181 research outputs found
A large time asymptotics for transparent potentials for the Novikov-Veselov equation at positive energy
In the present paper we begin studies on the large time asymptotic behavior
for solutions of the Cauchy problem for the Novikov--Veselov equation (an
analog of KdV in 2 + 1 dimensions) at positive energy. In addition, we are
focused on a family of reflectionless (transparent) potentials parameterized by
a function of two variables. In particular, we show that there are no isolated
soliton type waves in the large time asymptotics for these solutions in
contrast with well-known large time asymptotics for solutions of the KdV
equation with reflectionless initial data
Modulation instability in high power laser amplifiers
The modulation instability (MI) is one of the main factors responsible for the degradation of beam quality in high-power laser systems. The so-called B-integral restriction is commonly used as the criteria for MI control in passive optics devices. For amplifiers the adiabatic model, assuming locally the Bespalov-Talanov expression for MI growth, is commonly used to estimate the destructive impact of the instability. We present here the exact solution of MI development in amplifiers. We determine the parameters which control the effect of MI in amplifiers and calculate the MI growth rate as a function of those parameters. The safety range of operational parameters is presented. The results of the exact calculations are compared with the adiabatic model, and the range of validity of the latest is determined. We demonstrate that for practical situations the adiabatic approximation noticeably overestimates MI. The additional margin of laser system design is quantified
Some Recent Developments on Kink Collisions and Related Topics
We review recent works on modeling of dynamics of kinks in 1+1 dimensional
theory and other related models, like sine-Gordon model or
theory. We discuss how the spectral structure of small perturbations can affect
the dynamics of non-perturbative states, such as kinks or oscillons. We
describe different mechanisms, which may lead to the occurrence of the resonant
structure in the kink-antikink collisions. We explain the origin of the
radiation pressure mechanism, in particular, the appearance of the negative
radiation pressure in the and models. We also show that the
process of production of the kink-antikink pairs, induced by radiation is
chaotic.Comment: 26 pages, 9 figures; invited chapter to "A dynamical perspective on
the {\phi}4 model: Past, present and future", Eds. P.G. Kevrekidis and J.
Cuevas-Maraver; Springer book class with svmult.cls include
Leading Order Temporal Asymptotics of the Modified Non-Linear Schrodinger Equation: Solitonless Sector
Using the matrix Riemann-Hilbert factorisation approach for non-linear
evolution equations (NLEEs) integrable in the sense of the inverse scattering
method, we obtain, in the solitonless sector, the leading-order asymptotics as
tends to plus and minus infinity of the solution to the Cauchy
initial-value problem for the modified non-linear Schrodinger equation: also
obtained are analogous results for two gauge-equivalent NLEEs; in particular,
the derivative non-linear Schrodinger equation.Comment: 29 pages, 5 figures, LaTeX, revised version of the original
submission, to be published in Inverse Problem
Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent
We apply the method of nonlinear steepest descent to compute the long-time
asymptotics of the Korteweg-de Vries equation for decaying initial data in the
soliton and similarity region. This paper can be viewed as an expository
introduction to this method.Comment: 31 page
Solitary waves of nonlinear nonintegrable equations
Our goal is to find closed form analytic expressions for the solitary waves
of nonlinear nonintegrable partial differential equations. The suitable
methods, which can only be nonperturbative, are classified in two classes.
In the first class, which includes the well known so-called truncation
methods, one \textit{a priori} assumes a given class of expressions
(polynomials, etc) for the unknown solution; the involved work can easily be
done by hand but all solutions outside the given class are surely missed.
In the second class, instead of searching an expression for the solution, one
builds an intermediate, equivalent information, namely the \textit{first order}
autonomous ODE satisfied by the solitary wave; in principle, no solution can be
missed, but the involved work requires computer algebra.
We present the application to the cubic and quintic complex one-dimensional
Ginzburg-Landau equations, and to the Kuramoto-Sivashinsky equation.Comment: 28 pages, chapter in book "Dissipative solitons", ed. Akhmediev, to
appea
Long-Time Asymptotics of Perturbed Finite-Gap Korteweg-de Vries Solutions
We apply the method of nonlinear steepest descent to compute the long-time
asymptotics of solutions of the Korteweg--de Vries equation which are decaying
perturbations of a quasi-periodic finite-gap background solution. We compute a
nonlinear dispersion relation and show that the plane splits into
soliton regions which are interlaced by oscillatory regions, where
is the number of spectral gaps.
In the soliton regions the solution is asymptotically given by a number of
solitons travelling on top of finite-gap solutions which are in the same
isospectral class as the background solution. In the oscillatory region the
solution can be described by a modulated finite-gap solution plus a decaying
dispersive tail. The modulation is given by phase transition on the isospectral
torus and is, together with the dispersive tail, explicitly characterized in
terms of Abelian integrals on the underlying hyperelliptic curve.Comment: 45 pages. arXiv admin note: substantial text overlap with
arXiv:0705.034
Modulational and Parametric Instabilities of the Discrete Nonlinear Schr\"odinger Equation
We examine the modulational and parametric instabilities arising in a
non-autonomous, discrete nonlinear Schr{\"o}dinger equation setting. The
principal motivation for our study stems from the dynamics of Bose-Einstein
condensates trapped in a deep optical lattice. We find that under periodic
variations of the heights of the interwell barriers (or equivalently of the
scattering length), additionally to the modulational instability, a window of
parametric instability becomes available to the system. We explore this
instability through multiple-scale analysis and identify it numerically. Its
principal dynamical characteristic is that, typically, it develops over much
larger times than the modulational instability, a feature that is qualitatively
justified by comparison of the corresponding instability growth rates
Discrete breathers in classical spin lattices
Discrete breathers (nonlinear localised modes) have been shown to exist in
various nonlinear Hamiltonian lattice systems. In the present paper we study
the dynamics of classical spins interacting via Heisenberg exchange on spatial
-dimensional lattices (with and without the presence of single-ion
anisotropy). We show that discrete breathers exist for cases when the continuum
theory does not allow for their presence (easy-axis ferromagnets with
anisotropic exchange and easy-plane ferromagnets). We prove the existence of
localised excitations using the implicit function theorem and obtain necessary
conditions for their existence. The most interesting case is the easy-plane one
which yields excitations with locally tilted magnetisation. There is no
continuum analogue for such a solution and there exists an energy threshold for
it, which we have estimated analytically. We support our analytical results
with numerical high-precision computations, including also a stability analysis
for the excitations.Comment: 15 pages, 12 figure
Kinks in the Hartree approximation
The topological defects of the lambda phi^4 theory, kink and antikink, are
studied in the Hartree approximation. This allows us to discuss quantum effects
on the defects in both stationary and dynamical systems. The kink mass is
calculated for a number of parameters, and compared to classical, one loop and
Monte Carlo results known from the literature. We discuss the thermalization of
the system after a kink antikink collision. A classical result, the existence
of a critical speed, is rederived and shown for the first time in the quantum
theory. We also use kink antikink collisions as a very simple toy model for
heavy ion collisions and discuss the differences and similarities, for example
in the pressure. Finally, using the Hartree Ensemble Approximation allows us to
study kink antikink nucleation starting from a thermal (Bose Einstein)
distribution. In general our results indicate that on a qualitative level there
are few differences with the classical results, but on a quantitative level
there are some import ones.Comment: 20 pages REVTeX 4, 17 Figures. Uses amsmath.sty and subfigure.sty.
Final version, fixed typo in published versio
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