226 research outputs found
Quasiperiodic Solutions of the Fibre Optics Coupled Nonlinear Schr{\"o}dinger Equations
We consider travelling periodical and quasiperiodical waves in single mode
fibres, with weak birefringence and under the action of cross-phase modulation.
The problem is reduced to the ``1:2:1" integrable case of the two-particle
quartic potential. A general approach for finding elliptic solutions is given.
New solutions which are associated with two-gap Treibich-Verdier potentials are
found. General quasiperiodic solutions are given in terms of two dimensional
theta functions with explicit expressions for frequencies in terms of theta
constants. The reduction of quasiperiodic solutions to elliptic functions is
discussed.Comment: 24 page
Discrete soliton mobility in two-dimensional waveguide arrays with saturable nonlinearity
We address the issue of mobility of localized modes in two-dimensional
nonlinear Schr\"odinger lattices with saturable nonlinearity. This describes
e.g. discrete spatial solitons in a tight-binding approximation of
two-dimensional optical waveguide arrays made from photorefractive crystals. We
discuss numerically obtained exact stationary solutions and their stability,
focussing on three different solution families with peaks at one, two, and four
neighboring sites, respectively. When varying the power, there is a repeated
exchange of stability between these three solutions, with symmetry-broken
families of connecting intermediate stationary solutions appearing at the
bifurcation points. When the nonlinearity parameter is not too large, we
observe good mobility, and a well defined Peierls-Nabarro barrier measuring the
minimum energy necessary for rendering a stable stationary solution mobile.Comment: 19 pages, 4 figure
Quantum signatures of breather-breather interactions
The spectrum of the Quantum Discrete Nonlinear Schr\"odinger equation on a
periodic 1D lattice shows some interesting detailed band structure which may be
interpreted as the quantum signature of a two-breather interaction in the
classical case. We show that this fine structure can be interpreted using
degenerate perturbation theory.Comment: 4 pages, 4 fig
Biphonons in the Klein-Gordon lattice
A numerical approach is proposed for studying the quantum optical modes in
the Klein-Gordon lattices where the energy contribution of the atomic
displacements is non-quadratic. The features of the biphonon excitations are
investigated in detail for different non-quadratic contributions to the
Hamiltonian. The results are extended to multi-phonon bound states.Comment: Comments and suggestions are welcom
Hamiltonian Hopf bifurcations in the discrete nonlinear Schr\"odinger trimer: oscillatory instabilities, quasiperiodic solutions and a 'new' type of self-trapping transition
Oscillatory instabilities in Hamiltonian anharmonic lattices are known to
appear through Hamiltonian Hopf bifurcations of certain time-periodic solutions
of multibreather type. Here, we analyze the basic mechanisms for this scenario
by considering the simplest possible model system of this kind where they
appear: the three-site discrete nonlinear Schr\"odinger model with periodic
boundary conditions. The stationary solution having equal amplitude and
opposite phases on two sites and zero amplitude on the third is known to be
unstable for an interval of intermediate amplitudes. We numerically analyze the
nature of the two bifurcations leading to this instability and find them to be
of two different types. Close to the lower-amplitude threshold stable
two-frequency quasiperiodic solutions exist surrounding the unstable stationary
solution, and the dynamics remains trapped around the latter so that in
particular the amplitude of the originally unexcited site remains small. By
contrast, close to the higher-amplitude threshold all two-frequency
quasiperiodic solutions are detached from the unstable stationary solution, and
the resulting dynamics is of 'population-inversion' type involving also the
originally unexcited site.Comment: 25 pages, 11 figures, to be published in J. Phys. A: Math. Gen.
Revised and shortened version with few clarifying remarks adde
Properties of the series solution for Painlevé I
We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painlevé equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions of analogous formulae for the elliptic sigma function, as given by Weierstrass. Numerical and exact results on the symmetric solution which is singular at the origin are also presented
Discreteness-Induced Oscillatory Instabilities of Dark Solitons
We reveal that even weak inherent discreteness of a nonlinear model can lead
to instabilities of the localized modes it supports. We present the first
example of an oscillatory instability of dark solitons, and analyse how it may
occur for dark solitons of the discrete nonlinear Schrodinger and generalized
Ablowitz-Ladik equations.Comment: 11 pages, 4 figures, to be published in Physical Review Letter
Statics and Dynamics of an Inhomogeneously-Nonlinear Lattice
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.Comment: 14 pages, 4 figure
Comment on ‘‘Self-trapping on a dimer: Time-dependent solutions of a discrete nonlinear Schrödinger equation’’
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