415 research outputs found
Generalized Neighbor-Interaction Models Induced by Nonlinear Lattices
It is shown that the tight-binding approximation of the nonlinear
Schr\"odinger equation with a periodic linear potential and periodic in space
nonlinearity coefficient gives rise to a number of nonlinear lattices with
complex, both linear and nonlinear, neighbor interactions. The obtained
lattices present non-standard possibilities, among which we mention a
quasi-linear regime, where the pulse dynamics obeys essentially the linear
Schr{\"o}dinger equation. We analyze the properties of such models both in
connection with their modulational stability, as well as in regard to the
existence and stability of their localized solitary wave solutions
Compactons in Nonlinear Schr\"odinger Lattices with Strong Nonlinearity Management
The existence of compactons in the discrete nonlinear Schr\"odinger equation
in the presence of fast periodic time modulations of the nonlinearity is
demonstrated. In the averaged DNLS equation the resulting effective inter-well
tunneling depends on modulation parameters {\it and} on the field amplitude.
This introduces nonlinear dispersion in the system and can lead to a
prototypical realization of single- or multi-site stable discrete compactons in
nonlinear optical waveguide and BEC arrays. These structures can dynamically
arise out of Gaussian or compactly supported initial data.Comment: 4 pages, 4 figure
The frustrated Brownian motion of nonlocal solitary waves
We investigate the evolution of solitary waves in a nonlocal medium in the
presence of disorder. By using a perturbational approach, we show that an
increasing degree of nonlocality may largely hamper the Brownian motion of
self-trapped wave-packets. The result is valid for any kind of nonlocality and
in the presence of non-paraxial effects. Analytical predictions are compared
with numerical simulations based on stochastic partial differential equationComment: 4 pages, 3 figures
-symmetric coupler with nonlinearity
We introduce the notion of a -symmetric dimer with a
nonlinearity. Similarly to the Kerr case, we argue that such a nonlinearity
should be accessible in a pair of optical waveguides with quadratic
nonlinearity and gain and loss, respectively. An interesting feature of the
problem is that because of the two harmonics, there exist in general two
distinct gain/loss parameters, different values of which are considered herein.
We find a number of traits that appear to be absent in the more standard cubic
case. For instance, bifurcations of nonlinear modes from the linear solutions
occur in two different ways depending on whether the first or the second
harmonic amplitude is vanishing in the underlying linear eigenvector. Moreover,
a host of interesting bifurcation phenomena appear to occur including
saddle-center and pitchfork bifurcations which our parametric variations
elucidate. The existence and stability analysis of the stationary solutions is
corroborated by numerical time-evolution simulations exploring the evolution of
the different configurations, when unstable.Comment: 12 pages, 11 figure
Modulational instability in nonlocal Kerr-type media with random parameters
Modulational instability of continuous waves in nonlocal focusing and
defocusing Kerr media with stochastically varying diffraction (dispersion) and
nonlinearity coefficients is studied both analytically and numerically. It is
shown that nonlocality with the sign-definite Fourier images of the medium
response functions suppresses considerably the growth rate peak and bandwidth
of instability caused by stochasticity. Contrary, nonlocality can enhance
modulational instability growth for a response function with negative-sign
bands.Comment: 6 pages, 12 figures, revTeX, to appear in Phys. Rev.
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
A model for conservative chaos constructed from multi-component Bose-Einstein condensates with a trap in 2 dimensions
To show a mechanism leading to the breakdown of a particle picture for the
multi-component Bose-Einstein condensates(BECs) with a harmonic trap in high
dimensions, we investigate the corresponding 2- nonlinear Schr{\"o}dinger
equation (Gross-Pitaevskii equation) with use of a modified variational
principle. A molecule of two identical Gaussian wavepackets has two degrees of
freedom(DFs), the separation of center-of-masses and the wavepacket width.
Without the inter-component interaction(ICI) these DFs show independent regular
oscillations with the degenerate eigen-frequencies. The inclusion of ICI
strongly mixes these DFs, generating a fat mode that breaks a particle picture,
which however can be recovered by introducing a time-periodic ICI with zero
average. In case of the molecule of three wavepackets for a three-component
BEC, the increase of amplitude of ICI yields a transition from regular to
chaotic oscillations in the wavepacket breathing.Comment: 5 pages, 4 figure
Modulational instability and nonlocality management in coupled NLS system
The modulational instability of two interacting waves in a nonlocal Kerr-type
medium is considered analytically and numerically. For a generic choice of wave
amplitudes, we give a complete description of stable/unstable regimes for zero
group-velocity mismatch. It is shown that nonlocality suppresses considerably
the growth rate and bandwidth of instability. For nonzero group-velocity
mismatch we perform a geometrical analysis of a nonlocality management which
can provide stability of waves otherwise unstable in a local medium.Comment: 15 pages, 12 figures, to be published in Physica Script
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