13 research outputs found
Instabilities of dispersion-managed solitons in the normal dispersion regime
Dispersion-managed solitons are reviewed within a Gaussian variational
approximation and an integral evolution model. In the normal regime of the
dispersion map (when the averaged path dispersion is negative), there are two
solitons of different pulse duration and energy at a fixed propagation
constant. We show that the short soliton with a larger energy is linearly
(exponentially) unstable. The other (long) soliton with a smaller energy is
linearly stable but hits a resonance with excitations of the dispersion map.
The results are compared with the results from the recent publicationsComment: 20 figures, 20 pages. submitted to Phys. Rev.
A mode elimination technique to improve convergence of iteration methods for finding solitary waves
We extend the key idea behind the generalized Petviashvili method of Ref.
\cite{gP} by proposing a novel technique based on a similar idea. This
technique systematically eliminates from the iteratively obtained solution a
mode that is "responsible" either for the divergence or the slow convergence of
the iterations. We demonstrate, theoretically and with examples, that this mode
elimination technique can be used both to obtain some nonfundamental solitary
waves and to considerably accelerate convergence of various iteration methods.
As a collateral result, we compare the linearized iteration operators for the
generalized Petviashvili method and the well-known imaginary-time evolution
method and explain how their different structures account for the differences
in the convergence rates of these two methods.Comment: to appear in J. Comp. Phys.; 24 page
Conjugate gradient method for finding fundamental solitary waves
The Conjugate Gradient method (CGM) is known to be the fastest generic
iterative method for solving linear systems with symmetric sign definite
matrices. In this paper, we modify this method so that it could find
fundamental solitary waves of nonlinear Hamiltonian equations. The main
obstacle that such a modified CGM overcomes is that the operator of the
equation linearized about a solitary wave is not sign definite. Instead, it has
a finite number of eigenvalues on the opposite side of zero than the rest of
its spectrum. We present versions of the modified CGM that can find solitary
waves with prescribed values of either the propagation constant or power. We
also extend these methods to handle multi-component nonlinear wave equations.
Convergence conditions of the proposed methods are given, and their practical
implications are discussed. We demonstrate that our modified CGMs converge much
faster than, say, Petviashvili's or similar methods, especially when the latter
converge slowly.Comment: 44 pages, submitted to Physica
A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity
The Petviashvili's iteration method has been known as a rapidly converging
numerical algorithm for obtaining fundamental solitary wave solutions of
stationary scalar nonlinear wave equations with power-law nonlinearity: \
, where is a positive definite self-adjoint operator and . In this paper, we propose a systematic generalization of this method
to both scalar and vector Hamiltonian equations with arbitrary form of
nonlinearity and potential functions. For scalar equations, our generalized
method requires only slightly more computational effort than the original
Petviashvili method.Comment: to appear in J. Comp. Phys.; 35 page
On the statistical interpretation of optical rogue waves
Numerical simulations are used to discuss various aspects of "optical rogue
wave" statistics observed in noise-driven fiber supercontinuum generation
associated with highly incoherent spectra. In particular, we consider how long
wavelength spectral filtering influences the characteristics of the statistical
distribution of peak power, and we contrast the statistics of the spectrally
filtered SC with the statistics of both the peak power of the most red-shifted
soliton in the SC and the maximum peak power across the full temporal field
with no spectral selection. For the latter case, we show that the unfiltered
statistical distribution can still exhibit a long-tail, but the extreme-events
in this case correspond to collisions between solitons of different
frequencies. These results confirm the importance of collision dynamics in
supercontinuum generation. We also show that the collision-induced events
satisfy an extended hydrodynamic definition of "rogue wave" characteristics.Comment: Paper accepted for publication in the European Physical Journal ST,
Special Topics. Discussion and Debate: Rogue Waves - towards a unifying
concept? To appear 201
Experimental feasibility of measuring the gravitational redshift of light using dispersion in optical fibers
This paper describes a new class of experiments that use dispersion in
optical fibers to convert the gravitational frequency shift of light into a
measurable phase shift or time delay. Two conceptual models are explored. In
the first model, long counter-propagating pulses are used in a vertical fiber
optic Sagnac interferometer. The second model uses optical solitons in
vertically separated fiber optic storage rings. We discuss the feasibility of
using such an instrument to make a high precision measurement of the
gravitational frequency shift of light.Comment: 11 pages, 12 figure
A system of ODEs for a Perturbation of a Minimal Mass Soliton
We study soliton solutions to a nonlinear Schrodinger equation with a
saturated nonlinearity. Such nonlinearities are known to possess minimal mass
soliton solutions. We consider a small perturbation of a minimal mass soliton,
and identify a system of ODEs similar to those from Comech and Pelinovsky
(2003), which model the behavior of the perturbation for short times. We then
provide numerical evidence that under this system of ODEs there are two
possible dynamical outcomes, which is in accord with the conclusions of
Pelinovsky, Afanasjev, and Kivshar (1996). For initial data which supports a
soliton structure, a generic initial perturbation oscillates around the stable
family of solitons. For initial data which is expected to disperse, the finite
dimensional dynamics follow the unstable portion of the soliton curve.Comment: Minor edit