362 research outputs found
Dynamics of Solitons and Quasisolitons of Cubic Third-Order Nonlinear Schr\"odinger Equation
The dynamics of soliton and quasisoliton solutions of cubic third order
nonlinear Schr\"{o}dinger equation is studied. The regular solitons exist due
to a balance between the nonlinear terms and (linear) third order dispersion;
they are not important at small ( is the coefficient in
the third derivative term) and vanish at . The most essential,
at small , is a quasisoliton emitting resonant radiation (resonantly
radiating soliton). Its relationship with the other (steady) quasisoliton,
called embedded soliton, is studied analytically and in numerical experiments.
It is demonstrated that the resonantly radiating solitons emerge in the course
of nonlinear evolution, which shows their physical significance
Dynamics of shallow dark solitons in a trapped gas of impenetrable bosons
The dynamics of linear and nonlinear excitations in a Bose gas in the
Tonks-Girardeau (TG) regime with longitudinal confinement are studied within a
mean field theory of quintic nonlinearity. A reductive perturbation method is
used to demonstrate that the dynamics of shallow dark solitons, in the presence
of an external potential, can effectively be described by a
variable-coefficient Korteweg-de Vries equation. The soliton oscillation
frequency is analytically obtained to be equal to the axial trap frequency, in
agreement with numerical predictions obtained by Busch {\it et al.} [J. Phys. B
{\bf 36}, 2553 (2003)] via the Bose-Fermi mapping. We obtain analytical
expressions for the evolution of both soliton and emitted radiation (sound)
profiles.Comment: 4 pages, Phys. Rev. A (in press
Perturbation theory for localized solutions of sine-Gordon equation: decay of a breather and pinning by microresistor
We develop a perturbation theory that describes bound states of solitons
localized in a confined area. External forces and influence of inhomogeneities
are taken into account as perturbations to exact solutions of the sine-Gordon
equation. We have investigated two special cases of fluxon trapped by a
microresistor and decay of a breather under dissipation. Also, we have carried
out numerical simulations with dissipative sine-Gordon equation and made
comparison with the McLaughlin-Scott theory. Significant distinction between
the McLaughlin-Scott calculation for a breather decay and our numerical result
indicates that the history dependence of the breather evolution can not be
neglected even for small damping parameter
Dynamics of soliton-like solutions for slowly varying, generalized gKdV equations: refraction vs. reflection
In this work we continue the description of soliton-like solutions of some
slowly varying, subcritical gKdV equations.
In this opportunity we describe, almost completely, the allowed behaviors:
either the soliton is refracted, or it is reflected by the potential, depending
on its initial energy. This last result describes a new type of soliton-like
solution for gKdV equations, also present in the NLS case.
Moreover, we prove that the solution is not pure at infinity, unlike the
standard gKdV soliton.Comment: 51 pages, submitte
Solitons in cavity-QED arrays containing interacting qubits
We reveal the existence of polariton soliton solutions in the array of weakly
coupled optical cavities, each containing an ensemble of interacting qubits. An
effective complex Ginzburg-Landau equation is derived in the continuum limit
taking into account the effects of cavity field dissipation and qubit
dephasing. We have shown that an enhancement of the induced nonlinearity can be
achieved by two order of the magnitude with a negative interaction strength
which implies a large negative qubit-field detuning as well. Bright solitons
are found to be supported under perturbations only in the upper (optical)
branch of polaritons, for which the corresponding group velocity is controlled
by tuning the interacting strength. With the help of perturbation theory for
solitons, we also demonstrate that the group velocity of these polariton
solitons is suppressed by the diffusion process
Scattering and Trapping of Nonlinear Schroedinger Solitons in External Potentials
Soliton motion in some external potentials is studied using the nonlinear
Schr\"odinger equation. Solitons are scattered by a potential wall. Solitons
propagate almost freely or are trapped in a periodic potential. The critical
kinetic energy for reflection and trapping is evaluated approximately with a
variational method.Comment: 9 pages, 7 figure
Dust ion-acoustic shocks in quantum dusty pair-ion plasmas
The formation of dust ion-acoustic shocks (DIASs) in a four-component quantum
plasma whose constituents are electrons, both positive and negative ions and
immobile charged dust grains, is studied. The effects of both the dissipation
due to kinematic viscosity and the dispersion caused by the charge separation
as well as the quantum tunneling due to the Bohm potential are taken into
account. The propagation of small but finite amplitude dust ion-acoustic waves
(DIAWs) is governed by the Korteweg-de Vries-Burger (KdVB) equation which
exhibits both oscillatory and monotonic shocks depending not only on the
viscosity parameters, but also on the quantum parameter H (the ratio of the
electron plasmon to the electron Fermi energy) and the positive to negative ion
density ratio. Large amplitude stationary shocks are recovered for a Mach
number exceeding its critical value. Unlike the small amplitude shocks, quite a
smaller value of the viscosity parameter, H and the density ratio may lead to
the large amplitude monotonic shock strucutres. The results could be of
importance in astrophysical and laser produced plasmas.Comment: 15 pages, 5 figure
Nonlinear Schr\"odinger Equation with Spatio-Temporal Perturbations
We investigate the dynamics of solitons of the cubic Nonlinear Schr\"odinger
Equation (NLSE) with the following perturbations: non-parametric
spatio-temporal driving of the form , damping, and a
linear term which serves to stabilize the driven soliton. Using the time
evolution of norm, momentum and energy, or, alternatively, a Lagrangian
approach, we develop a Collective-Coordinate-Theory which yields a set of ODEs
for our four collective coordinates. These ODEs are solved analytically and
numerically for the case of a constant, spatially periodic force . The
soliton position exhibits oscillations around a mean trajectory with constant
velocity. This means that the soliton performs, on the average, a
unidirectional motion although the spatial average of the force vanishes. The
amplitude of the oscillations is much smaller than the period of . In
order to find out for which regions the above solutions are stable, we
calculate the time evolution of the soliton momentum and soliton
velocity : This is a parameter representation of a curve which is
visited by the soliton while time evolves. Our conjecture is that the soliton
becomes unstable, if this curve has a branch with negative slope. This
conjecture is fully confirmed by our simulations for the perturbed NLSE.
Moreover, this curve also yields a good estimate for the soliton lifetime: the
soliton lives longer, the shorter the branch with negative slope is.Comment: 21 figure
Variational approximation and the use of collective coordinates
We consider propagating, spatially localized waves in a class of equations that contain variational and nonvariational terms. The dynamics of the waves is analyzed through a collective coordinate approach. Motivated by the variational approximation, we show that there is a natural choice of projection onto collective variables for reducing the governing (nonlinear) partial differential equation (PDE) to coupled ordinary differential equations (ODEs). This projection produces ODEs whose solutions are exactly the stationary states of the effective Lagrangian that would be considered in applying the variational approximation method. We illustrate our approach by applying it to a modified Fisher equation for a traveling front, containing a non-constant-coefficient nonlinear term. We present numerical results that show that our proposed projection captures both the equilibria and the dynamics of the PDE much more closely than previously proposed projections. © 2013 American Physical Society
Spatial Solitons in Media with Delayed-Response Optical Nonlinearities
Near-soliton scanning light-beam propagation in media with both
delayed-response Kerr-type and thermal nonlinearities is analyzed. The
delayed-response part of the Kerr nonlinearity is shown to be competitive as
compared to the thermal nonlinearity, and relevant contributions to a
distortion of the soliton form and phase can be mutually compensated. This
quasi-soliton beam propagation regime keeps properties of the incli- ned
self-trapped channel.Comment: 7 pages, to be published in Europhys. Let
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