835 research outputs found
Embedded Eigenvalues and the Nonlinear Schrodinger Equation
A common challenge to proving asymptotic stability of solitary waves is
understanding the spectrum of the operator associated with the linearized flow.
The existence of eigenvalues can inhibit the dispersive estimates key to
proving stability. Following the work of Marzuola & Simpson, we prove the
absence of embedded eigenvalues for a collection of nonlinear Schrodinger
equations, including some one and three dimensional supercritical equations,
and the three dimensional cubic-quintic equation. Our results also rule out
nonzero eigenvalues within the spectral gap and, in 3D, endpoint resonances.
The proof is computer assisted as it depends on the sign of certain inner
products which do not readily admit analytic representations. Our source code
is available for verification at
http://www.math.toronto.edu/simpson/files/spec_prop_asad_simpson_code.zip.Comment: 29 pages, 27 figures: fixed a typo in an equation from the previous
version, and added two equations to clarif
Finite time collapse of N classical fields described by coupled nonlinear Schrodinger equations
We prove the finite-time collapse of a system of N classical fields, which
are described by N coupled nonlinear Schrodinger equations. We derive the
conditions under which all of the fields experiences this finite-time collapse.
Finally, for two-dimensional systems, we derive constraints on the number of
particles associated with each field that are necessary to prevent collapse.Comment: v2: corrected typo on equation
Phase Space Models for Stochastic Nonlinear Parabolic Waves: Wave Spread and Singularity
We derive several kinetic equations to model the large scale, low Fresnel
number behavior of the nonlinear Schrodinger (NLS) equation with a rapidly
fluctuating random potential. There are three types of kinetic equations the
longitudinal, the transverse and the longitudinal with friction. For these
nonlinear kinetic equations we address two problems: the rate of dispersion and
the singularity formation.
For the problem of dispersion, we show that the kinetic equations of the
longitudinal type produce the cubic-in-time law, that the transverse type
produce the quadratic-in-time law and that the one with friction produces the
linear-in-time law for the variance prior to any singularity.
For the problem of singularity, we show that the singularity and blow-up
conditions in the transverse case remain the same as those for the homogeneous
NLS equation with critical or supercritical self-focusing nonlinearity, but
they have changed in the longitudinal case and in the frictional case due to
the evolution of the Hamiltonian
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
Scaling Properties of Weak Chaos in Nonlinear Disordered Lattices
The Discrete Nonlinear Schroedinger Equation with a random potential in one
dimension is studied as a dynamical system. It is characterized by the length,
the strength of the random potential and by the field density that determines
the effect of nonlinearity. The probability of the system to be regular is
established numerically and found to be a scaling function. This property is
used to calculate the asymptotic properties of the system in regimes beyond our
computational power.Comment: 4 pages, 5 figure
Stability of Discrete Solitons in the Presence of Parametric Driving
In this brief report, we consider parametrically driven bright solitons in
the vicinity of the anti-continuum limit. We illustrate the mechanism through
which these solitons become unstable due to the collision of the phase mode
with the continuous spectrum, or eigenvelues bifurcating thereof. We show how
this mechanism typically leads to complete destruction of the bright solitary
wave.Comment: 4 pages, 4 figure
Analytical three-dimensional bright solitons and soliton-pairs in Bose-Einstein condensates with time-space modulation
We provide analytical three-dimensional bright multi-soliton solutions to the
(3+1)-dimensional Gross-Pitaevskii (GP) equation with time and space-dependent
potential, time-dependent nonlinearity, and gain/loss. The zigzag propagation
trace and the breathing behavior of solitons are observed. Different shapes of
bright solitons and fascinating interactions between two solitons can be
achieved with different parameters. The obtained results may raise the
possibility of relative experiments and potential applications.Comment: 5 pages, 4 figure
Nonlinear Schroedinger Equation in the Presence of Uniform Acceleration
We consider a recently proposed nonlinear Schroedinger equation exhibiting
soliton-like solutions of the power-law form , involving the
-exponential function which naturally emerges within nonextensive
thermostatistics [, with ]. Since
these basic solutions behave like free particles, obeying , and (), it is relevant to investigate how they
change under the effect of uniform acceleration, thus providing the first steps
towards the application of the aforementioned nonlinear equation to the study
of physical scenarios beyond free particle dynamics. We investigate first the
behaviour of the power-law solutions under Galilean transformation and discuss
the ensuing Doppler-like effects. We consider then constant acceleration,
obtaining new solutions that can be equivalently regarded as describing a free
particle viewed from an uniformly accelerated reference frame (with
acceleration ) or a particle moving under a constant force . The latter
interpretation naturally leads to the evolution equation with .
Remarkably enough, the potential couples to , instead of coupling
to , as happens in the familiar linear case ().Comment: 4 pages, no figure
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