150,314 research outputs found
Periodic solutions for a class of nonlinear partial differential equations in higher dimension
We prove the existence of periodic solutions in a class of nonlinear partial
differential equations, including the nonlinear Schroedinger equation, the
nonlinear wave equation, and the nonlinear beam equation, in higher dimension.
Our result covers cases where the bifurcation equation is infinite-dimensional,
such as the nonlinear Schroedinger equation with zero mass, for which solutions
which at leading order are wave packets are shown to exist.Comment: 34 page
Stability of narrow beams in bulk Kerr-type nonlinear media
We consider (2+1)-dimensional beams, whose transverse size may be comparable
to or smaller than the carrier wavelength, on the basis of an extended version
of the nonlinear Schr\"{o}dinger equation derived from the Maxwell`s equations.
As this equation is very cumbersome, we also study, in parallel to it, its
simplified version which keeps the most essential term: the term which accounts
for the {\it nonlinear diffraction}. The full equation additionally includes
terms generated by a deviation from the paraxial approximation and by a
longitudinal electric-field component in the beam. Solitary-wave stationary
solutions to both the full and simplified equations are found, treating the
terms which modify the nonlinear Schr\"{o}dinger equation as perturbations.
Within the framework of the perturbative approach, a conserved power of the
beam is obtained in an explicit form. It is found that the nonlinear
diffraction affects stationary beams much stronger than nonparaxiality and
longitudinal field. Stability of the beams is directly tested by simulating the
simplified equation, with initial configurations taken as predicted by the
perturbation theory. The numerically generated solitary beams are always stable
and never start to collapse, although they display periodic internal
vibrations, whose amplitude decreases with the increase of the beam power.Comment: 7 pages, 6 figures Accepted for publication in PR
Self consistent thermal wave model description of the transverse dynamics for relativistic charged particle beams in magnetoactive plasmas
Thermal Wave Model is used to study the strong self-consistent Plasma Wake
Field interaction (transverse effects) between a strongly magnetized plasma and
a relativistic electron/positron beam travelling along the external magnetic
field, in the long beam limit, in terms of a nonlocal NLS equation and the
virial equation. In the linear regime, vortices predicted in terms of
Laguerre-Gauss beams characterized by non-zero orbital angular momentum (vortex
charge). In the nonlinear regime, criteria for collapse and stable oscillations
is established and the thin plasma lens mechanism is investigated, for beam
size much greater than the plasma wavelength. The beam squeezing and the
self-pinching equilibrium is predicted, for beam size much smaller than the
plasma wavelength, taking the aberrationless solution of the nonlocal Nonlinear
Schroeding equation.Comment: Poster presentation P5.006 at the 38th EPS Conference on Plasma
Physics, Strasbourg, France, 26 June - 1 July, 201
Some non-linear s.p.d.e.'s that are second order in time
We extend Walsh's theory of martingale measures in order to deal with
hyperbolic stochastic partial differential equations that are second order in
time, such as the wave equation and the beam equation, and driven by spatially
homogeneous Gaussian noise. For such equations, the fundamental solution can be
a distribution in the sense of Schwartz, which appears as an integrand in the
reformulation of the s.p.d.e. as a stochastic integral equation. Our approach
provides an alternative to the Hilbert space integrals of Hilbert-Schmidt
operators. We give several examples, including the beam equation and the wave
equation, with nonlinear multiplicative noise terms
- …