406 research outputs found
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
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
New features of modulational instability of partially coherent light; importance of the incoherence spectrum
It is shown that the properties of the modulational instability of partially
coherent waves propagating in a nonlinear Kerr medium depend crucially on the
profile of the incoherent field spectrum. Under certain conditions, the
incoherence may even enhance, rather than suppress, the instability. In
particular, it is found that the range of modulationally unstable wave numbers
does not necessarily decrease monotonously with increasing degree of
incoherence and that the modulational instability may still exist even when
long wavelength perturbations are stable.Comment: 4 pages, 2 figures, submitted to Phys. Rev. Let
Singularites in the Bousseneq equation and in the generalized KdV equation
In this paper, two kinds of the exact singular solutions are obtained by the
improved homogeneous balance (HB) method and a nonlinear transformation. The
two exact solutions show that special singular wave patterns exists in the
classical model of some nonlinear wave problems
Regular spatial structures in arrays of Bose-Einstein condensates induced by modulational instability
We show that the phenomenon of modulational instability in arrays of
Bose-Einstein condensates confined to optical lattices gives rise to coherent
spatial structures of localized excitations. These excitations represent thin
disks in 1D, narrow tubes in 2D, and small hollows in 3D arrays, filled in with
condensed atoms of much greater density compared to surrounding array sites.
Aspects of the developed pattern depend on the initial distribution function of
the condensate over the optical lattice, corresponding to particular points of
the Brillouin zone. The long-time behavior of the spatial structures emerging
due to modulational instability is characterized by the periodic recurrence to
the initial low-density state in a finite optical lattice. We propose a simple
way to retain the localized spatial structures with high atomic concentration,
which may be of interest for applications. Theoretical model, based on the
multiple scale expansion, describes the basic features of the phenomenon.
Results of numerical simulations confirm the analytical predictions.Comment: 17 pages, 13 figure
Shock waves in the dissipative Toda lattice
We consider the propagation of a shock wave (SW) in the damped Toda lattice.
The SW is a moving boundary between two semi-infinite lattice domains with
different densities. A steadily moving SW may exist if the damping in the
lattice is represented by an ``inner'' friction, which is a discrete analog of
the second viscosity in hydrodynamics. The problem can be considered
analytically in the continuum approximation, and the analysis produces an
explicit relation between the SW's velocity and the densities of the two
phases. Numerical simulations of the lattice equations of motion demonstrate
that a stable SW establishes if the initial velocity is directed towards the
less dense phase; in the opposite case, the wave gradually spreads out. The
numerically found equilibrium velocity of the SW turns out to be in a very good
agreement with the analytical formula even in a strongly discrete case. If the
initial velocity is essentially different from the one determined by the
densities (but has the correct sign), the velocity does not significantly
alter, but instead the SW adjusts itself to the given velocity by sending
another SW in the opposite direction.Comment: 10 pages in LaTeX, 5 figures available upon regues
Adiabatic Compression of Soliton Matter Waves
The evolution of atomic solitary waves in Bose-Einstein condensate (BEC)
under adiabatic changes of the atomic scattering length is investigated. The
variations of amplitude, width, and velocity of soliton are found for both
spatial and time adiabatic variations. The possibility to use these variations
to compress solitons up to very high local matter densities is shown both in
absence and in presence of a parabolic confining potential.Comment: to appear in J.Phys.
Instability and Evolution of Nonlinearly Interacting Water Waves
We consider the modulational instability of nonlinearly interacting
two-dimensional waves in deep water, which are described by a pair of
two-dimensional coupled nonlinear Schroedinger equations. We derive a nonlinear
dispersion relation. The latter is numerically analyzed to obtain the regions
and the associated growth rates of the modulational instability. Furthermore,
we follow the long term evolution of the latter by means of computer
simulations of the governing nonlinear equations and demonstrate the formation
of localized coherent wave envelopes. Our results should be useful for
understanding the formation and nonlinear propagation characteristics of large
amplitude freak waves in deep water.Comment: 4 pages, 4 figures, to appear in Physical Review Letter
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
Compactons in PT-symmetric generalized Korteweg-de Vries Equations
In an earlier paper Cooper, Shepard, and Sodano introduced a generalized KdV
equation that can exhibit the kinds of compacton solitary waves that were first
seen in equations studied by Rosenau and Hyman. This paper considers the
PT-symmetric extensions of the equations examined by Cooper, Shepard, and
Sodano. From the scaling properties of the PT-symmetric equations a general
theorem relating the energy, momentum, and velocity of any solitary-wave
solution of the generalized KdV equation is derived, and it is shown that the
velocity of the solitons is determined by their amplitude, width, and momentum.Comment: 12 pages, 5 figure
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