disordered) systems: their moments have a power law dependence on a
semiclassical parameter, with exponents that are rational fractions. These
exponents are obtained from bifurcating periodic orbits trapped in the system.
They are universal in situations where sufficiently long orbits contribute. We
illustrate the influence of bifurcations on the time delay numerically using an
open quantum map.


title:
Delocalization of a Dynamic System with Preservation or Self-Resurection
  of Informational Functionality. Ghost machines or can a system survive its
  destruction

abstract:
  We analyzed the problem of a dynamic system delocalization due to changes in
the system environment - universe and system architecture. We developed a
Delocalization of Dynamic Cores model to analyze the migration of functional
properties in open information and dynamic systems undergoing architecture
transition and modifications. Information geometry and topological formalisms
are proposed to analyze informational dynamic systems. Different physical and
holographic models are proposed to construct systems able to conserve their
functional properties under delocalization transition. We found several
constraints for the system environment - universe which conserve the dynamic
core system functionality under transition from localized explicit
implementation of functions to the implicit distributed implementation.


title:
On a direct approach to quasideterminant solutions of a noncommutative
  KP equation

abstract:
  A noncommutative version of the KP equation and two families of its solutions
expressed as quasideterminants are discussed. The origin of these solutions is
explained by means of Darboux and binary Darboux transformations. Additionally,
it is shown that these solutions may also be verified directly. This approach
is reminiscent of the wronskian technique used for the Hirota bilinear form of
the regular, commutative KP equation but, in the noncommutative case, no
bilinearising transformation is available.


title: Escape Rates of the H\'enon-Heiles System

abstract:
  A particle in the H\'enon-Heiles potential can escape when its energy is
above the threshold value $E_{th}={1/6}$. We report a theoretical study on the
the escape rates near threshold. We derived an analytic formula for the escape
rate as a function of energy by exploring the property of chaos. We also
simulated the escaping process by following the motions of a large number of
particles. Two algorithms are employed to solve the equations of motion. One is
the Runge-Kutta-Fehlberg method, and another is a recently proposed fourth
order symplectic method. Our simulations show the escape of
H$\mathrm{\acute{e}}$non-Heiles system follows exponential laws. We extracted
the escape rates from the time dependence of particle numbers in the
H$\mathrm{\acute{e}}$non-Heiles potential. The extracted escape rates agree
with the analytic result.


title:
A remark on the Hankel determinant formula for solutions of the Toda
  equation

abstract:
  We consider the Hankel determinant formula of the $\tau$ functions of the
Toda equation. We present a relationship between the determinant formula and
the auxiliary linear problem, which is characterized by a compact formula for
the $\tau$ functions in the framework of the KP theory. Similar phenomena that
have been observed for the Painlev\'e II and IV equations are recovered. The
case of finite lattice is also discussed.


title: q-Boson in Quantum Integrable Systems

abstract:
  q-bosonic realization of the underlying Yang-Baxter algebra is identified for
a series of quantum integrable systems, including some new models like two-mode
q-bosonic model leading to a coupled two-component derivative NLS model, wide
range of q-deformed matter-radiation models, q-anyon model etc. Result on a new
exactly solvable interacting anyon gas, linked to q-anyons on the lattice is
reported.


title:
Dressing method based on homogeneous Fredholm equation: quasilinear PDEs
  in multidimensions

abstract:
  In this paper we develop a dressing method for constructing and solving some
classes of matrix quasi-linear Partial Differential Equations (PDEs) in
arbitrary dimensions. This method is based on a homogeneous integral equation
with a nontrivial kernel, which allows one to reduce the nonlinear PDEs to
systems of non-differential (algebraic or transcendental) equations for the
unknown fields. In the simplest examples, the above dressing scheme captures
matrix equations integrated by the characteristics method and nonlinear PDEs
associated with matrix Hopf-Cole transformations.


title: Universality in the flooding of regular islands by chaotic states

abstract:
  We investigate the structure of eigenstates in systems with a mixed phase
space in terms of their projection onto individual regular tori. Depending on
dynamical tunneling rates and the Heisenberg time, regular states disappear and
chaotic states flood the regular tori. For a quantitative understanding we
introduce a random matrix model. The resulting statistical properties of
eigenstates as a function of an effective coupling strength are in very good
agreement with numerical results for a kicked system. We discuss the
implications of these results for the applicability of the semiclassical
eigenfunction hypothesis.


title:
Toward a phenomenological approach to the clustering of heavy particles
  in turbulent flows

abstract:
  A simple model accounting for the ejection of heavy particles from the
vortical structures of a turbulent flow is introduced. This model involves a
space and time discretization of the dynamics and depends on only two
parameters: the fraction of space-time occupied by rotating structures of the
carrier flow and the rate at which particles are ejected from them. The latter
can be heuristically related to the response time of the particles and hence
measure their inertia. It is shown that such a model reproduces qualitatively
most aspects of the spatial distribution of heavy particles transported by
realistic flows. In particular the probability density function of the mass $m$
in a cell displays an power-law behavior at small values and decreases faster
than exponentially at large values. The dependence of the exponent of the first
tail upon the parameters of the dynamics is explicitly derived for the model.
The right tail is shown to decrease as $\exp (-C m \log m)$. Finally, the
distribution of mass averaged over several cells is shown to obey rescaling
properties as a function of the coarse-grain size and of the ejection rate of
the particles. Contrarily to what has been observed in direct numerical
simulations of turbulent flows (Bec et al., http://arxiv.org/nlin.CD/0608045),
such rescaling properties are only due in the model to the mass dynamics of the
particles and do not involve any scaling properties in the spatial structure of
the carrier flow.


title: Remarks on rational solutions for the Korteveg - de Vries hierarchy

abstract:
  Differential equations for the special polynomials associated with the
rational solutions of the second Painleve hierarchy are introduced. It is shown
rational solutions of the Korteveg - de Vries hierarchy can be found taking the
Yablonskii - Vorob'ev polynomials into account. Special polynomials associated
with rational solutions of the Korteveg - de Vries hierarchy are presented.


title:
Travelling Waves in Hamiltonian Systems on 2D Lattices with Nearest
  Neighbor Interactions

abstract:
  We study travelling waves on a two--dimensional lattice with linear and
nonlinear coupling between nearest particles and a periodic nonlinear substrate
potential. Such a discrete system can model molecules adsorbed on a substrate
crystal surface. We show the existence of both uniform sliding states and
periodic travelling waves as well in a two-dimensional sine-Gordon lattice
equation using topological and variational methods.


title: Andronov-Hopf Bifurcations in Planar, Piecewise-Smooth, Continuous Flows

abstract:
  An equilibrium of a planar, piecewise-$C^1$, continuous system of
differential equations that crosses a curve of discontinuity of the Jacobian of
its vector field can undergo a number of discontinuous or border-crossing
bifurcations. Here we prove that if the eigenvalues of the Jacobian limit to
$\lambda_L \pm {\rm i} \omega_L$ on one side of the discontinuity and
$-\lambda_R \pm {\rm i} \omega_R$ on the other, with $\lambda_L, \lambda_R >0$,
and the quantity $
  \Lambda = \lambda_L / \omega_L -\lambda_R / \omega_R $ is nonzero, then a
periodic orbit is created or destroyed as the equilibrium crosses the
discontinuity. This bifurcation is analogous to the classical Andronov-Hopf
bifurcation, and is supercritical if $\Lambda < 0$ and subcritical if $\Lambda
>0$.


title:
Non-finite-difference algorithm for integrating Newton's motion
  equations

abstract:
  We have presented some practical consequences on the molecular-dynamics
simulations arising from the numerical algorithm published recently in paper
Int. J. Mod. Phys. C 16, 413 (2005). The algorithm is not a finite-difference
method and therefore it could be complementary to the traditional numerical
integrating of the motion equations. It consists of two steps. First, an
analytic form of polynomials in some formal parameter $\lambda$ (we put
$\lambda=1$ after all) is derived, which approximate the solution of the system
of differential equations under consideration. Next, the numerical values of
the derived polynomials in the interval, in which the difference between them
and their truncated part of smaller degree does not exceed a given accuracy
$\epsilon$, become the numerical solution. The particular examples, which we
have considered, represent the forced linear and nonlinear oscillator and the
2D Lennard-Jones fluid. In the latter case we have restricted to the
polynomials of the first degree in formal parameter $\lambda$.
  The computer simulations play very important role in modeling materials with
unusual properties being contradictictory to our intuition. The particular
example could be the auxetic materials. In this case, the accuracy of the
applied numerical algorithms as well as various side-effects, which might
change the physical reality, could become important for the properties of the
simulated material.


title:
Isoperiodic deformations of the acoustic operator and periodic solutions
  of the Harry Dym equation

abstract:
  We consider the problem of describing the possible spectra of an acoustic
operator with a periodic finite-gap density. We construct flows on the moduli
space of algebraic Riemann surfaces that preserve the periods of the
corresponding operator. By a suitable extension of the phase space, these
equations can be written with quadratic irrationalities.


title:
$N$-soliton solutions and perturbation theory for the derivative
  nonlinear Scr\"{o}dinger equation with nonvanishing boundary conditions

abstract:
  We present a simple approach for finding $N$-soliton solution and the
corresponding Jost solutions of the derivative nonlinear Scr\"{o}dinger
equation with nonvanishing boundary conditions. Soliton perturbation theory
based on the inverse scattering transform method is developed. As an
application of the present theory we consider the action of the diffusive-type
perturbation on a single bright/dark soliton.


title: Relativistic Toda Chain with Boundary Interaction at Root of Unity

abstract:
  We apply the Separation of Variables method to obtain eigenvectors of
commuting Hamiltonians in the quantum relativistic Toda chain at a root of
unity with boundary interaction.


title:
Hamiltonian Dynamics of Thermostated Systems: Two-Temperature
  Heat-Conducting phi-4 Chains

abstract:
  We consider and compare four Hamiltonian formulations of thermostated
mechanics, three of them kinetic, and the other one configurational. Though all
four approaches ``work'' at equilibrium, their application to many-body
nonequilibrium simulations can fail to provide a proper flow of heat. All the
Hamiltonian formulations considered here are applied to the same prototypical
two-temperature "phi-4" model of a heat-conducting chain. This model
incorporates nearest-neighbor Hooke's-Law interactions plus a quartic tethering
potential. Physically correct results, obtained with the isokinetic Gaussian
and Nose-Hoover thermostats, are compared with two other Hamiltonian results.
The latter results, based on constrained Hamiltonian thermostats, fail
correctly to model the flow of heat.


title: 3D-vortex labyrinths in the near field of solid-state microchip laser

abstract:
  The spatiotemporal vortex lattices generated in high Fresnel number
solid-state microchip lasers are studied in connection with Talbot phenomenon
generic to spatially periodic electromagnetic fields. The spatial layout of
light field is obtained via dynamical model based on Maxwell-Bloch equations
for class-B laser, discrete Fox-Lee map with relaxation of inversion and static
model based on superposition of copropagating Gaussian beams. The spatial
patterns observed experimentally and obtained numerically are interpreted as
nonlinear superposition of vortices with helicoidal phase dislocations. The
usage of vortex labyrinths and Talbot lattices as optical dipole traps for
neutral atoms is considered for the wavelength of trapping radiation in the
range 0.98 $\div$ 2.79 $\mu m$. The separable optical trapping potential is
mounted as a sum of array of vortex lines and additional parabolic subtrap. The
factorization of macroscopic wavefunction have led to analytical solution of
Gross-Pitaevski equation for ground state of ensemble of quantum particles
trapped in vortex labyrinth and in spatially - periodic array of Gaussian
beams.

categories: nlin.PS cond-mat.mes-hall cond-mat.quant-gas physics.optics quant-ph


title:
Homology and symmetry breaking in Rayleigh-Benard convection:
  Experiments and simulations

abstract:
  Algebraic topology (homology) is used to analyze the weakly turbulent state
of spiral defect chaos in both laboratory experiments and numerical simulations
of Rayleigh-Benard convection.The analysis reveals topological asymmetries that
arise when non-Boussinesq effects are present.


title:
Trilinearization and Localized Coherent Structures and Periodic
  Solutions for the (2+1) dimensional K-dV and NNV equations

abstract:
  In this paper, using a novel approach involving the truncated Laurent
expansion in the Painlev\'e analysis of the (2+1) dimensional K-dV equation, we
have trilinearized the evolution equation and obtained rather general classes
of solutions in terms of arbitrary functions. The highlight of this method is
that it allows us to construct generalized periodic structures corresponding to
different manifolds in terms of Jacobian elliptic functions, and the
exponentially decaying dromions turn out to be special cases of these
solutions. We have also constructed multi-elliptic function solutions and
multi-dromions and analysed their interactions. The analysis is also extended
to the case of generalized Nizhnik-Novikov-Veselov (NNV) equation, which is
also trilinearized and general class of solutions obtained.


title:
Hierarchy of rational order families of chaotic maps with an invariant
  measure

abstract:
  We introduce an interesting hierarchy of rational order chaotic maps that
posses an invariant measure. In contrast to the previously introduced hierarchy
of chaotic maps \cite{J1,J2,J3,J4,J5}, with merely entropy production, the
rational order chaotic maps can simultaneously produce and consume entropy . We
compute the Kolmogorov-Sinai entropy of theses maps analytically and also their
Lyapunov exponent numerically, where that obtained numerical results support
the analytical calculations.


title:
Two novel classes of solvable many-body problems of goldfish type with
  constraints

abstract:
  Two novel classes of many-body models with nonlinear interactions "of
goldfish type" are introduced. They are solvable provided the initial data
satisfy a single constraint (in one case; in the other, two constraints): i.
e., for such initial data the solution of their initial-value problem can be
achieved via algebraic operations, such as finding the eigenvalues of given
matrices or equivalently the zeros of known polynomials. Entirely isochronous
versions of some of these models are also exhibited: i.e., versions of these
models whose nonsingular solutions are all completely periodic with the same
period.


title: On dual equation in theory of the second order ODE's

abstract:
  The relations between the second order ODE's cubical on the first derivative
and their dual equations are discussed


title: Defining Complexity: A Commentary to a paper by Charles H. Bennett

abstract:
  The letter by Mark Perakh entitled "DEFINING COMPLEXITY: A Commentary to a
paper by Charles H. Bennett" is here archived with the permission of the
author. This letter was downloaded from the site "On Talk Reason,
http://www.talkreason.org/articles/complexity.pdf, August (2004)".


title: On the Lax pairs of the sixth Painleve' equation

abstract:
  The dependence of the sixth equation of Painleve' on its four parameters $(2
\alpha,-2 \beta,2 \gamma,1-2 \delta)
=(\theta_{\infty}^2,\theta_{0}^2,\theta_{1}^2,\theta_{x}^2)$ is holomorphic,
therefore one expects all its Lax pairs to display such a dependence. This is
indeed the case of the second order scalar ``Lax'' pair of Fuchs, but the
second order matrix Lax pair of Jimbo and Miwa presents a meromorphic
dependence on $\theta_\infty$ (and a holomorphic dependence on the three other
$\theta_j$). We analyze the reason for this feature and make suggestions to
suppress it.


title: Chaos in a one-dimensional compressible flow

abstract:
  We study the dynamics of a one-dimensional discrete flow with open boundaries
- a series of moving point particles connected by ideal springs. These
particles flow towards an inlet at constant velocity, pass into a region where
they are free to move according to their nearest neighbor interactions, and
then pass an outlet where they travel with a sinusoidally varying velocity. As
the amplitude of the outlet oscillations is increased, we find that the
resident time of particles in the chamber follows a bifurcating (Feigenbaum)
route to chaos. This irregular dynamics may be related to the complex behavior
of many particle discrete flows or is possibly a low-dimensional analogue of
non-stationary flow in continuous systems.


title: The Peakon Limit of the N-Soliton Solution of the Camassa-Holm Equation

abstract:
  We show that the analytic N-soliton solution of the Camassa-Holm (CH)
shallow-water model equation converges to the nonanalytic N-peakon solution of
the dispersionless CH equation when the dispersion parameter tends to zero. To
demonstrate this, we develop a novel limiting procedure and apply it to the
parametric representation for the N-soliton solution of the CH equaiton. In the
process, we use Jacobi's formula for determinants as well as various identities
among the Hankel determinants to facilitate the asymptotic analysis. We also
provide a new representation of the N-peakon solution in terms of the Hankel
determinants.


title:
With a Cole-Hopf transformation to solutions of the noncommutative KP
  hierarchy in terms of Wronski matrices

abstract:
  In case of the KP hierarchy where the dependent variable takes values in an
(arbitrary) associative algebra A, it is known that there are solutions which
can be expressed in terms of quasideterminants of a Wronski matrix which solves
the linear heat hierarchy. We obtain these solutions without the help of
quasideterminants in a simple way via solutions of matrix KP hierarchies (over
A) and by use of a Cole-Hopf transformation. For this class of exact solutions
we work out a correspondence with 'weakly nonassociative' algebras.


title: Self-Replication of Mesa Patterns in Reaction-Diffusion Systems

abstract:
  Certain two-component reaction-diffusion systems on a finite interval are
known to possess mesa (box-like) steadystate patterns in the singularly
perturbed limit of small diffusivity for one of the two solution components. As
the diffusivity D of the second component is decreased below some critical
value Dc, with Dc = O(1), the existence of a steady-state mesa pattern is lost,
triggering the onset of a mesa self-replication event that ultimately leads to
the creation of additional mesas. The initiation of this phenomena is studied
in detail for a particular scaling limit of the Brusselator model. Near the
existence threshold Dc of a single steady-state mesa, it is shown that an
internal layer forms in the center of the mesa. The structure of the solution
within this internal layer is shown to be governed by a certain core problem,
comprised of a single non-autonomous second-order ODE. By analyzing this core
problem using rigorous and formal asymptotic methods, and by using the Singular
Limit Eigenvalue Problem (SLEP) method to asymptotically calculate small
eigenvalues, an analytical verification of the conditions of Nishiura and
Ueyema [Physica D, 130, No. 1, (1999), pp. 73-104], believed to be responsible
for self-replication, is given. These conditions include: (1) The existence of
a saddle-node threshold at which the steady-state mesa pattern disappears; (2)
the dimple-shaped eigenfunction at the threshold, believed to be responsible
for the initiation of the replication process; and (3) the stability of the
mesa pattern above the existence threshold. Finally, we show that the core
problem is universal in the sense that it pertains to a class of
reaction-diffusion systems, including the Gierer-Meinhardt model with
saturation, where mesa self-replication also occurs.


title: KP Trigonometric Solitons and an Adelic Flag Manifold

abstract:
  We show that the trigonometric solitons of the KP hierarchy enjoy a
differential-difference bispectral property, which becomes transparent when
translated on two suitable spaces of pairs of matrices satisfying certain rank
one conditions. The result can be seen as a non-self-dual illustration of
Wilson's fundamental idea [Invent. Math. 133 (1998), 1-41] for understanding
the (self-dual) bispectral property of the rational solutions of the KP
hierarchy. It also gives a bispectral interpretation of a (dynamical) duality
between the hyperbolic Calogero-Moser system and the rational
Ruijsenaars-Schneider system, which was first observed by Ruijsenaars [Comm.
Math. Phys. 115 (1988), 127-165].


title:
Exact shock solution of a coupled system of delay differential
  equations: a car-following model

abstract:
  In this paper, we present exact shock solutions of a coupled system of delay
differential equations, which was introduced as a traffic-flow model called
{\it the car-following model}. We use the Hirota method, originally developed
in order to solve soliton equations. %While, with a periodic boundary
condition, this system has % a traveling-wave solution given by elliptic
functions. The relevant delay differential equations have been known to allow
exact solutions expressed by elliptic functions with a periodic boundary
conditions. In the present work, however, shock solutions are obtained with
open boundary, representing the stationary propagation of a traffic jam.


title: Analytic Approach for Controlling Quantum States in Complex Systems

abstract:
  We examine random matrix systems driven by an external field in view of
optimal control theory (OCT). By numerically solving OCT equations, we can show
that there exists a smooth transition between two states called "moving bases"
which are dynamically related to initial and final states. In our previous work
[J. Phys. Soc. Jpn. 73 (2004) 3215-3216; Adv. Chem. Phys. 130A (2005) 435-458],
they were assumed to be orthogonal, but in this paper, we introduce orthogonal
moving bases. We can construct a Rabi-oscillation like representation of a
wavpacket using such moving bases, and derive an analytic optimal field as a
solution of the OCT equations. We also numerically show that the newly obtained
optimal field outperforms the previous one.


title:
Necessary conditions for partial and super-integrability of Hamiltonian
  systems with homogeneous potentia

abstract:
  We consider a natural Hamiltonian system of $n$ degrees of freedom with a
homogeneous potential. Such system is called partially integrable if it admits
$1<l<n$ independent and commuting first integrals, and it is called
super-integrable if it admits $n+l$, $0<l<n$ independent first integrals such
that $n$ of them commute. We formulate two theorems which give easily
computable and effective necessary conditions for partial and
super-integrability. These conditions are derived in the frame of the
Morales-Ramis theory, i.e., from an analysis of the differential Galois group
of variational equations along a particular solution of the system. To
illustrate an application of the formulated theorems, we investigete three and
four body problems on a line and the motion in a radial potential.


title:
Differential Galois obstructions for integrability of homogeneous Newton
  equations

abstract:
  n this paper we formulate necessary conditions for the integrability in the
Jacobi sense of Newton equations $\ddot \vq=-\vF(\vq)$, where $\vq\in\C^n$ and
all components of $\vF$ are polynomial and homogeneous of the same degree $l$.
These conditions are derived from an analysis of the differential Galois group
of the variational equations along special particular solutions of the Newton
equations. We show that, taking all admissible particular solutions, we
restrict considerably the set of Newton's equations satisfying the necessary
conditions for the integrability. Moreover, we apply the obtained conditions
for a detailed analysis of the Newton equations with two degrees of freedom
(i.e., $n=2$). We demonstrate the strength of the obtained results analyzing
general cases with $\deg F_i =l< 4$. For $l=3$ we found an integrable case when
the Newton equations have two polynomial first integrals and both of them are
of degree four in the momenta $p_1=\dot q_1$, and $p_2=\dot q_2$. Moreover, for
an arbitrary $k$, we found a family of Newton equations depending on one
parameter $\lambda$. For an arbitrary value of $\lambda$ one quadratic in the
momenta first integral exist. We distinguish infinitely many values of
$\lambda$ for which the system is integrable or superintegrable with additional
polynomial first integrals which seemingly can be of an arbitrarily high degree
with respect to the momenta.


title:
Finiteness of integrable $n$-dimensional homogeneous polynomial
  potentials

abstract:
  We consider natural Hamiltonian systems of $n>1$ degrees of freedom with
polynomial homogeneous potentials of degree $k$. We show that under a
genericity assumption, for a fixed $k$, at most only a finite number of such
systems is integrable. We also explain how to find explicit forms of these
integrable potentials for small $k$.


title: Saturable discrete vector solitons in one-dimensional photonic lattices

abstract:
  Localized vectorial modes, with equal frequencies and mutually orthogonal
polarizations, are investigated both analytically and experimentally in a
one-dimensional photonic lattice with saturable nonlinearity. It is shown that
these modes may span over many lattice elements and that energy transfer among
the two components is both phase and intensity dependent. The transverse
electrically polarized mode exhibits a single-hump structure and spreads in
cascades in saturation, while the transverse magnetically polarized mode
exhibits splitting into a two-hump structure. Experimentally such discrete
vector solitons are observed in lithium niobate lattices for both coherent and
mutually incoherent excitations.


title: Differential Galois obstructions for non-commutative integrability

abstract:
  We show that if a holomorphic Hamiltonian system is holomorphically
integrable in the non-commutative sense in a neighbourhood of a non-equilibrium
phase curve which is located at a regular level of the first integrals, then
the identity component of the differential Galois group of the variational
equations along the phase curve is Abelian. Thus necessary conditions for the
commutative and non-commutative integrability given by the differential Galois
approach are the same.


title: On the two different bi-Hamiltonian structures for the Toda lattice

abstract:
  We construct two different incompatible Poisson pencils for the Toda lattice
by using known variables of separation proposed by Moser and by Sklyanin.


title: Complex trajectories in chaotic dynamical tunneling

abstract:
  We develop the semiclassical method of complex trajectories in application to
chaotic dynamical tunneling. First, we suggest a systematic numerical technique
for obtaining complex tunneling trajectories by the gradual deformation of the
classical ones. This provides a natural classification of the tunneling
solutions. Second, we present a heuristic procedure for sorting out the least
suppressed trajectory. As an illustration, we apply our technique to the
process of chaotic tunneling in a quantum mechanical model with two degrees of
freedom. Our analysis reveals rich dynamics of the system. At the classical
level, there exists an infinite set of unstable solutions forming a fractal
structure. This structure is inherited by the complex tunneling paths and plays
the central role in the semiclassical study. The process we consider exhibits
the phenomenon of optimal tunneling: the suppression exponent of the tunneling
probability has a local minimum at a certain energy which is thus (locally) the
optimal energy for tunneling. We test the proposed method by comparison of the
semiclassical results with the results of the exact quantum computations and
find a good agreement.

categories: nlin.CD hep-th math-ph math.MP physics.chem-ph quant-ph


title: Bistability: a common feature in some "aggregates" of logistic maps

abstract:
  As it was argued by Anderson [Science 177, 393 (1972)], the "reductionist"
hypothesis does not by any means imply a "constructionist" one. Hence, in
general, the behavior of large and complex aggregates of elementary components
can not be understood nor extrapolated from the properties of a few components.
Following this insight, we have simulated different "aggregates" of logistic
maps according to a particular coupling scheme. All these aggregates show a
similar pattern of dynamical properties, concretely a bistable behavior, that
is also found in a network of many units of the same type, independently of the
number of components and of the interconnection topology. A qualitative
relationship with brain-like systems is suggested.


title: Delay-induced multistability near a global bifurcation

abstract:
  We study the effect of a time-delayed feedback within a generic model for a
saddle-node bifurcation on a limit cycle. Without delay the only attractor
below this global bifurcation is a stable node. Delay renders the phase space
infinite-dimensional and creates multistability of periodic orbits and the
fixed point. Homoclinic bifurcations, period-doubling and saddle-node
bifurcations of limit cycles are found in accordance with Shilnikov's theorems.


title: Importance of the Wick rotation on Tunnelling

abstract:
  A continuous complex rotation of time $t\mapsto t\EXP{-i\theta}$ is shown to
smooth out the huge fluctuations that characterise chaotic tunnelling. This is
illustrated in the kicked rotor model (quantum standard map) where the period
of the map is complexified: the associated chaotic classical dynamics, if
significant for $\theta=0$, is blurred out long before the Wick rotation is
completed ($\theta=\pi/2$). The influence of resonances on tunnelling rates
weakens exponentially as $\theta$ increases from zero, all the more rapidly the
sharper the fluctuations. The long range fluctuations can therefore be
identified in a deterministic way without ambiguity. When the last ones have
been washed out, tunnelling recovers the (quasi-)integrable exponential
behaviour governed by the action of a regular instanton.


title: Complexity of Polarized Spatial Patterns in Large Area Square VCSEL

abstract:
  We consider pattern selection process in a wide aperture VCSEL near
threshold. We show that for a square geometry of the laser aperture, the
patterns formed at lasing threshold can be very complicated because of a
possible misalignment between directions of an intrinsic spatial anisotropy of
VCSEL and lateral boundaries of its aperture. The analogy with quantum billiard
structures is established, and fingerprints of wave chaos are found. Influence
of localized inhomogeneous in the pump current is also considered.


title:
Chaotic fluctuation of temperature on environmental interface exchanging
  energy by visible and infrared radiation, convection and conduction

abstract:
  The concept of environmental interface is defined and analyzed from the point
of view of the possible source of non-standard behaviour. The energy balance
equation is written for the interface where all kinds of energy transfer occur.
It is shown that under certain conditions, the discrete version of the equation
for the temperature time rate turns in to the well-known logistic equation and
the conditions for chaotic behaviour are studied. They are determined by the
Lyapunov exponent. The realistic situation when the coefficients of the
equation vary with time, is studied for the Earth-environment general system.


title:
Competition and bistability of ordered undulations and undulation chaos
  in inclined layer convection

abstract:
  Experimental and theoretical investigations of undulation patterns in
high-pressure, inclined layer gas convection at a Prandtl number near unity are
reported. Particular focus is given to the competition between the
spatiotemporal chaotic state of undulation chaos and stationary patterns of
ordered undulations. In experiments a competition and bistability between the
two states is observed, with ordered undulations most prevalent at higher
Rayleigh number. The spectral pattern entropy, spatial correlation lengths, and
defect statistics are used to characterize the competing states. The
experiments are complemented by a theoretical analysis of the
Oberbeck-Boussinesq equations. The stability region of the ordered undulation
as a function of their wavevectors and the Rayleigh number is obtained with
Galerkin techniques. In addition, direct numerical simulations are used to
investigate the spatiotemporal dynamics. In the simulations both ordered
undulations and undulation chaos were observed dependent on initial conditions.
Experiment and theory are found to agree well.


title: On the speed of pulled fronts with a cutoff

abstract:
  We study the effect of a small cutoff $\epsilon$ on the velocity of a pulled
front in one dimension by means of a variational principle. We obtain a lower
bound on the speed dependent on the cutoff, and for which the two leading order
terms correspond to the Brunet Derrida expression. To do so we cast a known
variational principle for the speed of propagation of fronts in new variables
which makes it more suitable for applications.


title: Stochastic Resonance: from climate to biology

abstract:
  In this paper I will review some basic aspects of the mechanism of stochastic
resonance. Stochastic resonance was first introduced as a possible mechanism to
explain long term climatic variation. Since then, there have been many
applications of stochastic resonance in physical and biological systems. I will
show that in complex system, stochastic resonance can substantially change as a
function of the ``system complexity''. Also, I will briefly mention how to
apply stochastic resonance for the case of Brownian motors.


title: Ultrasonic cavity solitons

abstract:
  We report on a new type of localized structure, an ultrasonic cavity soliton,
supported by large aspect-ratio acoustic resonators containing viscous media.
The spatio-temporal dynamics of this system is analyzed on the basis of a
generalized Swift-Hohenberg equation, derived from the microscopic equations
under conditions close to nascent bistability. These states of the acoustic and
thermal fields are robust structures, existing whenever a spatially uniform
solution and a periodic pattern coexist. An analytical solution for the
ultrasonic cavity soliton is also presented.


title: Exact solutions to the focusing nonlinear Schrodinger equation

abstract:
  A method is given to construct globally analytic (in space and time) exact
solutions to the focusing cubic nonlinear Schrodinger equation on the line. An
explicit formula and its equivalents are presented to express such exact
solutions in a compact form in terms of matrix exponentials. Such exact
solutions can alternatively be written explicitly as algebraic combinations of
exponential, trigonometric, and polynomial functions of the spatial and
temporal coordinates.


title:
Dispersive destabilization of nonlinear light propagation in fiber Bragg
  gratings

abstract:
  The effect of retaining the material dispersion terms in the nonlinear
coupled mode equations (NLCME) that describe light propagation in fiber Bragg
gratings is analyzed. It is found that dispersion is responsible for new
instabilities of the uniform states and gives rise to new complex spatio
temporal dynamics that is not captured by the standard NLCME formulation. A
detailed analysis of the effect of dispersion on the linear stability
characteristics of the uniform solutions is presented and some numerical
integrations of the NLCME with dispersion are also performed in order to
corroborate the theoretical results.


title:
Dispersive destabilization of nonlinear light propagation in fiber Bragg
  gratings: a numerical verification

abstract:
  This paper presents some numerical simulations of the full one-dimensional
Maxwell-Lorentz equations that describe light propagation in fiber Bragg
gratings in order to confirm that the standard nonlinear coupled mode equations
fail to predict the weakly nonlinear dynamics of the system when dispersive
instabilities come into play, and that, in this case, the correct slow envelope
description of the system requires to consider higher order dispersion effects.


title:
Nonlinear oscillations and stability domains in fractional
  reaction-diffusion systems

abstract:
  We study a fractional reaction-diffusion system with two types of variables:
activator and inhibitor. The interactions between components are modeled by
cubical nonlinearity. Linearization of the system around the homogeneous state
provides information about the stability of the solutions which is quite
different from linear stability analysis of the regular system with integer
derivatives. It is shown that by combining the fractional derivatives index
with the ratio of characteristic times, it is possible to find the marginal
value of the index where the oscillatory instability arises. The increase of
the value of fractional derivative index leads to the time periodic solutions.
The domains of existing of periodic solutions for different parameters of the
problem are obtained. A computer simulation of the corresponding nonlinear
fractional ordinary differential equations is presented. It is established by
computer simulation that there exists a set of stable spatio-temporal tructures
of the one-dimensional system under the Neumann and periodic boundary
condition. The features of these solutions consist in the transformation of the
steady state dissipative structures to homogeneous oscillations or space
temporary structures at a certain value of fractional index and the ratio of
characteristic times of system.


title: On the water-bag model of dispersionless KP hierarchy (II)

abstract:
  We construct the bi-Hamiltonian structure of the waterbag model of dKP and
establish the third-order Hamiltonian operator associated with the waterbag
model. Also, the symmetries and conserved densities of rational type are
discussed.


title:
Model for the spatio-temporal intermittency of the energy dissipation in
  turbulent flows

abstract:
  Modeling the intermittent behavior of turbulent energy dissipation processes
both in space and time is often a relevant problem when dealing with phenomena
occurring in high Reynolds number flows, especially in astrophysical and space
fluids. In this paper, a dynamical model is proposed to describe the
spatio-temporal intermittency of energy dissipation rate in a turbulent system.
This is done by using a shell model to simulate the turbulent cascade and
introducing some heuristic rules, partly inspired by the well known $p$-model,
to construct a spatial structure of the energy dissipation rate. In order to
validate the model and to study its spatially intermittency properties, a
series of numerical simulations have been performed. These show that the level
of spatial intermittency of the system can be simply tuned by varying a single
parameter of the model and that scaling laws in agreement with those obtained
from experiments on fully turbulent hydrodynamic flows can be recovered. It is
finally suggested that the model could represent a useful tool to simulate the
spatio-temporal intermittency of turbulent energy dissipation in those high
Reynolds number astrophysical fluids where impulsive energy release processes
can be associated to the dynamics of the turbulent cascade.


title: Five Questions on Complexity

abstract:
  This introductory paper is structured in the form of an "interview", where
the author answers the following questions: Why did you begin working with
complex systems? How would you define complexity? What is your favourite
aspect/concept of complexity? In your opinion, what is the most problematic
aspect/concept of complexity? How do you see the future of complexity?


title:
Chirped chiral solitons in nonlinear Schr\"odinger equation with
  self-steepening and self-frequency shift

abstract:
  We find exact solutions to nonlinear Schr\"odinger equation in the presence
of self-steepening and self-frequency shift. These include periodic solutions
and localized solutions of dark-bright type which can be {\emph{chiral}}, and
chirality being controlled by sign of self steepening term. A new form of self
phase modulation, which can be tuned by higher order nonlinearities as also by
the initial conditions, distinct from nonlinear Schr\"odinger equation,
characterizes these solutions. In certain nontrivial parameter domain solutions
are found to satisfy {\emph{linear}} Schr\"odinger equation, indicating
possiblity of linear superposition in this nonlinear system. Dark and bright
solitons exist in both anomalous and normal dispersion regimes and a duality
between dark-bright type of solution and kinematic-higher order chirping is
also seen. Localized kink solutions similar to NLSE solitons, but with very
different self phase modulation, are identified.


title: Scaling of the dynamical properties of the Fermi-Ulam accelerator model

abstract:
  The chaotic sea below the lowest energy spanning curve of the complete
Fermi-Ulam model (FUM) is numerically investigated when the amplitude of
oscillation 'epsilon' of the moving wall is small. We use scaling analysis near
the integrable to non-integrable transition to describe the average energy as
function of time t and as function of iteration (or collision) number n. If t
is employed as independent variable, the exponents related to the energy
scaling properties of the FUM are different from the ones of a well known
simplification of this model (SFUM). However, if n is employed as independent
variable, the exponents are the same for both FUM and SFUM. In the collision
number analysis, we present analytical arguments supporting that the exponents
c* and b* related to the initial velocity and to 'epsilon' are given by c* =
-1/2 and b* = -1. We derive also a relation connecting the scaling exponents
related to the variables time and collision number. Moreover, we show that,
differently from the SFUM, the average energy in the FUM saturates for long
times and we justify the physical origins for some differences and similarities
observed between the FUM and its simplification.


title: Self-organized criticality in nonconservative mean-field sandpiles

abstract:
  A mean-field sandpile model that exhibits self-organized criticality (SOC)
despite violation of the grain-transfer conservation law during avalanches is
proposed. The sandpile consists of $N$ agents and possesses background activity
with intensity $\eta\in[0,1]$. Background activity is characterized by
transitions between two stable agent states. Analysis employing theories of
branching processes and fixed points reveals a transition from sub-critical to
SOC phase that is determined by $\eta N$. The model is used to explain the
school size distribution of free-swimming tuna as a result of population
depletion.


title: Quasideterminant solutions of a non-Abelian Hirota-Miwa equation

abstract:
  A non-Abelian version of the Hirota-Miwa equation is considered. In an
earlier paper [Nimmo (2006) J. Phys. A: Math. Gen. \textbf{39}, 5053-5065] it
was shown how solutions expressed as quasideterminants could be constructed for
this system by means of Darboux transformations. In this paper we discuss these
solutions from a different perspective and show that the solutions are
quasi-Pl\"{u}cker coordinates and that the non-Abelian Hirota-Miwa equation may
be written as a quasi-Pl\"{u}cker relation. The special case of the matrix
Hirota-Miwa equation is also considered using a more traditional, bilinear
approach and the techniques are compared.


title: The forget-remember mechanism for 2-state spreading

abstract:
  A new mechanism, the forget-remember mechanism, is proposed for studying the
spreading process in 2-state model. Such mechanism exhibits behaviors of
message spreading influenced by some kinds of functions about time and history
caring about the individuals of the spreading system, holding message or being
out of message. To demonstrate the mechanism, both linear and exponential forms
for forget-function and remember-function are simulated and show that a great
impact on the saturation of message-spreading and the relative phase
transformation.


title: Classical nonlinear response of a chaotic system: collective resonances

abstract:
  We consider the classical response in a chaotic system. In contrast to
behavior in integrable or almost integrable systems, the nonlinear classical
response in a chaotic system vanishes at long times. The response also reveals
certain features of collective resonances which do not correspond to any
periodic classical trajectories. The convergence of the response is shown to
hold due to the exponential time dependence of the stability matrix. The
growing exponentials corresponding to strong instability do not inhibit the
convergence. We calculate both linear and second-order response in one of the
simplest chaotic systems: free classical motion on a surface of constant
negative curvature. We demonstrate the relevance of the model for applications
to spectroscopic experiments.


title: Nonlinear optical effects in artificial materials

abstract:
  We consider some nonlinear phenomena in metamaterials with negative
refractive index properties. Our consideration includes a survey of previously
known results as well as identification of the phenomena that are important for
applications of this new field. We focus on optical behavior of thin films as
well as multi-wave interactions.


title: Advective collisions

abstract:
  Small particles advected in a fluid can collide (and therefore aggregate) due
to the stretching or shearing of fluid elements. This effect is usually
discussed in terms of a theory due to Saffman and Turner [J. Fluid Mech., 1,
16-30, (1956)]. We show that in complex or random flows the Saffman-Turner
theory for the collision rate describes only an initial transient (which we
evaluate exactly). We obtain precise expressions for the steady-state collision
rate for flows with small Kubo number, including the influence of fractal
clustering on the collision rate for compressible flows. For incompressible
turbulent flows, where the Kubo number is of order unity, the Saffman-Turner
theory is an upper bound.


title:
Quantization of a Free Particle Interacting Linearly with a Harmonic
  Oscillator

abstract:
  We investigate the quantization of a free particle coupled linearly to a
harmonic oscillator. This system, whose classical counterpart has clearly
separated regular and chaotic regions, provides an ideal framework for studying
the quantization of mixed systems. We identify key signatures of the
classically chaotic and regular portions in the quantum system by constructing
Husimi distributions and investigating avoided level crossings of eigenvalues
as functions of the strength and range of the interaction between the system's
two components. We show, in particular, that the Husimi structure becomes mixed
and delocalized as the classical dynamics becomes more chaotic.

categories: nlin.CD cond-mat.other math-ph math.DS math.MP


title:
Quantization of the Smoluchowski equation and the problem of quantum
  tunneling at zero temperature

abstract:
  In this article we address the problem of quantum tunneling of a
non-Markovian Brownian particle away from thermal equilibrium. We calculate the
Kramers escape rate at low temperature (including the zero temperature case) in
the Smoluchowski limit (strong friction regime). Our main findings are: (i) our
quantum escape rate is valid far from the thermal equilibrium and is
non-Markovian, but it becomes Markovian as the correlation time vanishes; (ii)
at thermodynamic equilibrium we obtain a non-Markovian quantum rate that
predicts a superfluidity phenomenon in the Markovian limit at low and zero
temperatures.


title: Jacobi Fields on Statistical Manifolds of Negative Curvature

abstract:
  Two entropic dynamical models are considered. The geometric structure of the
statistical manifolds underlying these models is studied. It is found that in
both cases, the resulting metric manifolds are negatively curved. Moreover, the
geodesics on each manifold are described by hyperbolic trajectories. A detailed
analysis based on the Jacobi equation for geodesic spread is used to show that
the hyperbolicity of the manifolds leads to chaotic exponential instability. A
comparison between the two models leads to a relation among statistical
curvature, stability of geodesics and relative entropy-like quantities.
Finally, the Jacobi vector field intensity and the entropy-like quantity are
suggested as possible indicators of chaoticity in the ED models due to their
similarity to the conventional chaos indicators based on the Riemannian
geometric approach and the Zurek-Paz criterion of linear entropy growth,
respectively.


title:
The Semiclassical Modified Nonlinear Schroedinger Equation I: Modulation
  Theory and Spectral Analysis

abstract:
  We study an integrable modification of the focusing nonlinear
  Schroedinger equation from the point of view of semiclassical asymptotics. In
particular, (i) we establish several important consequences of the mixed-type
limiting quasilinear system including the existence of maps that embed the
limiting forms of both the focusing and defocusing nonlinear Schroedinger
equations into the framework of a single limiting system for the modified
equation, (ii) we obtain bounds for the location of discrete spectrum for the
associated spectral problem that are particularly suited to the semiclassical
limit and that generalize known results for the spectrum of the nonselfadjoint
Zakharov-Shabat spectral problem, and (iii) we present a multiparameter family
of initial data for which we solve the associated spectral problem in terms of
special functions for all values of the semiclassical scaling parameter. We
view our results as part of a broader project to analyze the semiclassical
limit of the modified nonlinear Schroedinger equation via the noncommutative
steepest descent procedure of Deift and Zhou, and we also present a
self-contained development of a Riemann-Hilbert problem of inverse scattering
that differs from those given in the literature and that is well-adapted to
semiclassical asymptotics.


title:
Information-Geometric Indicators of Chaos in Gaussian Models on
  Statistical Manifolds of Negative Ricci Curvature

abstract:
  A new information-geometric approach to chaotic dynamics on curved
statistical manifolds based on Entropic Dynamics (ED) is proposed. It is shown
that the hyperbolicity of a non-maximally symmetric 6N-dimensional statistical
manifold M_{s} underlying an ED Gaussian model describing an arbitrary system
of 3N degrees of freedom leads to linear information-geometric entropy growth
and to exponential divergence of the Jacobi vector field intensity, quantum and
classical features of chaos respectively.


title:
The propagation of the light beam at the interface between uniform media
  and periodic media

abstract:
  We considered the wave propagation between two medias. For description of
physical model choose nonlinear Schredinger equation with saturation parameter.
The solutions of respectively equation are found and analyzed.


title:
Generalization of the classical Kramers rate for non-Markovian open
  systems out of equilibrium

abstract:
  We analyze the behavior of a Brownian particle moving in a double-well
potential. The escape probability of this particle over the potential barrier
from a metastable state toward another state is known as the Kramers problem.
In this work we generalize Kramers' rate theory to the case of an environment
always out of thermodynamic equilibrium reckoning with non-Markovian effects.


title:
Spectral analysis and an area-preserving extension of a piecewise linear
  intermittent map

abstract:
  We investigate spectral properties of a 1-dimensional piecewise linear
intermittent map, which has not only a marginal fixed point but also a singular
structure suppressing injections of the orbits into neighborhoods of the
marginal fixed point. We explicitly derive generalized eigenvalues and
eigenfunctions of the Frobenius--Perron operator of the map for classes of
observables and piecewise constant initial densities, and it is found that the
Frobenius--Perron operator has two simple real eigenvalues 1 and $\lambda_d \in
(-1,0)$, and a continuous spectrum on the real line $[0,1]$. From these
spectral properties, we also found that this system exhibits power law decay of
correlations. This analytical result is found to be in a good agreement with
numerical simulations. Moreover, the system can be extended to an
area-preserving invertible map defined on the unit square. This extended system
is similar to the baker transformation, but does not satisfy hyperbolicity. A
relation between this area-preserving map and a billiard system is also
discussed.


title:
Universally-convergent Squared-operator Iteration Methods for Solitary
  Waves in General Nonlinear Wave Equations

abstract:
  Three new iteration methods, namely the squared-operator method, the modified
squared-operator method, and the power-conserving squared-operator method, for
solitary waves in general scalar and vector nonlinear wave equations are
proposed. These methods are based on iterating new differential equations whose
linearization operators are squares of those for the original equations,
together with acceleration techniques. The first two methods keep the
propagation constants fixed, while the third method keeps the powers (or other
arbitrary functionals) of the solution fixed. It is proved that all these
methods are guaranteed to converge to any solitary wave (either ground state or
not) as long as the initial condition is sufficiently close to the
corresponding exact solution, and the time step in the iteration schemes is
below a certain threshold value. Furthermore, these schemes are
fast-converging, highly accurate, and easy to implement. If the solitary wave
exists only at isolated propagation constant values, the corresponding
squared-operator methods are developed as well. These methods are applied to
various solitary wave problems of physical interest, such as higher-gap vortex
solitons in the two-dimensional nonlinear Schr\"odinger equations with periodic
potentials, and isolated solitons in Ginzburg-Landau equations, and some new
types of solitary wave solutions are obtained. It is also demonstrated that the
modified squared-operator method delivers the best performance among the
methods proposed in this article.


title:
Colloquium: Theory of Drag Reduction by Polymers in Wall Bounded
  Turbulence

abstract:
  The flow of fluids in channels, pipes or ducts, as in any other wall-bounded
flow (like water along the hulls of ships or air on airplanes) is hindered by a
drag, which increases many-folds when the fluid flow turns from laminar to
turbulent. A major technological problem is how to reduce this drag in order to
minimize the expense of transporting fluids like oil in pipelines, or to move
ships in the ocean. It was discovered in the mid-twentieth century that minute
concentrations of polymers can reduce the drag in turbulent flows by up to 80%.
While experimental knowledge had accumulated over the years, the fundamental
theory of drag reduction by polymers remained elusive for a long time, with
arguments raging whether this is a "skin" or a "bulk" effect. In this
colloquium review we first summarize the phenomenology of drag reduction by
polymers, stressing both its universal and non-universal aspects, and then
proceed to review a recent theory that provides a quantitative explanation of
all the known phenomenology. We treat both flexible and rod-like polymers,
explaining the existence of universal properties like the Maximum Drag
Reduction (MDR) asymptote, as well as non-universal cross-over phenomena that
depend on the Reynolds number, on the nature of the polymer and on its
concentration. Finally we also discuss other agents for drag reduction with a
stress on the important example of bubbles.


title:
Coherent optical pulse dynamics in nanocomposite plasmonic Bragg
  gratings

abstract:
  The propagation of solitary waves in a Bragg grating formed by an array of
thin nanostructured dielectric films is considered. A system of equations of
Maxwell-Duffing type and describing forward- and backward-propagating waves in
such a grating, is derived. Exact solitary wave solutions are found, analyzed,
and compared with the results of direct numerical simulations.


title: Spatiotemporal pulses in a liquid crystal optical oscillator

abstract:
  A nonlinear optical medium results by the collective orientation of liquid
crystal molecules tightly coupled to a transparent photoconductive layer. We
show that such a medium can give a large gain, thus, if inserted in a ring
cavity, it results in an unidirectional optical oscillator. Dynamical regimes
with many interacting modes are made possible by the wide transverse size and
the high nonlinearity of the liquid crystals. We show the generation of
spatiotemporal pulses, coming from the random superposition of many coexisting
modes with different frequencies.


title: Universality of Level Spacing Distributions in Classical Chaos

abstract:
  We suggest that random matrix theory applied to a classical action matrix can
be used in classical physics to distinguish chaotic from non-chaotic behavior.
We consider the 2-D stadium billiard system as well as the 2-D anharmonic and
harmonic oscillator. By unfolding of the spectrum of such matrix we compute the
level spacing distribution, the spectral auto-correlation and spectral
rigidity. We observe Poissonian behavior in the integrable case and Wignerian
behavior in the chaotic case. We present numerical evidence that the action
matrix of the stadium billiard displays GOE behavior and give an explanation
for it. The findings present evidence for universality of level fluctuations -
known from quantum chaos - also to hold in classical physics.


title:
Symmetry breaking bifurcation in Nonlinear Schrodinger /Gross-Pitaevskii
  Equations

abstract:
  We consider a class of nonlinear Schrodinger / Gross-Pitaveskii (NLS-GP)
equations, i.e. NLS with a linear potential. We obtain conditions for a
symmetry breaking bifurcation in a symmetric family of states as N, the squared
L^2 norm
  (particle number, optical power), is increased. In the special case where the
linear potential is a double-well with well separation L, we estimate N_{cr},
the symmetry breaking threshold.
  Along the ``lowest energy'' symmetric branch, there is an exchange of
stability from the symmetric to asymmetric branch as N is increased beyond
N_{cr}.


title: Multiple Components in Narrow Planetary Rings

abstract:
  The phase-space volume of regions of regular or trapped motion, for bounded
or scattering systems with two degrees of freedom respectively, displays
universal properties. In particular, drastic reductions in the volume (gaps)
are observed at specific values of a control parameter. Using the stability
resonances we show that they, and not the mean-motion resonances, account for
the position of these gaps. For more degrees of freedom, exciting these
resonances divides the regions of trapped motion. For planetary rings, we
demonstrate that this mechanism yields rings with multiple components.


title:
Multidimensional integrable systems and deformations of Lie algebra
  homomorphisms

abstract:
  We use deformations of Lie algebra homomorphisms to construct deformations of
dispersionless integrable systems arising as symmetry reductions of
anti--self--dual Yang--Mills equations with a gauge group Diff$(S^1)$.


title: Quantization of three-wave equations

abstract:
  The subject of this paper is the consecutive procedure of discretization and
quantization of two similar classical integrable systems in three-dimensional
space-time: the standard three-wave equations and less known modified
three-wave equations. The quantized systems in discrete space-time may be
understood as the regularized integrable quantum field theories. Integrability
of the theories, and in particular the quantum tetrahedron equations for vertex
operators, follow from the quantum auxiliary linear problems. Principal object
of the lattice field theories is the Heisenberg discrete time evolution
operator constructed with the help of vertex operators.


title: Noise-induced Turbulence in Nonlocally Coupled Oscillators

abstract:
  We demonstrate that nonlocally coupled limit-cycle oscillators subject to
spatiotemporally white Gaussian noise can exhibit a noise-induced transition to
turbulent states. After illustrating noise-induced turbulent states with
numerical simulations using two representative models of limit-cycle
oscillators, we develop a theory that clarifies the effective dynamical
instabilities leading to the turbulent behavior using a hierarchy of dynamical
reduction methods. We determine the parameter region where the system can
exhibit noise-induced turbulent states, which is successfully confirmed by
extensive numerical simulations at each level of the reduction.


title: The Spectrum of the Partially Locked State for the Kuramoto Model

abstract:
  We solve a longstanding stability problem for the Kuramoto model of coupled
oscillators. This system has attracted mathematical attention, in part because
of its applications in fields ranging from neuroscience to condensed-matter
physics, and also because it provides a beautiful connection between nonlinear
dynamics and statistical mechanics. The model consists of a large population of
phase oscillators with all-to-all sinusoidal coupling. The oscillators'
intrinsic frequencies are randomly distributed across the population according
to a prescribed probability density, here taken to be unimodal and symmetric
about its mean. As the coupling between the oscillators is increased, the
system spontaneously synchronizes: the oscillators near the center of the
frequency distribution lock their phases together and run at the same
frequency, while those in the tails remain unlocked and drift at different
frequencies. Although this ``partially locked'' state has been observed in
simulations for decades, its stability has never been analyzed mathematically.
Part of the difficulty is in formulating a reasonable infinite-N limit of the
model. Here we describe such a continuum limit, and prove that the
corresponding partially locked state is, in fact, neutrally stable, contrary to
what one might have expected. The possible implications of this result are
discussed.


title: Dynamical Systems, Stability, and Chaos

abstract:
  In this expository and resources chapter we review selected aspects of the
mathematics of dynamical systems, stability, and chaos, within a historical
framework that draws together two threads of its early development: celestial
mechanics and control theory, and focussing on qualitative theory. From this
perspective we show how concepts of stability enable us to classify dynamical
equations and their solutions and connect the key issues of nonlinearity,
bifurcation, control, and uncertainty that are common to time-dependent
problems in natural and engineered systems. We discuss stability and
bifurcations in three simple model problems, and conclude with a survey of
recent extensions of stability theory to complex networks.


title: Symmetry structure of integrable non-evolutionary equations

abstract:
  We study a class of evolutionary partial differential systems with two
components related to second order (in time) non-evolutionary equations of odd
order in spatial variable. We develop the formal diagonalisation method in
symbolic representation, which enables us to derive an explicit set of
necessary conditions of existence of higher symmetries. Using these conditions
we globally classify all such homogeneous integrable systems, i.e. systems
which possess a hierarchy of infinitely many higher symmetries.


title: Randomized Cellular Automata

abstract:
  We define and study a few properties of a class of random automata networks.
While regular finite one-dimensional cellular automata are defined on periodic
lattices, these automata networks, called randomized cellular automata, are
defined on random directed graphs with constant out-degrees and evolve
according to cellular automaton rules. For some families of rules, a few
typical a priori unexpected results are presented.


title:
Robust oscillations in SIS epidemics on adaptive networks:
  Coarse-graining by automated moment closure

abstract:
  We investigate the dynamics of an epidemiological
susceptible-infected-susceptible (SIS) model on an adaptive network. This model
combines epidemic spreading (dynamics on the network) with rewiring of network
connections (topological evolution of the network). We propose and implement a
computational approach that enables us to study the dynamics of the network
directly on an emergent, coarse-grained level. The approach sidesteps the
derivation of closed low-dimensional approximations. Our investigations reveal
that global coupling, which enters through the awareness of the population to
the disease, can result in robust large-amplitude oscillations of the state and
topology of the network.


title:
Chaotic Scattering and the $n$-bounce Resonance in Solitary Wave
  Interactions

abstract:
  We present a new and complete analysis of the n-bounce resonance and chaotic
scattering in solitary wave collisions. In these phenomena, the speed at which
a wave exits a collision depends in a complicated fractal way on its input
speed. We present a new asymptotic analysis of collective-coordinate ODEs,
reduced models that reproduce the dynamics of these systems. We reduce the ODEs
to discrete-time iterated separatrix maps and obtain new quantitative results
unraveling the fractal structure of the scattering behavior. These phenomena
have been observed repeatedly in many solitary-wave systems over 25 years.


title: Solitary waves in plasmonic Bragg gratings

abstract:
  Light propagation in a Bragg periodic structure containing thin films with
metallic nanoparticles is studied. Plasmonic resonance frequency, Bragg
frequency, and light carrier frequency are assumed to be close. Exact solutions
describing solitary gap-waves are found, and a light arrest phenomenon due to
nonlinearity of plasmonic oscillations is studied.


title: Modeling of Kundu-Eckhaus equation

abstract:
  In work the numerical solutions of Kundu-Eckhaus equation are presented. The
conditions of dominate nonlinearity or disperse are cleared up.


title:
Two-component Analogue of Two-dimensional Long Wave-Short Wave Resonance
  Interaction Equations: A Derivation and Solutions

abstract:
  The two-component analogue of two-dimensional long wave-short wave resonance
interaction equations is derived in a physical setting. Wronskian solutions of
the integrable two-component analogue of two-dimensional long wave-short wave
resonance interaction equations are presented.

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


title: A possible mechanism of generation of extreme waves in shallow water

abstract:
  We consider a mechanism of generation of huge waves by multi-soliton resonant
interactions. A non-stationary wave amplification phenomenon is found in some
exact solutions of the Kadomtsev-Petviashvili (KP) equation. The mechanism
proposed here explains the character of extreme waves and of those in Tsunami.


title: A new organizational metrics: A quantum approach

abstract:
  A future goal of robot teams and agent-based models (ABMs) is to field
organizations and systems based on first principles derived from human
counterparts. Forestalling that opportunity, the failure of traditional
organizational theory has at the same time opened the way to innovative
theories of organizations and change. Inspired by Bohr and Heisenberg about the
application of interdependent uncertainty in the interaction between action and
observation, making organizations bistable, we have begun to construct a theory
of organizations based on the uncertainty of energy level (resources) and
belief/action consensus, leading to preliminary metrics of organizational
performance that we have discovered in field studies. Our goal in this project
is to address the problem posed by organizations with: the development of new
theory; field tests of new metrics for organizations; and the development of
quantum ABMs set within a social circuit as a building block for an
organization. Should we be successful, our research would represent a
fundamental departure from traditional observational methods of social science
by forming the basis of a predictive science of organizations. We expect that
replacing the traditional method of observation with a predictive science must
account for when cognitive observations work and when they do not (illusions).


title: Giant acceleration in slow-fast space-periodic Hamiltonian systems

abstract:
  Motion of an ensemble of particles in a space-periodic potential well with a
weak wave-like perturbation imposed is considered. We found that slow
oscillations of wavenumber of the perturbation lead to occurrence of directed
particle current. This current is amplifying with time due to giant
acceleration of some particles. It is shown that giant acceleration is linked
with the existence of resonant channels in phase space.


title: By Magri's Theorem, Self-Dual Gravity is Completely Integrable

abstract:
  By Magri's theorem the bi-Hamiltonian structure of Plebanski's second
heavenly equation proves that (anti)-self-dual gravity is a completely
integrable system in four dimensions.


title: Quantum Baker Maps for Spiraling Chaotic Motion

abstract:
  We define a coupling of two baker maps through a pi/2 rotation both in
position and in momentum. The classical trajectories thus exhibit spiraling, or
loxodromic motion, which is only possible for conservative maps of at least two
degrees of freedom. This loxodromic baker map is still hyperbolic, that is,
fully chaotic. Quantization of this map follows on similar lines to other
generalized baker maps. It is found that the eigenvalue spectrum for quantum
loxodromic baker map is far removed from those of the canonical random matrix
ensembles. An investigation of the symmetries of the loxodromic baker map
reveals the cause of this deviation from the Bohigas-Giannoni-Schmit
conjecture.


title:
Universal shape law of stochastic supercritical bifurcations: Theory and
  experiments

abstract:
  A universal law for the supercritical bifurcation shape of transverse
one-dimensional (1D) systems in presence of additive noise is given. The
stochastic Langevin equation of such systems is solved by using a Fokker-Planck
equation leading to the expression for the most probable amplitude of the
critical mode. From this universal expression, the shape of the bifurcation,
its location and its evolution with the noise level are completely defined.
Experimental results obtained for a 1D transverse Kerr-like slice subjected to
optical feedback are in excellent agreement.


title: The Ablowitz-Ladik Hierarchy Revisited

abstract:
  We provide a detailed recursive construction of the Ablowitz-Ladik (AL)
hierarchy and its zero-curvature formalism. The two-coefficient AL hierarchy
under investigation can be considered a complexified version of the discrete
nonlinear Schr\"odinger equation and its hierarchy of nonlinear evolution
equations.
  Specifically, we discuss in detail the stationary Ablowitz-Ladik formalism in
connection with the underlying hyperelliptic curve and the stationary
Baker-Akhiezer function and separately the corresponding time-dependent
Ablowitz-Ladik formalism.


title:
Localized Asymmetric Atomic Matter Waves in Two-Component Bose-Einstein
  Condensates Coupled with Two Photon Microwave Field

abstract:
  We investigate localized atomic matter waves in two-component Bose-Einstein
condensates coupled by the two photon microwave field. Interestingly, the
oscillations of localized atomic matter waves will gradually decay and finally
become non-oscillating behavior even if existing coupling field. In particular,
atom numbers occupied in two different hyperfine spin states will appear
asymmetric occupations after some time evolution.


title:
N-Wave Equations with Orthogonal Algebras: Z_2 and Z_2 \times Z_2
  Reductions and Soliton Solutions

abstract:
  We consider $N$-wave type equations related to the orthogonal algebras
obtained from the generic ones via additional reductions. The first
$\mathbb{Z}_2$-reduction is the canonical one. We impose a second
$\mathbb{Z}_2$-reduction and consider also the combined action of both
reductions. For all three types of $N$-wave equations we construct the soliton
solutions by appropriately modifying the Zakharov-Shabat dressing method. We
also briefly discuss the different types of one-soliton solutions. Especially
rich are the types of one-soliton solutions in the case when both reductions
are applied. This is due to the fact that we have two different configurations
of eigenvalues for the Lax operator $L$: doublets, which consist of pairs of
purely imaginary eigenvalues, and quadruplets. Such situation is analogous to
the one encountered in the sine-Gordon case, which allows two types of
solitons: kinks and breathers. A new physical system, describing Stokes-anti
Stokes Raman scattering is obtained. It is represented by a 4-wave equation
related to the ${\bf B}_2$ algebra with a canonical $\mathbb{Z}_2$ reduction.


title:
Spectrum of the supersymmetric t-J model with non-diagonal open
  boundaries

abstract:
  In this work we diagonalize the double-row transfer matrix of the
supersymmetric t-J model with non-diagonal boundary terms by means of the
algebraic Bethe ansatz. The corresponding reflection equations are studied and
two distinct classes of solutions are found, one diagonal solution and other
non-diagonal. In the non-diagonal case the eigenvalues in the first sectors are
given for arbitrary values of the boundary parameters.


title: Consistency in the driven butterfly

abstract:
  We study the consistency of the responses in the chaotic Lorenz model driven
by chaotic and colored-noise signals. A measure is introduced for the
consistency of the response in the system for repeated applications of the same
drive signal, starting at different initial conditions of the response system.
The convergence of responses is analyzed in dynamical phase space in terms of
the local conditional Lyapunov exponent. The dependence of the local
conditional Lyapunov exponent on drive signal, and the existence of multiple
basins of consistency depending on strength of drive signal, are shown.


title: Directed deterministic classical transport: symmetry breaking and beyond

abstract:
  We consider transport properties of a double delta-kicked system, in a regime
where all the symmetries (spatial and temporal) that could prevent directed
transport are removed. We analytically investigate the (non trivial) behavior
of the classical current and diffusion properties and show that the results are
in good agreement with numerical computations. The role of dissipation for a
meaningful classical ratchet behavior is also discussed.


title:
Nonlinear modes for the Gross-Pitaevskii equation -- demonstrative
  computation approach

abstract:
  A method for the study of steady-state nonlinear modes for Gross-Pitaevskii
equation (GPE) is described. It is based on exact statement about coding of the
steady-state solutions of GPE which vanish as $x\to+\infty$ by reals. This
allows to fulfill {\it demonstrative computation} of nonlinear modes of GPE
i.e. the computation which allows to guarantee that {\it all} nonlinear modes
within a given range of parameters have been found. The method has been applied
to GPE with quadratic and double-well potential, for both, repulsive and
attractive nonlinearities. The bifurcation diagrams of nonlinear modes in these
cases are represented. The stability of these modes has been discussed.


title: Reentrant and Whirling Hexagons in Non-Boussinesq convection

abstract:
  We review recent computational results for hexagon patterns in non-Boussinesq
convection. For sufficiently strong dependence of the fluid parameters on the
temperature we find reentrance of steady hexagons, i.e. while near onset the
hexagon patterns become unstable to rolls as usually, they become again stable
in the strongly nonlinear regime. If the convection apparatus is rotated about
a vertical axis the transition from hexagons to rolls is replaced by a Hopf
bifurcation to whirling hexagons. For weak non-Boussinesq effects they display
defect chaos of the type described by the two-dimensional complex
Ginzburg-Landau equation. For stronger non-Boussinesq effects the Hopf
bifurcation becomes subcritical and localized bursting of the whirling
amplitude is found. In this regime the coupling of the whirling amplitude to
(small) deformations of the hexagon lattice becomes important. For yet stronger
non-Boussinesq effects this coupling breaks up the hexagon lattice and strongly
disordered states characterized by whirling and lattice defects are obtained.


title: Stability of Waves in Multi-component DNLS system

abstract:
  In this work, we systematically generalize the Evans function methodology to
address vector systems of discrete equations. We physically motivate and
mathematically use as our case example a vector form of the discrete nonlinear
Schrodinger equation with both nonlinear and linear couplings between the
components. The Evans function allows us to qualitatively predict the stability
of the nonlinear waves under the relevant perturbations and to quantitatively
examine the dependence of the corresponding point spectrum eigenvalues on the
system parameters. These analytical predictions are subsequently corroborated
by numerical computations.


title: Sublattice synchronization of chaotic networks with delayed couplings

abstract:
  Synchronization of chaotic units coupled by their time delayed variables are
investigated analytically. A new type of cooperative behavior is found:
sublattice synchronization. Although the units of one sublattice are not
directly coupled to each other, they completely synchronize without time delay.
The chaotic trajectories of different sublattices are only weakly correlated
but not related by generalized synchronization. Nevertheless, the trajectory of
one sublattice is predictable from the complete trajectory of the other one.
The spectra of Lyapunov exponents are calculated analytically in the limit of
infinite delay times, and phase diagrams are derived for different topologies.


title: Chaotic Phenomenon in Nonlinear Gyrotropic Medium

abstract:
  Nonlinear gyrotropic medium is a medium, whose natural optical activity
depends on the intensity of the incident light wave. The Kuhn's model is used
to study nonlinear gyrotropic medium with great success. The Kuhn's model
presents itself a model of nonlinear coupled oscillators. This article is
devoted to the study of the Kuhn's nonlinear model. In the first paragraph of
the paper we study classical dynamics in case of weak as well as strong
nonlinearity. In case of week nonlinearity we have obtained the analytical
solutions, which are in good agreement with the numerical solutions. In case of
strong nonlinearity we have determined the values of those parameters for which
chaos is formed in the system under study. The second paragraph of the paper
refers to the question of the Kuhn's model integrability. It is shown, that at
the certain values of the interaction potential this model is exactly
integrable and under certain conditions it is reduced to so-called universal
Hamiltonian. The third paragraph of the paper is devoted to quantum-mechanical
consideration. It shows the possibility of stochastic absorption of external
field energy by nonlinear gyrotropic medium. The last forth paragraph of the
paper is devoted to generalization of the Kuhn's model for infinite chain of
interacting oscillators.


title: Phase-space structure of two-dimensional excitable localized structures

abstract:
  In this work we characterize in detail the bifurcation leading to an
excitable regime mediated by localized structures in a dissipative nonlinear
Kerr cavity with a homogeneous pump. Here we show how the route can be
understood through a planar dynamical system in which a limit cycle becomes the
homoclinic orbit of a saddle point (saddle-loop bifurcation). The whole picture
is unveiled, and the mechanism by which this reduction occurs from the full
infinite-dimensional dynamical system is studied. Finally, it is shown that the
bifurcation leads to an excitability regime, under the application of suitable
perturbations. Excitability is an emergent property for this system, as it
emerges from the spatial dependence since the system does not exhibit any
excitable behavior locally.


title: Emergence of order in selection-mutation dynamics

abstract:
  We characterize the time evolution of a d-dimensional probability
distribution by the value of its final entropy. If it is near the
maximally-possible value we call the evolution mixing, if it is near zero we
say it is purifying. The evolution is determined by the simplest non-linear
equation and contains a d times d matrix as input. Since we are not interested
in a particular evolution but in the general features of evolutions of this
type, we take the matrix elements as uniformly-distributed random numbers
between zero and some specified upper bound. Computer simulations show how the
final entropies are distributed over this field of random numbers. The result
is that the distribution crowds at the maximum entropy, if the upper bound is
unity. If we restrict the dynamical matrices to certain regions in matrix
space, for instance to diagonal or triangular matrices, then the entropy
distribution is maximal near zero, and the dynamics typically becomes
purifying.


title:
Experimental study of imperfect phase synchronization in the forced
  Lorenz system

abstract:
  In this work we demonstrate for an experimental system, that exhibits the
Lorenz butterfly attractor behavior, that perfect chaotic phase synchronization
cannot be achieved in systems with an unbounded distribution of intrinsic time
scales. Instead, imperfect phase synchronization is characterized by the
occurrence of phase slips, associated to epochs of time during which the
chaotic oscillator exhibits a slower time scale. Interestingly, during phase
slips the chaotic oscillator keeps in sync with the drive, but with a different
locking ratio.


title:
Classical nonlinear response of a chaotic system: Langevin dynamics and
  spectral decomposition

abstract:
  We consider the classical response of a strongly chaotic Hamiltonian system.
The spectrum of such a system consists of discrete complex Ruelle-Pollicott
(RP) resonances which manifest themselves in the behavior of the correlation
and response functions. We interpret the RP resonances as the eigenstates and
eigenvalues of the Fokker-Planck operator obtained by adding an infinitesimal
noise term to the first-order Liouville operator. We demonstrate how the
deterministic expression for the linear response is reproduced in the limit of
vanishing noise. For the second-order response we establish an equivalence of
the spectral decomposition with infinitesimal noise and the long-time
asymptotic expansion for the deterministic case.


title: Chimera Ising Walls in Forced Nonlocally Coupled Oscillators

abstract:
  Nonlocally coupled oscillator systems can exhibit an exotic spatiotemporal
structure called chimera, where the system splits into two groups of
oscillators with sharp boundaries, one of which is phase-locked and the other
is phase-randomized. Two examples of the chimera states are known: the first
one appears in a ring of phase oscillators, and the second one is associated
with the two-dimensional rotating spiral waves. In this article, we report yet
another example of the chimera state that is associated with the so-called
Ising walls in one-dimensional spatially extended systems, which is exhibited
by a nonlocally coupled complex Ginzburg-Landau equation with external forcing.


title: Phase Space of Rolling Solutions of the Tippe Top

abstract:
  Equations of motion of an axially symmetric sphere rolling and sliding on a
plane are usually taken as model of the tippe top. We study these equations in
the nonsliding regime both in the vector notation and in the Euler angle
variables when they admit three integrals of motion that are linear and
quadratic in momenta. In the Euler angle variables $(\theta,\phi,\psi)$ these
integrals give separation equations that have the same structure as the
equations of the Lagrange top. It makes it possible to describe the whole space
of solutions by representing them in the space of parameters $(D,\lambda,E)$
being constant values of the integrals of motion.

categories: nlin.SI math-ph math.MP physics.class-ph


title:
Defining relations on the Hamiltonians of $XXX$ and $XXZ$ $R$-matrices
  and new integrable spin-orbital chains

abstract:
  Several complete systems of integrability conditions on a spin chain
Hamiltonian density matrix are presented. The corresponding formulas for
$R$-matrices are also given. The latter is expressed via the local Hamiltonian
density in the form similar to spin one half $XXX$ and $XXZ$ models. The result
is applied to the problem of integrability of $SU(2)\times SU(2)$- and
$SU(2)\times U(1)$-invariant spin-orbital chains (the Kugel-Homskii-Inagaki
model). The eight new integrable cases are found. One of them corresponds to
the Temperley-Lieb algebra, the others three to the algebra associated with the
$XXX$, $XXZ$ and graded $XXZ$ models. The last two $R$-matrices are also
presented.


title:
Commutator identities on associative algebras and integrability of
  nonlinear pde's

abstract:
  It is shown that commutator identities on associative algebras generate
solutions of linearized integrable equations. Next, a special kind of the
dressing procedure is suggested that in a special class of integral operators
enables to associate to such commutator identity both nonlinear equation and
its Lax pair. Thus problem of construction of new integrable pde's reduces to
construction of commutator identities on associative algebras.


title: Weak compressibility of surface wave turbulence

abstract:
  We study the growth of small-scale inhomogeneities of the density of
particles floating in weakly nonlinear, small-amplitude, surface waves. Despite
the amplitude smallness, the accumulated effect of the long-time evolution may
produce strongly inhomogeneous distribution of the floaters: density
fluctuations grow exponentially with a small but finite exponent. We show that
the exponent is of sixth or higher order in wave amplitude. As a result, the
inhomogeneities do not form within typical time-scales of the natural
environment. We conclude that turbulence of surface waves is weakly
compressible and alone it cannot be a realistic mechanism of the clustering of
matter on liquid surfaces.


title:
Existence of multi-site intrinsic localized modes in one-dimensional
  Debye crystals

abstract:
  The existence of highly localized multi-site oscillatory structures (discrete
multibreathers) in a nonlinear Klein-Gordon chain which is characterized by an
inverse dispersion law is proven and their linear stability is investigated.
The results are applied in the description of vertical (transverse, off-plane)
dust grain motion in dusty plasma crystals, by taking into account the lattice
discreteness and the sheath electric and/or magnetic field nonlinearity.
Explicit values from experimental plasma discharge experiments are considered.
The possibility for the occurrence of multibreathers associated with vertical
charged dust grain motion in strongly-coupled dusty plasmas (dust crystals) is
thus established. From a fundamental point of view, this study aims at
providing a first rigorous investigation of the existence of intrinsic
localized modes in Debye crystals and/or dusty plasma crystals and, in fact,
suggesting those lattices as model systems for the study of fundamental crystal
properties.


title: Three-dimensional vortex solitons in self-defocusing media

abstract:
  We demonstrate that families of vortex solitons are possible in a
bi-dispersive three-dimensional nonlinear Schrodinger equation. These solutions
can be considered as extensions of two-dimensional dark vortex solitons which,
along the third dimension, remain localized due to the interplay between
dispersion and nonlinearity. Such vortex solitons can be observed in optical
media with normal dispersion, normal diffraction, and defocusing nonlinearity.


title: Directional emission from weakly eccentric resonators

abstract:
  It is shown that when a circular resonator is deformed in a nonintegrable
way, a symmetry breaking of escaping rays occurs which can dramatically
modulate the outgoing wave even for small perturbations. The underlying
mechanism does not occur in integrable models for which the ray families can be
computed exactly and is described in this Letter on the basis of canonical
perturbation theory. Emission from deformed resonators is currently of immense
practical interest in the context of whispering-gallery optical resonances of
dielectric cavities and the approach outlined here promises simple analytical
characterisations in the important case of small deformations.


title: Exceptional Points in Atomic Spectra

abstract:
  We report the existence of exceptional points for the hydrogen atom in
crossed magnetic and electric fields in numerical calculations. The resonances
of the system are investigated and it is shown how exceptional points can be
found by exploiting characteristic properties of the degeneracies, which are
branch point singularities. A possibility for the observation of exceptional
points in an experiment with atoms is proposed.


title: Spectral properties of Bunimovich mushroom billiards

abstract:
  Properties of a quantum mushroom billiard in the form of a superconducting
microwave resonator have been investigated. They reveal unexpected nonuniversal
features such as, e.g., a supershell effect in the level density and a dip in
the nearest-neighbor spacing distribution. Theoretical predictions for the
quantum properties of mixed systems rely on the sharp separability of phase
space - an unusual property met by mushroom billiards. We however find
deviations which are ascribed to the presence of dynamic tunneling.


title: Some Remarks on the KP System of the Camassa-Holm Hierarchy

abstract:
  We study a Kadomtsev-Petviashvili system for the local Camassa-Holm hierarchy
obtaining a candidate to the Baker-Akhiezer function for its first reduction
generalizing the local Camassa-Holm. We focus our attention on the differences
with the standard KdV-KP case.