title: N-soliton solutions to the DKP equation and Weyl group actions

abstract:
  We study soliton solutions to the DKP equation which is defined by the Hirota
bilinear form, \[ {\begin{array}{llll} (-4D_xD_t+D_x^4+3D_y^2)
\tau_n\cdot\tau_n=24\tau_{n-1}\tau_{n+1}, (2D_t+D_x^3\mp 3D_xD_y) \tau_{n\pm
1}\cdot\tau_n=0 \end{array} \quad n=1,2,.... \] where $\tau_0=1$. The
$\tau$-functions $\tau_n$ are given by the pfaffians of certain skew-symmetric
matrix. We identify one-soliton solution as an element of the Weyl group of
D-type, and discuss a general structure of the interaction patterns among the
solitons. Soliton solutions are characterized by $4N\times 4N$ skew-symmetric
constant matrix which we call the $B$-matrices. We then find that one can have
$M$-soliton solutions with $M$ being any number from $N$ to $2N-1$ for some of
the $4N\times 4N$ $B$-matrices having only $2N$ nonzero entries in the upper
triangular part (the number of solitons obtained from those $B$-matrices was
previously expected to be just $N$).


title:
A Bethe ansatz solvable model for superpositions of Cooper pairs and
  condensed molecular bosons

abstract:
  We introduce a general Hamiltonian describing coherent superpositions of
Cooper pairs and condensed molecular bosons. For particular choices of the
coupling parameters, the model is integrable. One integrable manifold, as well
as the Bethe ansatz solution, was found by Dukelsky et al., Phys. Rev. Lett. 93
(2004) 050403. Here we show that there is a second integrable manifold,
established using the boundary Quantum Inverse Scattering Method. In this
manner we obtain the exact solution by means of the algebraic Bethe ansatz. In
the case where the Cooper pair energies are degenerate we examine the
relationship between the spectrum of these integrable Hamiltonians and the
quasi-exactly solvable spectrum of particular Schrodinger operators. For the
solution we derive here the potential of the Schrodinger operator is given in
terms of hyperbolic functions. For the solution derived by Dukelsky et al.,
loc. cit. the potential is sextic and the wavefunctions obey PT-symmetric
boundary conditions. This latter case provides a novel example of an integrable
Hermitian Hamiltonian acting on a Fock space whose states map in to a Hilbert
space of PT-symmetric wavefunctions defined on a contour in the complex plane.


title: Traveling ion channel density waves affected by a conservation law

abstract:
  A model of mobile, charged ion channels embedded in a biomembrane is
investigated. The ion channels fluctuate between an opened and a closed state
according to a simple two-state reaction scheme whereas the total number of ion
channels is a conserved quantity. Local transport mechanisms suggest that the
ion channel densities are governed by electrodiffusion-like equations that have
to be supplemented by a cable-type equation describing the dynamics of the
transmembrane voltage. It is shown that the homogeneous distribution of ion
channels may become unstable to either a stationary or an oscillatory
instability. The nonlinear behavior immediately above threshold of an
oscillatory bifurcation occuring at finite wave number is analyzed in terms of
amplitude equations. Due to the conservation law imposed on ion channels
large-scale modes couple to the finite wave number instability and have thus to
be included in the asymptotic analysis near onset of pattern formation. A
modified Ginzburg-Landau equation extended by long-wavelength stationary
excitations is established and it is highlighted how the global conservation
law affects the stability of traveling ion channel density waves.


title: Renormalization group theory for the phase field crystal equation

abstract:
  We derive a set of rotationally covariant amplitude equations for use in
multiscale simulation of the two dimensional phase field crystal (PFC) model by
a variety of renormalization group (RG) methods. We show that the presence of a
conservation law introduces an ambiguity in operator ordering in the RG
procedure, which we show how to resolve. We compare our analysis with standard
multiple scales techniques, where identical results can be obtained with
greater labor, by going to sixth order in perturbation theory, and by assuming
the correct scaling of space and time.


title: From chaos of lines to Lagrangian structures in flux conservative fields

abstract:
  A numerical method is proposed in order to track field lines of
three-dimensional divergence free fields. Field lines are computed by a locally
valid Hamiltonian mapping, which is computed using a symplectic scheme. The
method is theoretically valid everywhere but at points where the field is null
or infinite. For any three dimensional flux conservative field for which
problematic points are sufficiently sparse, a systematic procedure is proposed
and implemented. Construction of field lines is achieved by means of tracers
and the introduction of various Hamiltonians adapted to the "geometrical state"
each line or tracer is. The states are artificially defined by an a priori
given frame of reference and Cartesian coordinates, and refer to a Hamiltonian
which is locally valid at the time step to be computed. This procedure ensures
the preservation of the volume (flux condition) during the iteration. This
method is first tested with an ABC-type flow. Its benefits when compared to
typical Runge-Kutta scheme are demonstrated. Potential use of the method to
exhibit "coherent" Lagrangian structures in a chaotic setting is shown. An
illustration to the computation of magnetic field lines resulting from a
three-dimensional MHD simulation is also provided.


title: On Decompositions of the KdV 2-Soliton

abstract:
  The KdV equation is the canonical example of an integrable non-linear partial
differential equation supporting multi-soliton solutions. Seeking to understand
the nature of this interaction, we investigate different ways to write the KdV
2-soliton solution as a sum of two or more functions. The paper reviews
previous work of this nature and introduces new decompositions with unique
features, putting it all in context and in a common notation for ease of
comparison.


title: Perturbed soliton excitations in DNA molecular chain

abstract:
  We study nonlinear dynamics of a periodic inhomogeneous DNA double helical
chain under dynamic plane-base rotator model by considering angular rotation of
bases in a plane normal to the helical axis. The dynamics is governed by a
perturbed sine-Gordon equation. The perturbed soliton solution is obtained
using a multiple scale soliton perturbation theory. The perturbed kink-antikink
solitons represent formation of open state configuration with fluctuation in
DNA.


title:
Real Hamiltonian Forms of Affine Toda Models Related to Exceptional Lie
  Algebras

abstract:
  The construction of a family of real Hamiltonian forms (RHF) for the special
class of affine 1+1-dimensional Toda field theories (ATFT) is reported. Thus
the method, proposed in [1] for systems with finite number of degrees of
freedom is generalized to infinite-dimensional Hamiltonian systems. The
construction method is illustrated on the explicit nontrivial example of RHF of
ATFT related to the exceptional algebras E_6 and E_7. The involutions of the
local integrals of motion are proved by means of the classical R-matrix
approach.


title: Statics and Dynamics of an Inhomogeneously-Nonlinear Lattice

abstract:
  We introduce an inhomogeneously-nonlinear Schr{\"o}dinger lattice, featuring
a defocusing segment, a focusing segment and a transitional interface between
the two. We illustrate that such inhomogeneous settings present vastly
different dynamical behavior than the one expected in their homogeneous
counterparts in the vicinity of the interface. We analyze the relevant
stationary states, as well as their stability by means of perturbation theory
and linear stability analysis. We find good agreement with the numerical
findings in the vicinity of the anti-continuum limit. For larger values of the
coupling, we follow the relevant branches numerically and show that they
terminate at values of the coupling strength which are larger for more extended
solutions. The dynamical development of relevant instabilities is also
monitored in the case of unstable solutions.


title:
Exponentially growing solutions in homogeneous Rayleigh-Benard
  convection

abstract:
  It is shown that homogeneous Rayleigh-Benard flow, i.e., Rayleigh-Benard
turbulence with periodic boundary conditions in all directions and a volume
forcing of the temperature field by a mean gradient, has a family of exact,
exponentially growing, separable solutions of the full non-linear system of
equations. These solutions are clearly manifest in numerical simulations above
a computable critical value of the Rayleigh number. In our numerical
simulations they are subject to secondary numerical noise and resolution
dependent instabilities that limit their growth to produce statistically steady
turbulent transport.


title: Spatial synchronization and extinction of species under external forcing

abstract:
  We study the interplay between synchronization and extinction of a species.
Using a general model we show that under a common external forcing, the species
with a quadratic saturation term in the population dynamics first undergoes
spatial synchronization and then extinction, thereby avoiding the rescue
effect. This is because the saturation term reduces the synchronization time
scale but not the extinction time scale. The effect can be observed even when
the external forcing acts only on some locations provided there is a
synchronizing term in the dynamics. Absence of the quadratic saturation term
can help the species to avoid extinction.


title: On Reductions and Real Hamiltonian Forms of Affine Toda Field Theories

abstract:
  A family of real Hamiltonian forms (RHF) for the special class of affine 1+1
- dimensional Toda field theories is constructed. Thus the method, proposed in
[Mikhailov;1981] for systems with finite number of degrees of freedom is
generalized to infinite-dimensional Hamiltonian systems. We show that each of
these RHF is related to a special Z_2-symmetry of the system of roots for the
relevant Kac-Moody algebra. A number of explicit nontrivial examples of RHF of
ATFT are presented.


title: Elliptic Schlesinger system and Painlev{\'e} VI

abstract:
  We construct an elliptic generalization of the Schlesinger system (ESS) with
positions of marked points on an elliptic curve and its modular parameter as
independent variables (the parameters in the moduli space of the complex
structure). ESS is a non-autonomous Hamiltonian system with pair-wise commuting
Hamiltonians. The system is bihamiltonian with respect to the linear and the
quadratic Poisson brackets. The latter are the multi-color generalization of
the Sklyanin-Feigin-Odeskii classical algebras. We give the Lax form of the
ESS. The Lax matrix defines a connection of a flat bundle of degree one over
the elliptic curve with first order poles at the marked points.
  The ESS is the monodromy independence condition on the complex structure for
the linear systems related to the flat bundle.
  The case of four points for a special initial data is reduced to the
Painlev{\'e} VI equation in the form of the Zhukovsky-Volterra gyrostat,
proposed in our previous paper.


title:
Two-soliton solution for the derivative nonlinear Schr\"odinger equation
  with nonvanishing boundary conditions

abstract:
  An explicit two-soliton solution for the derivative nonlinear Schr\"odinger
equation with nonvanishing boundary conditions is derived, demonstrating
details of interactions between two bright solitons, two dark solitons, as well
as one bright soliton and one dark soliton. Shifts of soliton positions due to
collisions are analytically obtained, which are irrespective of the bright or
dark characters of the participating solitons.


title: Chaotic Jets

abstract:
  The problem of characterizing the origin of the non-Gaussian properties of
transport resulting from Hamiltonian dynamics is addressed. For this purpose
the notion of chaotic jet is revisited and leads to the definition of a
diagnostic able to capture some singular properties of the dynamics. This
diagnostic is applied successfully to the problem of advection of passive
tracers in a flow generated by point vortices. We present and discuss this
diagnostic as a result of which clues on the origin of anomalous transport in
these systems emerge.


title: Evidence of fractional transport in point-vortex flow

abstract:
  Advection properties of passive particles in flows generated by point
vortices are considered. Transport properties are anomalous with characteristic
transport exponent $\mu \sim 1.5$. This behavior is linked back to the presence
of coherent fractal structures within the flow. A fractional kinetic analysis
allows to link the characteristic transport exponent $\mu $ to the trapping
time exponent $\gamma =1+\mu $. The quantitative agreement is found for
different systems of vortices investigated and a clear signature is obtained of
the fractional nature of transport in these flows.


title:
Pattern stabilization through parameter alternation in a nonlinear
  optical system

abstract:
  We report the first experimental realization of pattern formation in a
spatially extended nonlinear system when the system is alternated between two
states, neither of which exhibits patterning. Dynamical equations modeling the
system are used for both numerical simulations and a weakly nonlinear analysis
of the patterned states. The simulations show excellent agreement with the
experiment. The nonlinear analysis provides an explanation of the patterning
under alternation and accurately predicts both the observed dependence of the
patterning on the frequency of alternation, and the measured spatial
frequencies of the patterns.


title: A Unified Approach to Attractor Reconstruction

abstract:
  In the analysis of complex, nonlinear time series, scientists in a variety of
disciplines have relied on a time delayed embedding of their data, i.e.
attractor reconstruction. The process has focused primarily on heuristic and
empirical arguments for selection of the key embedding parameters, delay and
embedding dimension. This approach has left several long-standing, but common
problems unresolved in which the standard approaches produce inferior results
or give no guidance at all. We view the current reconstruction process as
unnecessarily broken into separate problems. We propose an alternative approach
that views the problem of choosing all embedding parameters as being one and
the same problem addressable using a single statistical test formulated
directly from the reconstruction theorems. This allows for varying time delays
appropriate to the data and simultaneously helps decide on embedding dimension.
A second new statistic, undersampling, acts as a check against overly long time
delays and overly large embedding dimension. Our approach is more flexible than
those currently used, but is more directly connected with the mathematical
requirements of embedding. In addition, the statistics developed guide the user
by allowing optimization and warning when embedding parameters are chosen
beyond what the data can support. We demonstrate our approach on uni- and
multivariate data, data possessing multiple time scales, and chaotic data. This
unified approach resolves all the main issues in attractor reconstruction.


title:
Poisson structure and Action-Angle variables for the Camassa-Holm
  equation

abstract:
  The Poisson brackets for the scattering data of the Camassa-Holm equation are
computed. Consequently, the action-angle variables are expressed in terms of
the scattering data.


title: Runup of nonlinear deformed waves on a beach

abstract:
  The problem of the sea wave runup on a beach is discussed in the framework of
the rigorous solutions of the nonlinear shallow-water theory. Key and novel
moment here is the accounting of the asymmetric waves when the face slope
steepness exceeds the back slope steepness. It is shown that the runup height
growth with increase of the face slope steepness meanwhile the rundown
characteristics are weakly depend from the face slope steepness. These results
explain why the tsunami waves with steep front (as during the 2004 tsunami in
the Indian Ocean) penetrate inland on large distances to compare with wave with
symmetrical shape.


title: Steepness and spectrum of nonlinear deformed wave in shallow water

abstract:
  Process of the nonlinear deformation of the surface wave in shallow water is
studied. Main attention is paid to the relation between the Fourier-spectrum
and wave steepness. It is shown that the spectral harmonics of the initially
sine wave can be expressed through the wave steepness, and this is important
for applications.


title:
Two-dimensional solitons in saturable media with a quasi-one-dimensional
  lattice potential

abstract:
  We study families of solitons in a two-dimensional (2D) model of the light
transmission through a photorefractive medium equipped with a
(quasi-)one-dimensional photonic lattice. The soliton families are bounded from
below by finite minimum values of the peak and total power. Narrow solitons
have a single maximum, while broader ones feature side lobes. Stability of the
solitons is checked by direct simulations. The solitons can be set in motion
across the lattice (actually, made tilted in the spatial domain), provided that
the respective boost parameter does not exceed a critical value. Collisions
between moving solitons are studied too. Collisions destroy the solitons,
unless their velocities are sufficiently small. In the latter case, the
colliding solitons merge into a single stable pulse.


title: Time scale synchronization of chaotic oscillators

abstract:
  This paper presents the result of the investigation of chaotic oscillator
synchronization. A new approach for detecting of synchronized behaviour of
chaotic oscillators has been proposed. This approach is based on the analysis
of different time scales in the time series generated by the coupled chaotic
oscillators. This approach has been applied for the coupled Rossler and Lorenz
systems.


title: Mechanisms Behind the Generalized Synchronization Conditions

abstract:
  A universal mechanism underlying generalized synchronization conditions in
unidirectionally coupled stochastic oscillators is considered. The
consideration is carried out in the framework of a modified system with
additional dissipation. The approach developed is illustrated with model
examples. The conclusion is reached that two types of the behavior of nonlinear
dynamic systems known as generalized synchronization and noise-induced
synchronization, which are viewed as different phenomena, actually represent a
unique type of the synchronous behavior of stochastic oscillators and are
caused by the same mechanism.


title:
Turbulent Drag Reduction by Flexible and Rodlike Polymers: Crossover
  Effects at Small Concentrations

abstract:
  Drag reduction by polymers is bounded between two universal asymptotes, the
von-K\'arm\'an log-law of the law and the Maximum Drag Reduction (MDR)
asymptote. It is theoretically understood why the MDR asymptote is universal,
independent of whether the polymers are flexible or rodlike. The cross-over
behavior from the Newtonian von-K\'arm\'an log-law to the MDR is however not
universal, showing different characteristics for flexible and rodlike polymers.
In this paper we provide a theory for this cross-over phenomenology.


title:
Discrete solitons and nonlinear surface modes in semi-infinite waveguide
  arrays

abstract:
  We discuss the formation of self-trapped localized states near the edge of a
semi-infinite array of nonlinear waveguides. We study a crossover from
nonlinear surface states to discrete solitons by analyzing the families of odd
and even modes centered at different distances from the surface, and reveal the
physical mechanism of the nonlinearity-induced stabilization of surface modes.


title:
Two-color discrete localized modes and resonant scattering in arrays of
  nonlinear quadratic optical waveguides

abstract:
  We analyze the properties and stability of two-color discrete localized modes
in arrays of channel waveguides where tunable quadratic nonlinearity is
introduced as a nonlinear defect by periodic poling of a single waveguide in
the array. We show that, depending on the value of the phase mismatch and the
input power, such two-color defect modes can be realized in three different
localized states. We also study resonant light scattering in the arrays with
the defect waveguide.


title:
Composite solitary waves in three-component scalar field theory: I. The
  kink variety

abstract:
  We study the structure of the manifold of solitary waves in a particular
three-component scalar field theoretical model in two-dimensional Minkowski
space. These solitary waves involve one, two, three, four, six or seven lumps
of energy.


title:
Composite solitary waves in three-component scalar field theory: II.
  Three-body low-energy scattering

abstract:
  We discuss time evolution of some solitary waves described in the first part
of this work. The adiabatic motion of the non-linear non-dispersive waves
composed of three lumps is interpreted as three-body low energy scattering of
these particle-like kinks.


title:
On elliptic solutions of the quintic complex one-dimensional
  Ginzburg-Landau equation

abstract:
  The Conte-Musette method has been modified for the search of only elliptic
solutions to systems of differential equations. A key idea of this a priory
restriction is to simplify calculations by means of the use of a few Laurent
series solutions instead of one and the use of the residue theorem. The
application of our approach to the quintic complex one-dimensional
Ginzburg-Landau equation (CGLE5) allows to find elliptic solutions in the wave
form. We also find restrictions on coefficients, which are necessary conditions
for the existence of elliptic solutions for the CGLE5. Using the investigation
of the CGLE5 as an example, we demonstrate that to find elliptic solutions the
analysis of a system of differential equations is more preferable than the
analysis of the equivalent single differential equation.


title: Coarse-grained description of a passive scalar

abstract:
  The issue of the parameterization of small-scale dynamics is addressed in the
context of passive-scalar turbulence. The basic idea of our strategy is to
identify dynamical equations for the coarse-grained scalar dynamics starting
from closed equations for two-point statistical indicators. With the aim of
performing a fully-analytical study, the Kraichnan advection model is
considered. The white-in-time character of the latter model indeed leads to
closed equations for the equal-time scalar correlation functions. The classical
closure problem however still arises if a standard filtering procedure is
applied to those equations in the spirit of the large-eddy-simulation strategy.
We show both how to perform exact closures and how to identify the
corresponding coarse-grained scalar evolution.


title: Fractional Liouville and BBGKI Equations

abstract:
  We consider the fractional generalizations of Liouville equation. The
normalization condition, phase volume, and average values are generalized for
fractional case.The interpretation of fractional analog of phase space as a
space with fractal dimension and as a space with fractional measure are
discussed. The fractional analogs of the Hamiltonian systems are considered as
a special class of non-Hamiltonian systems. The fractional generalization of
the reduced distribution functions are suggested. The fractional analogs of the
BBGKI equations are derived from the fractional Liouville equation.

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


title: Hierarchical Self-Organization in the Finitary Process Soup

abstract:
  Current analyses of genomes from numerous species show that the diversity of
organism's functional and behavioral characters is not proportional to the
number of genes that encode the organism. We investigate the hypothesis that
the diversity of organismal character is due to hierarchical organization. We
do this with the recently introduced model of the finitary process soup, which
allows for a detailed mathematical and quantitative analysis of the population
dynamics of structural complexity. Here we show that global complexity in the
finitary process soup is due to the emergence of successively higher levels of
organization, that the hierarchical structure appears spontaneously, and that
the process of structural innovation is facilitated by the discovery and
maintenance of relatively noncomplex, but general individuals in a population.


title:
Using the Hadamard and related transforms for simplifying the spectrum
  of the quantum baker's map

abstract:
  We rationalize the somewhat surprising efficacy of the Hadamard transform in
simplifying the eigenstates of the quantum baker's map, a paradigmatic model of
quantum chaos. This allows us to construct closely related, but new, transforms
that do significantly better, thus nearly solving for many states of the
quantum baker's map. These new transforms, which combine the standard Fourier
and Hadamard transforms in an interesting manner, are constructed from
eigenvectors of the shift permutation operator that are also simultaneous
eigenvectors of bit-flip (parity) and possess bit-reversal (time-reversal)
symmetry.


title: Topology, Braids, and Mixing in Fluids

abstract:
  Stirring of fluid with moving rods is necessary in many practical
applications to achieve homogeneity. These rods are topological obstacles that
force stretching of fluid elements. The resulting stretching and folding is
commonly observed as filaments and striations, and is a precursor to mixing. In
a space-time diagram, the trajectories of the rods form a braid, and the
properties of this braid impose a minimal complexity in the flow. We review the
topological viewpoint of fluid mixing, and discuss how braids can be used to
diagnose mixing and construct efficient mixing devices. We introduce a new,
realisable design for a mixing device, the silver mixer, based on these
principles.


title: Surrogate data method applied to nonlinear time series

abstract:
  The surrogate data method is widely applied as a data dependent technique to
test observed time series against a barrage of hypotheses. However, often the
hypotheses one is able to address are not those of greatest interest,
particularly for system known to be nonlinear. In the review we focus on
techniques which overcome this shortcoming. We summarize a number of recently
developed surrogate data methods. While our review of surrogate methods is not
exhaustive, we do focus on methods which may be applied to experimental, and
potentially nonlinear, data. In each case, the hypothesis being tested is one
of the interests to the experimental scientist.


title:
Discrete mappings with an explicit discrete Lyapunov function related to
  integrable mappings

abstract:
  We propose discrete mappings of second order that have a discrete analogue of
Lyapunov function. The mappings are extensions of the integrable
Quispel-Roberts-Thompson (QRT) mapping, and a discrete Lyapunov function of the
mappings is identical to an explicit conserved quantity of the QRT mapping.
Moreover we can obtain a differential and an ultradiscrete limit of the
mappings preserving the existence of Lyapunov function. We also give
applications of a mapping with an adjusted parameter, a probabilistic mapping
and coupled mappings.


title: Evidence for a Binary origin of the Young Planetary Nebula HB 12

abstract:
  Young planetary nebulae play an important role in stellar evolution when
intermediate- to low-mass stars (0.8 ~ 8 M) evolve from the proto-planetary
nebulae phase to the planetary nebulae phase. Many young planetary nebulae
display distinct bipolar structures as they evolve away from the
proto-planetary nebulae phase. One possible cause of their bipolarity could be
due to a binary origin of its energy source. Here we report our detailed
investigation of the young planetary nebula, Hubble 12, which is well-known for
its extended hourglass-like envelope. We present evidence with time-series
photometric observations the existence of an eclipsing binary at the center of
Hubble 12. Low-resolution spectra of the central source show, on the other
hand, absorption features such as CN, G-band & Mg b, which can be suggestive of
a low-mass nature of the secondary component.


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

abstract:
  We investigate the bi-Hamiltonian structure of the waterbag model of dKP for
two component case. One can establish the third-order and first-order
Hamiltonian operator associated with the waterbag model. Also, the dispersive
corrections are discussed.


title: Notes on solutions in Wronskian form to soliton equations: KdV-type

abstract:
  This paper can be an overview on solutions in Wronskian/Casoratian form to
soliton equations with KdV-type bilinear forms. We first investigate properties
of matrices commuting with a Jordan block, by which we derive explicit general
solutions to equations satisfied by Wronskian/Casoratian entry vectors, which
we call condition equations. These solutions are given according to the
coefficient matrix in the condition equations taking diagonal or Jordan block
form. Limit relations between these different solutions are described. We take
the KdV equation and the Toda lattice to serve as two examples for solutions in
Wronskian form and Casoratian form, respectively. We also discuss Wronskian
solutions for the KP equation. Finally, we formulate the Wronskian technique as
four steps.


title: Localized Induction Equation for Stretched Vortex Filament

abstract:
  We study numerically the motion of the stretched vortex filaments by using
the localized induction equation with the stretch and that without the stretch.


title:
Inverse scattering method for square matrix nonlinear Schr\"odinger
  equation under nonvanishing boundary conditions

abstract:
  Matrix generalization of the inverse scattering method is developed to solve
the multicomponent nonlinear Schr\"odinger equation with nonvanishing boundary
conditions. It is shown that the initial value problem can be solved exactly.
The multi-soliton solution is obtained from the Gel'fand--Levitan--Marchenko
equation.


title: Lattice hyper-solitons in photorrefractive materials

abstract:
  We show a novel kind of nonlinear waves in two-dimensional photonic lattices.
This waves take the form of light clusters that may fill an arbitrary number of
lattice sites. We have demonstrated by numerical simulations that stable
propagation can be achieved under adequate conditions and we have described the
unstable patterns developed otherwise. Our results show that these new kind of
nonlinear waves can be easily found in current experiments.


title:
Stochastic resonance with different periodic forces in overdamped two
  coupled anharmonic oscillators

abstract:
  We study the stochastic resonance phenomenon in the overdamped two coupled
anharmonic oscillators with Gaussian noise and driven by different external
periodic forces. We consider (i) sine, (ii) square, (iii) symmetric saw-tooth,
(iv) asymmetric saw-tooth, (v) modulus of sine and (vi) rectified sinusoidal
forces. The external periodic forces and Gaussian noise term are added to one
of the two state variables of the system. The effect of each force is studied
separately. In the absence of noise term, when the amplitude $f$ of the applied
periodic force is varied cross-well motion is realized above a critical value
($f_{\mathrm{c}}$) of $f$. This is found for all the forces except the modulus
of sine and rectified sinusoidal forces.Stochastic resonance is observed in the
presence of noise and periodic forces. The effect of different forces is
compared. The logarithmic plot of mean residence time $\tau_{\mathrm{MR}}$
against $ 1/(D - D_{\mathrm{c}})$ where $D$ is the intensity of the noise and
$D_{\mathrm{c}}$ is the value of $D$ at which cross-well motion is initiated
shows a sharp knee-like structure for all the forces. Signal-to-noise ratio is
found to be maximum at the noise intensity $D=D_{\mathrm{max}}$ at which mean
residence time is half of the period of the driving force for the forces such
as sine, square, symmetric saw-tooth and asymmetric saw-tooth waves. With
modulus of sine wave and rectified sine wave, the $SNR$ peaks at a value of $D$
for which sum of $\tau_{MR}$ in two wells of the potential of the system is
half of the period of the driving force. For the chosen values of $f$ and
$\omega$, signal-to-noise ratio is found to be maximum for square wave while it
is minimum for modulus of sine and rectified sinusoidal waves.


title:
Harmonic vs. subharmonic patterns in a spatially forced oscillating
  chemical reaction

abstract:
  The effects of a spatially periodic forcing on an oscillating chemical
reaction as described by the Lengyel-Epstein model are investigated. We find a
surprising competition between two oscillating patterns, where one is harmonic
and the other subharmonic with respect to the spatially periodic forcing. The
occurrence of a subharmonic pattern is remarkable as well as its preference up
to rather large values of the modulation amplitude. For small modulation
amplitudes we derive from the model system a generic equation for the envelope
of the oscillating reaction that includes an additional forcing contribution,
compared to the amplitude equations known from previous studies in other
systems. The analysis of this amplitude equation allows the derivation of
analytical expressions even for the forcing corrections to the threshold and to
the oscillation frequency, which are in a wide range of parameters in good
agreement with the numerical analysis of the complete reaction equations. In
the nonlinear regime beyond threshold, the subharmonic solutions exist in a
finite range of the control parameter that has been determined by solving the
reaction equations numerically for various sets of parameters.


title:
Stability of two-dimensional spatial solitons in nonlocal nonlinear
  media

abstract:
  We discuss existence and stability of two-dimensional solitons in media with
spatially nonlocal nonlinear response. We show that such systems, which include
thermal nonlinearity and dipolar Bose Einstein condensates, may support a
variety of stationary localized structures - including rotating spatial
solitons. We also demonstrate that the stability of these structures critically
depends on the spatial profile of the nonlocal response function.


title:
A geometric framework for phase synchronization in coupled noisy
  nonlinear systems

abstract:
  A geometric approach is introduced for understanding the phenomenon of phase
synchronization in coupled nonlinear systems in the presence of additive noise.
We show that the emergence of cooperative behaviour through a change of
stability via a Hopf bifurcation entails the spontaneous appearance of a gauge
structure in the system, arising from the evolution of the slow dynamics, but
induced by the fast variables. The conditions for the oscillators to be
synchronised in phase are obtained. The role of weak noise appears to be to
drive the system towards a more synchronized behaviour. Our analysis provides a
framework to explain recent experimental observations on noise-induced phase
synchronization in coupled nonlinear systems.


title: Lattice complexity and fine graining of symbolic sequence

abstract:
  A new complexity measure named as Lattice Complexity is presented for finite
symbolic sequences. This measure is based on the symbolic dynamics of
one-dimensional iterative maps and Lempel-Ziv Complexity. To make Lattice
Complexity distinguishable from Lempel-Ziv Complexity, an approach called
fine-graining process is also proposed. When the control parameter
fine-graining order is small enough, the two measures are almost equal. While
the order increases, the difference between the two measures becomes more and
more significant. Applying Lattice Complexity to logistic map with a proper
order, we find that the sequences that are regarded as complex are roughly at
the edges of chaotic regions. Further derived properties of the two measures
concerning the fine-graining process are also discussed.


title: Transition function for the Toda chain model

abstract:
  The method of Lambda-operators developed by S. Derkachov, G. Korchemsky, A.
Manashov is applied to a derivation of eigenfunctions for the open Toda chain.
The Sklyanin measure is reproduced using diagram technique developed for these
Lambda-operators. The properties of the Lambda-operators are studied. This
approach to the open Toda chain eigenfunctions reproduces Gauss-Givental
representation for these eigenfunctions.


title: Functional representations of integrable hierarchies

abstract:
  We consider a general framework for integrable hierarchies in Lax form and
derive certain universal equations from which `functional representations' of
particular hierarchies (like KP, discrete KP, mKP, AKNS), i.e. formulations in
terms of functional equations, are systematically and quite easily obtained.
The formalism genuinely applies to hierarchies where the dependent variables
live in a noncommutative (typically matrix) algebra. The obtained functional
representations can be understood as `noncommutative' analogs of `Fay
identities' for the KP hierarchy.


title: Inverse Scattering Transform for the Camassa-Holm equation

abstract:
  An Inverse Scattering Method is developed for the Camassa-Holm equation. As
an illustration of our approach the solutions corresponding to the
reflectionless potentials are explicitly constructed in terms of the scattering
data. The main difference with respect to the standard Inverse Scattering
Transform lies in the fact that we have a weighted spectral problem. We
therefore have to develop different asymptotic expansions.


title: Dynamo in Helical MHD Turbulence: Quantum Field Theory Approach

abstract:
  A quantum field model of helical MHD stochastically forced by gaussian
hydrodynamic, magnetic and mixed noices is investigated. These helical noises
lead to an exponential increase of magnetic fluctuations in the large scale
range. Instabilities, which are produced in this process, are eliminated by
spontaneous symmetry breaking mechanism accompanied by creation of the
homogeneous stationary magnetic field.


title:
Equivalence of kinetic-theory and random-matrix approaches to Lyapunov
  spectra of hard-sphere systems

abstract:
  In the study of chaotic behaviour of systems of many hard spheres, Lyapunov
exponents of small absolute value exhibit interesting characteristics leading
to speculations about connections to non-equilibrium statistical mechanics.
Analytical approaches to these exponents so far can be divided into two groups,
macroscopically oriented approaches, using kinetic theory or hydrodynamics, and
more microscopically oriented random-matrix approaches in quasi-one-dimensional
systems. In this paper, I present an approach using random matrices and weak
disorder expansion in an arbitrary number of dimensions. Correlations between
subsequent collisions of a particle are taken into account. It is shown that
the results are identical to those of a previous approach based on an extended
Enskog-equation. I conclude that each approach has its merits, and provides
different insights into the approximations made, which include the
Sto{\ss}zahlansatz, the continuum limit, and the long-wavelength approximation.
The comparison also gives insight into possible connections between Lyapunov
exponents and fluctuations.


title: Translationally invariant nonlinear Schrodinger lattices

abstract:
  Persistence of stationary and traveling single-humped localized solutions in
the spatial discretizations of the nonlinear Schrodinger (NLS) equation is
addressed. The discrete NLS equation with the most general cubic polynomial
function is considered. Constraints on the nonlinear function are found from
the condition that the second-order difference equation for stationary
solutions can be reduced to the first-order difference map. The discrete NLS
equation with such an exceptional nonlinear function is shown to have a
conserved momentum but admits no standard Hamiltonian structure. It is proved
that the reduction to the first-order difference map gives a sufficient
condition for existence of translationally invariant single-humped stationary
solutions and a necessary condition for existence of single-humped traveling
solutions. Other constraints on the nonlinear function are found from the
condition that the differential advance-delay equation for traveling solutions
admits a reduction to an integrable normal form given by a third-order
differential equation. This reduction also gives a necessary condition for
existence of single-humped traveling solutions. The nonlinear function which
admits both reductions defines a two-parameter family of discrete NLS equations
which generalizes the integrable Ablowitz--Ladik lattice.


title: Evolution equation for bidirectional surface waves in a convecting fluid

abstract:
  Surface waves in a heated viscous fluid exhibit a long wave oscillatory
instability. The nonlinear evolution of unidirectional waves is known to be
described by a modified Korteweg-deVries-Kuramoto-Sivashinsky equation. In the
present work we eliminate the restriction of unidirectional waves and find that
the evolution of the wave is governed by a modified Boussinesq system . A
perturbed Boussinesq equation of the form $y_{tt}-y_{xx} -\epsilon^2(y_{xxtt} +
(y^2)_{xx})+ \epsilon^3(y_{xxt}+y_{xxxxt} + (y^2)_{xxt}) =0 $ which includes
instability and dissipation is derived from this system.


title:
A non subjective approach to the GP algorithm for analysing noisy time
  series

abstract:
  We present an adaptation of the standard Grassberger-Proccacia (GP) algorithm
for estimating the Correlation Dimension of a time series in a non subjective
manner. The validity and accuracy of this approach is tested using different
types of time series, such as, those from standard chaotic systems, pure white
and colored noise and chaotic systems added with noise. The effectiveness of
the scheme in analysing noisy time series, particularly those involving colored
noise, is investigated. An interesting result we have obtained is that, for the
same percentage of noise addition, data with colored noise is more
distinguishable from the corresponding surrogates, than data with white noise.
As examples for real life applications, analysis of data from an astrophysical
X-ray object and human brain EEG, are presented.


title: Electronic Circuit Realization of the Logistic Map

abstract:
  An electronic circuit realization of the logistic difference equation is
presented using analog electronics. The behavior of the realized system is
evaluated against computer simulations of the same. The circuit is found to
exhibit the entire range of dynamics of the logistic equation: fixed points,
periodicity, period doubling, chaos and intermittency. Quantitative
measurements of the dynamics of the realized system are presented and are found
to be in good agreement with the theoretical values. Some possible applications
of such a realization are briefly discussed.


title: Construction Theory, Self-Replication, and the Halting Problem

abstract:
  This essay aims to propose construction theory, a new domain of theoretical
research on machine construction, and use it to shed light on a fundamental
relationship between living and computational systems. Specifically, we argue
that self-replication of von Neumann's universal constructors holds a close
similarity to circular computational processes of universal computers that
appear in Turing's original proof of the undecidability of the halting problem.
The result indicates the possibility of reinterpreting a self-replicating
biological organism as embodying an attempt to solve the halting problem for a
{\em diagonal} input in the context of construction. This attempt will never be
completed because of the indefinite cascade of self-computation/construction,
which accounts for the undecidability of the halting problem and also agrees
well with the fact that life has maintained its reproductive activity for an
indefinitely long period of time.


title:
Exact soliton solutions and nonlinear modulation instability in spinor
  Bose-Einstein condensates

abstract:
  We find one-, two-, and three-component solitons of the polar and
ferromagnetic (FM) types in the general (non-integrable) model of a spinor
(three-component) model of the Bose-Einstein condensate (BEC), based on a
system of three nonlinearly coupled Gross-Pitaevskii equations. The stability
of the solitons is studied by means of direct simulations, and, in a part,
analytically, using linearized equations for small perturbations. Global
stability of the solitons is considered by means of the energy comparison. As a
result, ground-state and metastable soliton states of the FM and polar types
are identified. For the special integrable version of the model, we develop the
Darboux transformation (DT). As an application of the DT, analytical solutions
are obtained that display full nonlinear evolution of the modulational
instability (MI) of a continuous-wave (CW) state seeded by a small spatially
periodic perturbation. Additionally, by dint of direct simulations, we
demonstrate that solitons of both the polar and FM types, found in the
integrable system, are structurally stable, i.e., they are robust under random
changes of the relevant nonlinear coefficient in time.


title:
Baxter-Bazhanov-Stroganov model: Separation of Variables and Baxter
  Equation

abstract:
  The Baxter-Bazhanov-Stroganov model (also known as the \tau^(2) model) has
attracted much interest because it provides a tool for solving the integrable
chiral Z_N-Potts model. It can be formulated as a face spin model or via cyclic
L-operators. Using the latter formulation and the Sklyanin-Kharchev-Lebedev
approach, we give the explicit derivation of the eigenvectors of the component
B_n(\lambda) of the monodromy matrix for the fully inhomogeneous chain of
finite length. For the periodic chain we obtain the Baxter T-Q-equations via
separation of variables. The functional relations for the transfer matrices of
the \tau^(2) model guarantee non-trivial solutions to the Baxter equations. For
the N=2 case, which is free fermion point of a generalized Ising model, the
Baxter equations are solved explicitly.

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


title: Gardner's deformations of the Boussinesq equations

abstract:
  Using the algebraic method of Gardner's deformations for completely
integrable systems, we construct the recurrence relations for densities of the
Hamiltonians for the Boussinesq and the Kaup-Boussinesq equations. By extending
the Magri schemes for these systems, we obtain new integrable equations adjoint
with respect to the initial ones and describe their Hamiltonian structures and
symmetry properties.


title:
Influence of helicity on anomalous scaling of a passive scalar advected
  by the turbulent velocity field with finite correlation time: Two-loop
  approximation

abstract:
  The influence of helicity on the stability of scaling regimes, on the
effective diffusivity, and on the anomalous scaling of structure functions of a
passive scalar advected by a Gaussian solenoidal velocity field with finite
correlation time is investigated by the field theoretic renormalization group
and operator product expansion within two-loop approximation. The influence of
helicity on the scaling regimes is discussed and shown in the plane of
exponents $\epsilon-\eta$, where $\epsilon$ characterizes the energy spectrum
of the velocity field in the inertial range $E\propto k^{1-2\epsilon}$, and
$\eta$ is related to the correlation time at the wave number $k$ which is
scaled as $k^{-2+\eta}$. The restrictions given by nonzero helicity on the
regions with stable fixed points which correspond to the scaling regimes are
analyzed in detail. The dependence of the effective diffusivity on the helicity
parameter is discussed. The anomalous exponents of the structure functions of
the passive scalar field which define their anomalous scaling are calculated
and it is shown that although the separate composite operators which define
them strongly depend on the helicity parameter the resulting two-loop
contributions to the critical dimensions of the structure functions are
independent of helicity. Details of calculations are shown.


title:
Scaling, renormalization and statistical conservation laws in the
  Kraichnan model of turbulent advection

abstract:
  We present a systematic way to compute the scaling exponents of the structure
functions of the Kraichnan model of turbulent advection in a series of powers
of $\xi$, adimensional coupling constant measuring the degree of roughness of
the advecting velocity field. We also investigate the relation between standard
and renormalization group improved perturbation theory. The aim is to shed
light on the relation between renormalization group methods and the statistical
conservation laws of the Kraichnan model, also known as zero modes.


title: A normal form for excitable media

abstract:
  We present a normal form for travelling waves in one-dimensional excitable
media in form of a differential delay equation. The normal form is built around
the well-known saddle-node bifurcation generically present in excitable media.
Finite wavelength effects are captured by a delay. The normal form describes
the behaviour of single pulses in a periodic domain and also the richer
behaviour of wave trains. The normal form exhibits a symmetry preserving Hopf
bifurcation which may coalesce with the saddle-node in a Bogdanov-Takens point,
and a symmetry breaking spatially inhomogeneous pitchfork bifurcation. We
verify the existence of these bifurcations in numerical simulations. The
parameters of the normal form are determined and its predictions are tested
against numerical simulations of partial differential equation models of
excitable media with good agreement.


title:
Infinitesimal symmetries and conservation laws of the DNLSE hierarchy
  and the Noether's theorem

abstract:
  The hierarchy of the integrable nonlinear equations associated with the
quadratic bundle is considered. The expressions for the solution of the
linearization of these equations and their conservation law in the terms of the
solutions of the corresponding Lax pairs are found. It is shown for the first
member of the hierarchy that the conservation law is connected with the
solution of the linearized equation due to the Noether's theorem. The local
hierarchy and three nonlocal ones of the infinitesimal symmetries and the
conservation laws that are explicitly expressed through the variables of the
nonlinear equations are derived.


title:
Discrete Nonlinear Schrodinger Equations with arbitrarily high order
  nonlinearities

abstract:
  A class of discrete nonlinear Schrodinger equations with arbitrarily high
order nonlinearities is introduced. These equations are derived from the same
Hamiltonian using different Poisson brackets and include as particular cases
the saturable discrete nonlinear Schrodinger equation and the Ablowitz-Ladik
equation. As a common property, these equations possess three kinds of exact
analytical stationary solutions for which the Peierls-Nabarro barrier is zero.
Several properties of these solutions, including stability, discrete breathers
and moving solutions, are investigated.


title:
Field momentum and gyroscopic dynamics of classical systems with
  topological defects

abstract:
  The standard relation between the field momentum and the force is generalized
for the system with a field singularity: in addition to the regular force,
there appear the singular one. This approach is applied to the description of
the gyroscopic dynamics of the classical field with topological defects. The
collective variable Lagrangian description is considered for gyroscopical
systems with account of singularities. Using this method we describe the
dynamics of two-dimensional magnetic solitons. We establish a relation between
the gyroscopic force and the singular one. An effective Lagrangian description
is discussed for the magnetic soliton dynamics.


title:
Turbulent Relative Dispersion in Two-Dimensional Free Convection
  Turbulence

abstract:
  The relative dispersion process in two-dimensional free convection turbulence
is investigated by direct numerical simulation. In the inertial range, the
growth of relative separation, $r$, is expected as $<r^2(t)>\propto t^5$
according to the Bolgiano-Obukhov scaling. The result supporting the scaling is
obtained with exit-time statistics. Detailed investigation of exit-time PDF
shows that the PDF is divided into two regions, the Region-I and -II,
reflecting two types of separating processes: persistent expansion and random
transitions between expansion and compression of relative separation. This is
consistent with the physical picture of the self-similar telegraph model. In
addition, a method for estimating the parameters of the model are presented.
Comparing two turbulence cases, two-dimensional free convection and inverse
cascade turbulence, the relation between the drift term of the model and nature
of coherent structures is discussed.


title:
Inverse problem of variational calculus for nonlinear evolution
  equations

abstract:
  We couple a nonlinear evolution equation with an associated one and derive
the action principle. This allows us to write the Lagrangian density of the
system in terms of the original field variables rather than Casimir potentials.
We find that the corresponding Hamiltonian density provides a natural basis to
recast the pair of equations in the canonical form. Amongst the case studies
presented the KdV and modified KdV pairs exhibit bi-Hamiltonian structure and
allow one to realize the associated fields in physical terms


title:
On a Generalized Fifth-Order Integrable Evolution Equation and its
  Hierarchy

abstract:
  A general form of the fifth-order nonlinear evolution equation is considered.
Helmholtz solution of the inverse variational problem is used to derive
conditions under which this equation admits an analytic representation. A
Lennard type recursion operator is then employed to construct a hierarchy of
Lagrangian equations. It is explicitly demonstrated that the constructed system
of equations has a Lax representation and two compatible Hamiltonian
structures. The homogeneous balance method is used to derive analytic soliton
solutions of the third- and fifth-order equations.


title: Collection of Mutually Synchronized Chaotic Systems

abstract:
  A general explicit coupling for mutual synchronization of two arbitrary
identical continuous systems is proposed. The synchronization is proved
analytically. The coupling is given for all 19 systems from Sprott's
collection. For one of the systems the numerical results are shown in detail.
The method could be adopted for the teaching of the topic.


title:
Modes of nonlinear acoustic transparency in the strained paramagnetic
  crystal

abstract:
  The propagation of the transverse-longitudinal acoustic pulses through a
strained cubic crystal containing the resonant paramagnetic impurities with
effective spin S=1 is investigated. It is supposed that the pulses propagate
under arbitrary angle with respect to the direction of the external static
deformation parallel to the fourth-order symmetry axis. In this geometry, both
the transverse and longitudinal components of the acoustic field have the
high-frequency and zero-frequency spectral components. We show that a pulse can
propagate in the modes different from the acoustic self-induced transparency.
In particular, a pulse propagating in the mode of acoustic self-induced
supertransparency substantially changes the populations of the spin sublevels,
but its group velocity remains almost equal to the linear velocity of the
sound. If a pulse propagates in the acoustic extraordinary transparency mode,
then its group velocity is substantially lower while the sublevel populations
remain virtually invariant. Also, the modes of propagation under conditions of
weakly excited spin transitions and large detuning of the pulse high-frequency
components are identified.


title:
Anomalous scaling in two and three dimensions for a passive vector field
  advected by a turbulent flow

abstract:
  A model of the passive vector field advected by the uncorrelated in time
Gaussian velocity with power-like covariance is studied by means of the
renormalization group and the operator product expansion. The structure
functions of the admixture demonstrate essential power-like dependence on the
external scale in the inertial range (the case of an anomalous scaling). The
method of finding of independent tensor invariants in the cases of two and
three dimensions is proposed to eliminate linear dependencies between the
operators entering into the operator product expansions of the structure
functions. The constructed operator bases, which include the powers of the
dissipation operator and the enstrophy operator, provide the possibility to
calculate the exponents of the anomalous scaling.


title: Stability of axial orbits in galactic potentials

abstract:
  We investigate the dynamics in a galactic potential with two reflection
symmetries. The phase-space structure of the real system is approximated with a
resonant detuned normal form constructed with the method based on the Lie
transform. Attention is focused on the stability properties of the axial
periodic orbits that play an important role in galactic models. Using energy
and ellipticity as parameters, we find analytical expressions of bifurcations
and compare them with numerical results available in the literature.


title: Periodic orbits in scattering from elastic voids

abstract:
  The scattering determinant for the scattering of waves from several obstacles
is considered in the case of elastic solids with voids. The multi-scattering
determinant displays contributions from periodic ray-splitting orbits. A
discussion of the weights of such orbits is presented.

categories: nlin.CD cond-mat.mtrl-sci math-ph math.MP


title: Explicit form of the Yablonskii - Vorob'ev polynomials

abstract:
  Special polynomials associated with rational solutions of the second
Painlev\'{e} equation and other members of its hierarchy are discussed. New
approach, which allows one to construct each polynomial is presented. The
structure of the polynomials is established. Formulas of their coefficients are
found. Correlations between the roots of every polynomial are obtained.


title: Towards Self-organizing Bureaucracies

abstract:
  This paper proposes self-organization as a method to improve the efficiency
and adaptability of bureaucracies and similar social systems. Bureaucracies are
described as networks of agents, where the main design principle is to reduce
local "friction" to increase local and global "satisfaction". Following this
principle, solutions are proposed for improving communication within
bureaucracies, sensing public satisfaction, dynamic modification of
hierarchies, and contextualization of procedures. Each of these reduces
friction between agents (internal or external), increasing the efficiency of
bureaucracies. "Random agent networks" (RANs), novel computational models, are
introduced to illustrate the benefits of self-organizing bureaucracies.


title:
On a Class of Spatial Discretizations of Equations of the Nonlinear
  Schrodinger Type

abstract:
  We demonstrate the systematic derivation of a class of discretizations of
nonlinear Schr{\"o}dinger (NLS) equations for general polynomial nonlinearity
whose stationary solutions can be found from a reduced two-point algebraic
condition. We then focus on the cubic problem and illustrate how our class of
models compares with the well-known discretizations such as the standard
discrete NLS equation, or the integrable variant thereof. We also discuss the
conservation laws of the derived generalizations of the cubic case, such as the
lattice momentum or mass and the connection with their corresponding continuum
siblings.


title:
Discrete Nonlinear Schrodinger Equations Free of the Peierls-Nabarro
  Potential

abstract:
  We derive a class of discrete nonlinear Schr{\"o}dinger (DNLS) equations for
general polynomial nonlinearity whose stationary solutions can be found from a
reduced two-point algebraic problem. It is demonstrated that the derived class
of discretizations contains subclasses conserving classical norm or a modified
norm and classical momentum. These equations are interesting from the physical
standpoint since they support stationary discrete solitons free of the
Peierls-Nabarro potential. As a consequence, even in highly-discrete regimes,
solitons are not trapped by the lattice and they can be accelerated by even
weak external fields. Focusing on the cubic nonlinearity we then consider a
small perturbation around stationary soliton solutions and, solving
corresponding eigenvalue problem, we (i) demonstrate that solitons are stable;
(ii) show that they have two additional zero-frequency modes responsible for
their effective translational invariance; (iii) derive semi-analytical
solutions for discrete solitons moving at slow speed. To highlight the unusual
properties of solitons in the new discrete models we compare them with that of
the classical DNLS equation giving several numerical examples.


title:
From AKNS to derivative NLS hierarchies via deformations of associative
  products

abstract:
  Using deformations of associative products, derivative nonlinear Schrodinger
(DNLS) hierarchies are recovered as AKNS-type hierarchies. Since the latter can
also be formulated as Gelfand-Dickey-type Lax hierarchies, a recently developed
method to obtain 'functional representations' can be applied. We actually
consider hierarchies with dependent variables in any (possibly noncommutative)
associative algebra, e.g., an algebra of matrices of functions. This also
covers the case of hierarchies of coupled DNLS equations.


title:
Quantitative laboratory observations of internal wave reflection on
  ascending slopes

abstract:
  Internal waves propagate obliquely through a stratified fluid with an angle
that is fixed with respect to gravity. Upon reflection on a sloping bed,
striking phenomena are expected to occur close to the slope. We present here
laboratory observations at moderately large Reynolds number. A particle image
velocimetry (PIV) technique is used to provide time resolved velocity fields in
large volumes. The generation of the second and third harmonic frequencies are
clearly demonstrated in the impact zone. The mechanism for nonlinear wavelength
selection is also discussed. Evanescent waves with frequency larger than the
Brunt-V\"ais\"al\"a frequency are detected and experimental results agree very
well with theoretical predictions. The amplitude of the different harmonics
after reflection are also obtained.


title: Discrete Solitons in the Salerno Model with competing nonlinearities

abstract:
  We consider a lattice equation (Salerno model) combining onsite self-focusing
and intersite self-defocusing cubic terms, which may describe a Bose-Einstein
condensate of dipolar atoms trapped in a strong periodic potential. In the
continuum approximation, the model gives rise to solitons in a finite band of
frequencies, with sechlike solitons near one edge, and an exact peakon solution
at the other. A similar family of solitons is found in the discrete system,
including a peakon; beyond the peakon, the family continues in the form of
cuspons. Stability of the lattice solitons is explored through computation of
eigenvalues for small perturbations, and by direct simulations. A small part of
the family is unstable (in that case, the discrete solitons transform into
robust pulsonic excitations); both peakons and cuspons are stable. The
Vakhitov-Kolokolov criterion precisely explains the stability of regular
solitons and peakons, but does not apply to cuspons. In-phase and out-of-phase
bound states of solitons are also constructed. They exchange their stability at
a point where the bound solitons are peakons. Mobile solitons, composed of a
moving core and background, exist up to a critical value of the strength of the
self-defocusing intersite nonlinearity. Colliding solitons always merge into a
single pulse.


title: Synchronization on fast and slow dynamics in drive-response systems

abstract:
  Two types of synchronization, Achronal Synchronization and Isochronous
synchronization are investigated numerically when unidirectionally coupled
laser systems are considered both on fast and slow dynamics by studying the
correlation function. Although the synchronization behaviors are found to be
separated on relatively fast dynamics of chaotic fluctuations while blended
together on slow dynamics of dropout events, their main features revealed are
in a good agreement, which most probably suggests that the observation based on
detections on slow dynamics is of importance when it is the only choice as
detection on fast dynamics is usually extremely hard to be done.


title: Excitable Dynamics in the Presence of Time Delay

abstract:
  The spiking properties of a subcritical Hopf oscillator with a time delayed
nonlinear feedback is investigated. Finite time delay is found to significantly
affect both the statistics and the fine structure of the spiking behavior.
These dynamical changes are explained in terms of the fundamental modifications
occurring in the bifurcation scenario of the system. Our mathematical model can
find useful applications in understanding the dynamical behaviour of various
real life excitable systems where propagation delay effects are ubiquitous.


title: Bursting in a Subcritical Hopf Oscillator with a Nonlinear Feedback

abstract:
  Bursting is a periodic transition between a quiescent state and a state of
repetitive spiking. The phenomenon is ubiquitous in a variety of neurophysical
systems. We numerically study the dynamical properties of a normal form of
subcritical Hopf oscillator (at the bifurcation point) subjected to a nonlinear
feedback. This dynamical system shows an infinite-period or a saddle-node on a
limit cycle (SNLC) bifurcation for certain strengths of the nonlinear feedback.
When the feedback is time delayed, the bifurcation scenario changes and the
limit cycle terminates through a homoclinic or a saddle separatrix loop (SSL)
bifurcation. This system when close to the bifurcation point exhibits various
types of bursting phenomenon when subjected to a slow periodic external
stimulus of an appropriate strength. The time delay in the feedback enhances
the spiking rate i.e. reduces the interspike interval in a burst and also
increases the width or the duration of a burst.


title:
Algebro-geometric approach in the theory of integrable hydrodynamic type
  systems

abstract:
  The algebro-geometric approach for integrability of semi-Hamiltonian
hydrodynamic type systems is presented. This method is significantly simplified
for so-called symmetric hydrodynamic type systems. Plenty interesting and
physically motivated examples are investigated.


title:
Classification of integrable hydrodynamic chains and generating
  functions of conservation laws

abstract:
  New approach to classification of integrable hydrodynamic chains is
established. Generating functions of conservation laws are classified by the
method of hydrodynamic reductions. N parametric family of explicit hydrodynamic
reductions allows to reconstruct corresponding hydrodynamic chains. Plenty new
hydrodynamic chains are found.


title: Hydrodynamic chains and a classification of their Poisson brackets

abstract:
  Necessary and sufficient conditions for an existence of the Poisson brackets
significantly simplify in the Liouville coordinates. The corresponding
equations can be integrated. Thus, a description of local Hamiltonian
structures is a first step in a description of integrable hydrodynamic chains.
The concept of $M$ Poisson bracket is introduced. Several new Poisson brackets
are presented.


title:
The Hamiltonian approach in classification and integrability of
  hydrodynamic chains

abstract:
  New approach in classification of integrable hydrodynamic chains is
established. This is the method of the Hamiltonian hydrodynamic reductions.
Simultaneously, this approach yields explicit Hamiltonian hydrodynamic
reductions of the Hamiltonian hydrodynamic chains. The concept of reducible
Poisson brackets is established. Also this approach is useful for
non-Hamiltonian hydrodynamic chains. The deformed Benney hydrodynamic chain is
considered.


title: Space-time directional Lyapunov exponents for cellular automata

abstract:
  Space-time directional Lyapunov exponents are introduced. They describe the
maximal velocity of propagation to the right or to the left of fronts of
perturbations in a frame moving with a given velocity. The continuity of these
exponents as function of the velocity and an inequality relating them to the
directional entropy is proved.


title:
Length-scale estimates for the LANS-alpha equations in terms of the
  Reynolds number

abstract:
  Foias, Holm & Titi \cite{FHT2} have settled the problem of existence and
uniqueness for the 3D \lans equations on periodic box $[0,L]^{3}$. There still
remains the problem, first introduced by Doering and Foias \cite{DF} for the
Navier-Stokes equations, of obtaining estimates in terms of the Reynolds number
$\Rey$, whose character depends on the fluid response, as opposed to the
Grashof number, whose character depends on the forcing. $\Rey$ is defined as
$\Rey = U\ell/\nu$ where $U$ is a bounded spatio-temporally averaged
Navier-Stokes velocity field and $\ell$ the characteristic scale of the
forcing. It is found that the inverse Kolmogorov length is estimated by
$\ell\lambda_{k}^{-1} \leq c (\ell/\alpha)^{1/4}\Rey^{5/8}$. Moreover, the
estimate of Foias, Holm & Titi for the fractal dimension of the global
attractor, in terms of $\Rey$, comes out to be $$ d_{F}(\mathcal{A}) \leq c
\frac{V_{\alpha}V_{\ell}^{1/2}}{(L^{2}\lambda_{1})^{9/8}} \Rey^{9/4} $$ where
$V_{\alpha} = (L/(\ell\alpha)^{1/2})^{3}$ and $V_{\ell} = (L/\ell)^{3}$. It is
also shown that there exists a series of time-averaged inverse squared length
scales whose members, $\left<\kappa_{n,0}^2\right>$, %, are related to the
$2n$th-moments of the energy spectrum when $\alpha\to 0$. are estimated as
$(n\geq 1)$ $$ \ell^{2}\left<\kappa_{n,0}^2\right> \leq
c_{n,\alpha}V_{\alpha}^{\frac{n-1}{n}} \Rey^{{11/4} -
\frac{7}{4n}}(\ln\Rey)^{\frac{1}{n}} + c_{1}\Rey(\ln\Rey) . $$ The upper bound
on the first member of the hierarchy $\left<\kappa_{1,0}^2\right>$ coincides
with the inverse squared Taylor micro-scale to within log-corrections.


title: Singular solutions for geodesic flows of Vlasov moments

abstract:
  The Vlasov equation for the collisionless evolution of the single-particle
probability distribution function (PDF) is a well-known example of coadjoint
motion. Remarkably, the property of coadjoint motion survives the process of
taking moments. That is, the evolution of the moments of the Vlasov PDF is also
coadjoint motion. We find that {\it geodesic} coadjoint motion of the Vlasov
moments with respect to powers of the single-particle momentum admits singular
(weak) solutions concentrated on embedded subspaces of physical space. The
motion and interactions of these embedded subspaces are governed by canonical
Hamiltonian equations for their geodesic evolution.


title: Evolving momentum-projected densities in billiards with quantum states

abstract:
  The classical Liouville density on the constant energy surface reveals a
number of interesting features when the initial density has no directional
preference. It has been shown (Physical Review Letters, 93 (2004) 204102) that
the eigenvalues and eigenfunctions of the momentum-projected density evolution
operator have a correspondence with the quantum Neumann energy eigenstates in
billiard systems. While the classical eigenfunctions are well approximated by
the quantum Neumann eigenfunctions, the classical eigenvalues are of the form
f(\sqrt{E_n} vt) where {E_n} are close to the quantum Neumann eigenvalues and v
is the speed of the classical particle. Despite the approximate nature of the
correspondence, we demonstrate here that the exact quantum Neumann eigenstates
can be used to expand and evolve an arbitrary classical density on the energy
surface projected on to the configuration space. For the rectangular and
stadium billiards, results are compared with the actual evolution of the
density using classical trajectories.


title: High frequency integrable regimes in nonlocal nonlinear optics

abstract:
  We consider an integrable model which describes light beams propagating in
nonlocal nonlinear media of Cole-Cole type. The model is derived as high
frequency limit of both Maxwell equations and the nonlocal nonlinear
Schroedinger equation. We demonstrate that for a general form of nonlinearity
there exist selfguided light beams. In high frequency limit nonlocal
perturbations can be seen as a class of phase deformation along one direction.
We study in detail nonlocal perturbations described by the dispersionless
Veselov-Novikov (dVN) hierarchy. The dVN hierarchy is analyzed by the reduction
method based on symmetry constraints and by the quasiclassical Dbar-dressing
method. Quasiclassical Dbar-dressing method reveals a connection between
nonlocal nonlinear geometric optics and the theory of quasiconformal mappings
of the plane.


title:
Synchronization and time shifts of dynamical patterns for mutually
  delay-coupled fiber ring lasers

abstract:
  A pair of coupled erbium doped fiber ring lasers is used to explore the
dynamics of coupled spatiotemporal systems. The lasers are mutually coupled
with a coupling delay less than the cavity round-trip time. We study
synchronization between the two lasers in the experiment and in a delay
differential equation model of the system. Because the lasers are internally
perturbed by spontaneous emission, we include a noise source in the model to
obtain stochastic realizations of the deterministic equations. Both amplitude
synchronization and phase synchronization are considered. We use the Hilbert
transform to define the phase variable and compute phase synchronization. We
find that synchronization increases with coupling strength in the experiment
and the model. When the time series from two lasers are time-shifted in either
direction by the delay time, approximately equal synchronization is frequently
observed, so that a clear leader and follower cannot be identified. We define
an algorithm to determine which laser leads the other when the synchronization
is sufficiently different with one direction of time shift, and statistics of
switches in leader and follower are studied. The frequency of switching between
leader and follower increases with coupling strength, as might be expected
since the lasers mutually influence each other more effectively with stronger
coupling.


title: Semi-direct sums of Lie algebras and discrete integrable couplings

abstract:
  A relation between semi-direct sums of Lie algebras and integrable couplings
of lattice equations is established, and a practicable way to construct
integrable couplings is further proposed. An application of the resulting
general theory to the generalized Toda spectral problem yields two classes of
integrable couplings for the generalized Toda hierarchy of lattice equations.
The construction of integrable couplings using semi-direct sums of Lie algebras
provides a good source of information on complete classification of integrable
lattice equations.


title: Sawtooth disruptions and limit cycle oscillations

abstract:
  A minimal (low-dimensional) dynamical model of the sawtooth oscillations is
presented. It is assumed that the sawtooth is triggered by a thermal
instability which causes the plasma temperature in the central part of the
plasma to drop suddenly, leading to the sawtooth crash. It is shown that this
model possesses an isolated limit cycle which exhibits relaxation oscillation,
in the appropriate parameter regime, which is the typical characteristics of
sawtooth oscillations. It is further shown that the invariant manifold of the
model is actually the slow manifold of the relaxation oscillation.


title:
New Integrable Multi-Component NLS Type Equations on Symmetric Spaces:
  Z_4 and Z_6 Reductions

abstract:
  The reductions of the multi-component nonlinear Schrodinger (MNLS) type
models related to C.I and D.III type symmetric spaces are studied. We pay
special attention to the MNLS related to the sp(4), so(10) and so(12) Lie
algebras. The MNLS related to sp(4) is a three-component MNLS which finds
applications to Bose-Einstein condensates. The MNLS related to so(12) and
so(10) Lie algebras after convenient Z_6 or Z_4 reductions reduce to three and
four-component MNLS showing new types of chi ^(3)-interactions that are
integrable. We briefly explain how these new types of MNLS can be integrated by
the inverse scattering method. The spectral properties of the Lax operators L
and the corresponding recursion operator Lambda are outlined. Applications to
spinor model of Bose-Einstein condensates are discussed.


title: Scaling behaviour of non-hyperbolic coupled map lattices

abstract:
  Coupled map lattices of non-hyperbolic local maps arise naturally in many
physical situations described by discretised reaction diffusion equations or
discretised scalar field theories. As a prototype for these types of lattice
dynamical systems we study diffusively coupled Tchebyscheff maps of N-th order
which exhibit strongest possible chaotic behaviour for small coupling constants
a. We prove that the expectations of arbitrary observables scale with \sqrt{a}
in the low-coupling limit, contrasting the hyperbolic case which is known to
scale with a. Moreover we prove that there are log-periodic oscillations of
period \log N^2 modulating the \sqrt{a}-dependence of a given expectation
value. We develop a general 1st order perturbation theory to analytically
calculate the invariant 1-point density, show that the density exhibits
log-periodic oscillations in phase space, and obtain excellent agreement with
numerical results.


title: Nodal Domain Statistics for Quantum Maps, Percolation and SLE

abstract:
  We develop a percolation model for nodal domains in the eigenvectors of
quantum chaotic torus maps. Our model follows directly from the assumption that
the quantum maps are described by random matrix theory. Its accuracy in
predicting statistical properties of the nodal domains is demonstrated by
numerical computations for perturbed cat maps and supports the use of
percolation theory to describe the wave functions of general hamiltonian
systems, where the validity of the underlying assumptions is much less clear.
We also demonstrate that the nodal domains of the perturbed cat maps obey the
Cardy crossing formula and find evidence that the boundaries of the nodal
domains are described by SLE with $\kappa$ close to the expected value of 6,
suggesting that quantum chaotic wave functions may exhibit conformal invariance
in the semiclassical limit.


title: On the integrable inhomogeneous Myrzakulov I equation

abstract:
  By using the prolongation structure theory proposed by Morris, we give a
(2+1)-dimensional integrable inhomogeneous Heisenberg Ferromagnet models,
namely, the inhomogeneous Myrzakulov I equation. Through the motion of space
curves endowed with an additional spatial variable, its geometrical equivalent
counterpart is also presented.


title: Beyond scaling and locality in turbulence

abstract:
  An analytic perturbation theory is suggested in order to find finite-size
corrections to the scaling power laws. In the frame of this theory it is shown
that the first order finite-size correction to the scaling power laws has
following form $S(r) \cong cr^{\alpha_0}[\ln(r/\eta)]^{\alpha_1}$, where $\eta$
is a finite-size scale (in particular for turbulence, it can be the Kolmogorov
dissipation scale). Using data of laboratory experiments and numerical
simulations it is shown shown that a degenerate case with $\alpha_0 =0$ can
describe turbulence statistics in the near-dissipation range $r > \eta$, where
the ordinary (power-law) scaling does not apply. For moderate Reynolds numbers
the degenerate scaling range covers almost the entire range of scales of
velocity structure functions (the log-corrections apply to finite Reynolds
number). Interplay between local and non-local regimes has been considered as a
possible hydrodynamic mechanism providing the basis for the degenerate scaling
of structure functions and extended self-similarity. These results have been
also expanded on passive scalar mixing in turbulence. Overlapping phenomenon
between local and non-local regimes and a relation between position of maximum
of the generalized energy input rate and the actual crossover scale between
these regimes are briefly discussed.


title:
Radiationless Travelling Waves In Saturable Nonlinear Schr\"odinger
  Lattices

abstract:
  The longstanding problem of moving discrete solitary waves in nonlinear
Schr{\"o}dinger lattices is revisited. The context is photorefractive crystal
lattices with saturable nonlinearity whose grand-canonical energy barrier
vanishes for isolated coupling strength values. {\em Genuinely localised
travelling waves} are computed as a function of the system parameters {\it for
the first time}. The relevant solutions exist only for finite velocities.


title: Solutions of Non-Integrable Equations by the Hirota Direct Method

abstract:
  We show that we can also apply the Hirota method to some non-integrable
equations. For this purpose, we consider the extensions of the
Kadomtsev-Petviashvili (KP) and the Boussinesq (Bo) equations. We present
several solutions of these equations.


title: Towards a classification of natural bi-hamiltonian systems

abstract:
  For construction and classification of the natural integrable systems we
propose to use a criterion of separability in Darboux--Nijenhuis coordinates,
which can be tested without an a priori explicit knowledge of these
coordinates.


title:
Exact static solutions for discrete $\phi^4$ models free of the
  Peierls-Nabarro barrier: Discretized first integral approach

abstract:
  We propose a generalization of the discrete Klein-Gordon models free of the
Peierls-Nabarro barrier derived in Nonlinearity {\bf 12}, 1373 (1999) and Phys.
Rev. E {\bf 72}, 035602(R) (2005), such that they support not only kinks but a
one-parameter set of exact static solutions. These solutions can be obtained
iteratively from a two-point nonlinear map whose role is played by the
discretized first integral of the static Klein-Gordon field, as suggested in J.
Phys. A {\bf 38}, 7617 (2005). We then discuss some discrete $\phi^4$ models
free of the Peierls-Nabarro barrier and identify for them the full space of
available static solutions, including those derived recently in Phys. Rev. E
{\bf 72} 036605 (2005) but not limited to them. These findings are also
relevant to standing wave solutions of discrete nonlinear Schr{\"o}dinger
models. We also study stability of the obtained solutions. As an interesting
aside, we derive the list of solutions to the continuum $\phi^4$ equation that
fill the entire two-dimensional space of parameters obtained as the continuum
limit of the corresponding space of the discrete models.


title: Fast Ensemble Smoothing

abstract:
  Smoothing is essential to many oceanographic, meteorological and hydrological
applications. The interval smoothing problem updates all desired states within
a time interval using all available observations. The fixed-lag smoothing
problem updates only a fixed number of states prior to the observation at
current time. The fixed-lag smoothing problem is, in general, thought to be
computationally faster than a fixed-interval smoother, and can be an
appropriate approximation for long interval-smoothing problems. In this paper,
we use an ensemble-based approach to fixed-interval and fixed-lag smoothing,
and synthesize two algorithms. The first algorithm produces a linear time
solution to the interval smoothing problem with a fixed factor, and the second
one produces a fixed-lag solution that is independent of the lag length.
Identical-twin experiments conducted with the Lorenz-95 model show that for lag
lengths approximately equal to the error doubling time, or for long intervals
the proposed methods can provide significant computational savings. These
results suggest that ensemble methods yield both fixed-interval and fixed-lag
smoothing solutions that cost little additional effort over filtering and model
propagation, in the sense that in practical ensemble application the additional
increment is a small fraction of either filtering or model propagation costs.
We also show that fixed-interval smoothing can perform as fast as fixed-lag
smoothing and may be advantageous when memory is not an issue.


title:
Approximating the Amplitude and Form of Limit Cycles in the Weakly
  Nonlinear Regime of Lienard Systems

abstract:
  Li\'{e}nard equations, $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with $f(x)$ an
even continuous function are considered. In the weakly nonlinear regime
($\epsilon\to 0$), the number and an order zero in $\epsilon$ approximation of
the amplitude of limit cycles present in this type of systems can be obtained
by applying a methodology recently proposed by the authors [L\'opez-Ruiz R,
L\'opez JL. Bifurcation curves of limit cycles in some Li\'enard systems. Int J
Bifurcat Chaos 2000; 10:971-980]. In the present work, that method is carried
forward to higher orders in $\epsilon$ and is embedded in a general recursive
algorithm capable to approximate the form of the limit cycles and to correct
their amplitudes as an expansion in powers of $\epsilon$. Several examples
showing the application of this scheme are given.


title: Two-component soliton systems and the Painlev\'e equations

abstract:
  We give an extension of the two-component KP hierarchy by considering
additional time variables. We obtain the linear $2\times 2$ system by taking
into consideration the hierarchy through a reduction procedure. The Lax pair of
the Schlesinger system and the sixth Painlev\'e equation is given from this
linear system. A unified approach to treat the other Painlev\'e equations from
the usual two-component KP hierarchy is also considered.


title:
Correlations and periodicities in Himalayan tree ring widths and
  temperature anomalies through wavelets

abstract:
  We have studied periodicities and correlation properties of tree ring width
chronology of deodar tree from Joshimath (1584 - 1999 years) and Uttarkashi
(1500 - 2002 years) in the western Himalayas and the pre-monsoon
(March-April-May) temperature anomalies (1876 - 2003) relative to 1961 -1990
mean, through wavelet analysis. Periodic behavior is observed in the tree ring
chronology with periodicity in the form 11, 22, and 42 years. The analysis of
the self-similar nature reveals long-range correlation with a Hurst exponent,
$H >0.5$. These are anti-correlated with the temperature anomalies. An
interesting inversion behavior is observed around the year 1750. The power
spectral analysis of the time series corroborate the results of wavelet method.


title: Dispersionless Hirota equations of two-component BKP hierarchy

abstract:
  The BKP hierarchy has a two-component analogue (the 2-BKP hierarchy).
Dispersionless limit of this multi-component hierarchy is considered on the
level of the $\tau$-function. The so called dispersionless Hirota equations are
obtained from the Hirota equations of the $\tau$-function. These dispersionless
Hirota equations turn out to be equivalent to a system of Hamilton-Jacobi
equations. Other relevant equations, in particular, dispersionless Lax
equations, can be derived from these fundamental equations. For comparison,
another approach based on auxiliary linear equations is also presented.


title: Basic aspects of soliton theory

abstract:
  This is a review of the main ideas of the inverse scattering method (ISM) for
solving nonlinear evolution equations (NLEE), known as soliton equations. As a
basic tool we use the fundamental analytic solutions (FAS) of the Lax operator
L. Then the inverse scattering problem for L reduces to a Riemann-Hilbert
problem. Such construction cab be applied to wide class of Lax operators,
related to the simple Lie algebras g. We construct the kernel of the resolvent
of L in terms of FAS and derive the spectral decompositions of L. Thus we can
solve the relevant classes of NLEE which include the NLS eq. and its
multi-component generalizations, the N-wave equations etc. Using the dressing
method of Zakharov and Shabat we derive the N-soliton solutions of these
equations. We explain that the ISM is a natural generalization of the Fourier
transform method. As appropriate generalizations of the usual exponential
function we use the so-called "squared solutions" which are constructed again
in terms of FAS and the Cartan-Weyl basis of the relevant Lie algebra. One can
prove the completeness relations for the "squared solutions" which in fact
provide the spectral decompositions of the recursion operator \Lambda. These
decompositions can be used to derive all fundamental properties of the
corresponding NLEE in terms of \Lambda: i) the explicit form of the class of
integrable NLEE; ii) the generating functionals of integrals of motion; iii)
the hierarchies of Hamiltonian structures. We outline the importance of the
classical R-matrices for extracting the involutive integrals of motion.


title: Selected Aspects of Soliton Theory. Constant boundary conditions

abstract:
  Brief review of the methods for solving the multicomponent nonlinear
Schrodinger (MNLS) equations and analysis of their Hamiltonian structures is
given. Main attention is paid to the MNLS related to the C.II- and D.III-types
symmetric spaces with nonvanishing (constant) boundary conditions. The spectral
properties of their Lax operators are described. The derivation of the trace
identities is outlined. The involutivity of their integrals of motion is proved
using the method of the classical R-matrix.


title: Delay Induced Oscillations in a Fundamental Power System Model

abstract:
  In this paper, we study the dynamics and stability of a fundamental power
system model when a time delay is imposed on the excitation of the generator.
It is observed that sustained oscillations can arise in an otherwise stable
power system through a delay induced Andronov-Hopf bifurcation. Numerical
simulations are conducted to explore the dynamics of the time delayed system
after the bifurcation which indicate period doublings culminating in a strange
attractor.


title: Fractional Generalization of Gradient Systems

abstract:
  We consider a fractional generalization of gradient systems. We use
differential forms and exterior derivatives of fractional orders. Examples of
fractional gradient systems are considered. We describe the stationary states
of these systems.

categories: nlin.CD math-ph math.DS math.MP physics.chem-ph physics.class-ph q-bio.OT


title: Dressing chain for the acoustic spectral problem

abstract:
  The iterations are studied of the Darboux transformation for the generalized
Schroedinger operator. The applications to the Dym and Camassa-Holm equations
are considered.


title:
The Surface Topography of a Magnetic Fluid -- a Quantitative Comparison
  between Experiment and Numerical Simulation

abstract:
  The normal field instability in magnetic liquids is investigated
experimentally by means of a radioscopic technique which allows a precise
measurement of the surface topography. The dependence of the topography on the
magnetic field is compared to results obtained by numerical simulations via the
finite element method. Quantitative agreement has been found for the critical
field of the instability, the scaling of the pattern amplitude and the detailed
shape of the magnetic spikes. The fundamental Fourier mode approximates the
shape to within 10% accuracy for a range of up to 40% of the bifurcation
parameter of this subcritical bifurcation. The measured control parameter
dependence of the wavenumber differs qualitatively from analytical predictions
obtained by minimization of the free energy.


title:
Periodic waves of a discrete higher order nonlinear Schroedinger
  equation

abstract:
  The Hirota equation is a higher order extension of the nonlinear Schroedinger
equation by incorporating third order dispersion and one form of self
steepening effect. New periodic waves for the discrete Hirota equation are
given in terms of elliptic functions. The continuum limit converges to the
analogous result for the continuous Hirota equation, while the long wave limit
of these discrete periodic patterns reproduces the known result of the
integrable Ablowitz-Ladik system.


title:
A reduction of the resonant three-wave interaction to the generic sixth
  Painleve' equation

abstract:
  Among the reductions of the resonant three-wave interaction system to
six-dimensional differential systems, one of them has been specifically
mentioned as being linked to the generic sixth Painleve' equation P6. We derive
this link explicitly, and we establish the connection to a three-degree of
freedom Hamiltonian previously considered for P6.


title: Painleve' property of the He'non-Heiles Hamiltonians

abstract:
  Time independent Hamiltonians of the physical type H =
(P_1^2+P_2^2)/2+V(Q_1,Q_2) pass the Painleve' test for only seven potentials
$V$, known as the He'non-Heiles Hamiltonians, each depending on a finite number
of free constants. Proving the Painleve' property was not yet achieved for
generic values of the free constants. We integrate each missing case by
building a birational transformation to some fourth order first degree ordinary
differential equation in the classification (Cosgrove, 2000) of such polynomial
equations which possess the Painleve' property. The properties common to each
Hamiltonian are: (i) the general solution is meromorphic and expressed with
hyperelliptic functions of genus two, (ii) the Hamiltonian is complete (the
addition of any time independent term would ruin the Painleve' property).


title: Nonlinear waves and related nonintegrable and integrable systems

abstract:
  Spectral method related to Lame equation with finite-gap potential is used to
study the optical cascading equations. These equations are known not to be
integrable by inverse scattering method. Due to "partial integrability" two-gap
solutions are obtained in terms of products of elliptic functions and are
classified in five different families related to eigenvalues of appropriate
spectral problem. In special cases, when periodic solutions reduce to localized
solitary waves, previously known phase-locked solutions are recovered, and
additional one solution is obtained. For vector nonlinear Schrodinger equation
n=3 we present exact solutions in a form of multicomponent cnoidal waves.


title: Shil'nikov Chaos control using Homoclinic orbits and the Newhouse region

abstract:
  A method of controlling Shil'nikov's type chaos using windows that appear in
the 1 dimensional bifurcation diagram when perturbations are applied, and using
existence of stable homoclinic orbits near the unstable one is presented and
applied to the electronic Chua's circuit. A demonstration of the chaos control
in the electronic circuit experiments and their simulations and bifurcation
analyses are given.


title: Quadratic Solitons in Negative Refractive Index Medium

abstract:
  We are considered the propagation of the fundamental and second harmonic
solitary waves under the slowly varying envelope pulses approximation in a
quadratic nonlinear medium that characterized by negative refraction index at
the frequency of fundamental wave and by positive refractive index at the
second harmonic frequency. We find the some solutions of the evolution
equations which described the steady state simulton, coupled second harmonic
and fundamental frequency waves propagation in this materials.


title:
Symbolic Synchronization and the Detection of Global Properties of
  Coupled Dynamics from Local Information

abstract:
  We study coupled dynamics on networks using symbolic dynamics. The symbolic
dynamics is defined by dividing the state space into a small number of regions
(typically 2), and considering the relative frequencies of the transitions
between those regions. It turns out that the global qualitative properties of
the coupled dynamics can be classified into three different phases based on the
synchronization of the variables and the homogeneity of the symbolic dynamics.
Of particular interest is the {\it homogeneous unsynchronized phase} where the
coupled dynamics is in a chaotic unsynchronized state, but exhibits (almost)
identical symbolic dynamics at all the nodes in the network. We refer to this
dynamical behaviour as {\it symbolic synchronization}. In this phase, the local
symbolic dynamics of any arbitrarily selected node reflects global properties
of the coupled dynamics, such as qualitative behaviour of the largest Lyapunov
exponent and phase synchronization. This phase depends mainly on the network
architecture, and only to a smaller extent on the local chaotic dynamical
function. We present results for two model dynamics, iterations of the
one-dimensional logistic map and the two-dimensional H\'enon map, as local
dynamical function.


title:
Nonintegrability of (2+1)-dimensional continuum isotropic Heisenberg
  spin system: Painlev\'e analysis

abstract:
  While many integrable spin systems are known to exist in (1+1) and (2+1)
dimensions, the integrability property of the physically important (2+1)
dimensional isotropic Heisenberg ferromagnetic spin system in the continuum
limit has not been investigated in the literature. In this paper, we show
through a careful singularity structure analysis of the underlying nonlinear
evolution equation that the system admits logarithmic type singular manifolds
and so is of non-Painlev\'e type and is expected to be nonintegrable.


title: Baxter operators for the quantum sl(3) invariant spin chain

abstract:
  The noncompact homogeneous sl(3) invariant spin chains are considered. We
show that the transfer matrix with generic auxiliary space is factorized into
the product of three sl(3) invariant commuting operators. These operators
satisfy the finite difference equations in the spectral parameters which follow
from the structure of the reducible sl(3) modules.