162 research outputs found
Linear and nonlinear waves in surface and wedge index potentials
We study optical beams that are supported at the surface of a medium with a linear index potential and by a piecewise linear wedge type potential. In the linear limit the modes are described by Airy functions. In the nonlinear regime we find families of solutions that bifurcate from the linear modes and study their stability for both self-focusing and self-defocusing Kerr nonlinearity. The total power of such nonlinear waves is finite without the need for apodization
Fourier mode dynamics for the nonlinear Schroedinger equation in one-dimensional bounded domains
We analyze the 1D focusing nonlinear Schr\"{o}dinger equation in a finite
interval with homogeneous Dirichlet or Neumann boundary conditions. There are
two main dynamics, the collapse which is very fast and a slow cascade of
Fourier modes. For the cubic nonlinearity the calculations show no long term
energy exchange between Fourier modes as opposed to higher nonlinearities. This
slow dynamics is explained by fairly simple amplitude equations for the
resonant Fourier modes. Their solutions are well behaved so filtering high
frequencies prevents collapse. Finally these equations elucidate the unique
role of the zero mode for the Neumann boundary conditions
Hysteresis and metastability of Bose-Einstein condensed clouds of atoms confined in ring potentials
We consider a Bose-Einstein condensed cloud of atoms which rotate in a
toroidal/annular potential. Assuming one-dimensional motion, we evaluate the
critical frequencies associated with the effect of hysteresis and the critical
coupling for stability of the persistent currents. We perform these
calculations using both the mean-field approximation and the method of
numerical diagonalization of the many-body Hamiltonian which includes
corrections due to the finiteness of the atom number.Comment: 7 pages, 5 figures, section on experimental relevance added, final
versio
Stability of vortex solitons in a photorefractive optical lattice
Stability of off-site vortex solitons in a photorefractive optical lattice is
analyzed. It is shown that such solitons are linearly unstable in both the high
and low intensity limits. In the high-intensity limit, the vortex looks like a
familiar ring vortex, and it suffers oscillatory instabilities. In the
low-intensity limit, the vortex suffers both oscillatory and Vakhitov-Kolokolov
instabilities. However, in the moderate-intensity regime, the vortex becomes
stable if the lattice intensity or the applied voltage is above a certain
threshold value. Stability regions of vortices are also determined at typical
experimental parameters.Comment: 3 pages, 5 figure
Nonlinear optics and light localization in periodic photonic lattices
We review the recent developments in the field of photonic lattices
emphasizing their unique properties for controlling linear and nonlinear
propagation of light. We draw some important links between optical lattices and
photonic crystals pointing towards practical applications in optical
communications and computing, beam shaping, and bio-sensing.Comment: to appear in Journal of Nonlinear Optical Physics & Materials (JNOPM
Analytic theory of narrow lattice solitons
The profiles of narrow lattice solitons are calculated analytically using
perturbation analysis. A stability analysis shows that solitons centered at a
lattice (potential) maximum are unstable, as they drift toward the nearest
lattice minimum. This instability can, however, be so weak that the soliton is
``mathematically unstable'' but ``physically stable''. Stability of solitons
centered at a lattice minimum depends on the dimension of the problem and on
the nonlinearity. In the subcritical and supercritical cases, the lattice does
not affect the stability, leaving the solitons stable and unstable,
respectively. In contrast, in the critical case (e.g., a cubic nonlinearity in
two transverse dimensions), the lattice stabilizes the (previously unstable)
solitons. The stability in this case can be so weak, however, that the soliton
is ``mathematically stable'' but ``physically unstable''
Two-soliton collisions in a near-integrable lattice system
We examine collisions between identical solitons in a weakly perturbed
Ablowitz-Ladik (AL) model, augmented by either onsite cubic nonlinearity (which
corresponds to the Salerno model, and may be realized as an array of strongly
overlapping nonlinear optical waveguides), or a quintic perturbation, or both.
Complex dependences of the outcomes of the collisions on the initial phase
difference between the solitons and location of the collision point are
observed. Large changes of amplitudes and velocities of the colliding solitons
are generated by weak perturbations, showing that the elasticity of soliton
collisions in the AL model is fragile (for instance, the Salerno's perturbation
with the relative strength of 0.08 can give rise to a change of the solitons'
amplitudes by a factor exceeding 2). Exact and approximate conservation laws in
the perturbed system are examined, with a conclusion that the small
perturbations very weakly affect the norm and energy conservation, but
completely destroy the conservation of the lattice momentum, which is explained
by the absence of the translational symmetry in generic nonintegrable lattice
models. Data collected for a very large number of collisions correlate with
this conclusion. Asymmetry of the collisions (which is explained by the
dependence on the location of the central point of the collision relative to
the lattice, and on the phase difference between the solitons) is investigated
too, showing that the nonintegrability-induced effects grow almost linearly
with the perturbation strength. Different perturbations (cubic and quintic
ones) produce virtually identical collision-induced effects, which makes it
possible to compensate them, thus finding a special perturbed system with
almost elastic soliton collisions.Comment: Phys. Rev. E, in pres
Surface Gap Soliton Ground States for the Nonlinear Schr\"{o}dinger Equation
We consider the nonlinear Schr\"{o}dinger equation , with and and with periodic in each coordinate direction. This problem
describes the interface of two periodic media, e.g. photonic crystals. We study
the existence of ground state solutions (surface gap soliton ground
states) for . Using a concentration compactness
argument, we provide an abstract criterion for the existence based on ground
state energies of each periodic problem (with and ) as well as a more practical
criterion based on ground states themselves. Examples of interfaces satisfying
these criteria are provided. In 1D it is shown that, surprisingly, the criteria
can be reduced to conditions on the linear Bloch waves of the operators
and .Comment: definition of ground and bound states added, assumption (H2) weakened
(sign changing nonlinearity is now allowed); 33 pages, 4 figure
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