10 research outputs found
Open Boundaries for the Nonlinear Schrodinger Equation
We present a new algorithm, the Time Dependent Phase Space Filter (TDPSF)
which is used to solve time dependent Nonlinear Schrodinger Equations (NLS).
The algorithm consists of solving the NLS on a box with periodic boundary
conditions (by any algorithm). Periodically in time we decompose the solution
into a family of coherent states. Coherent states which are outgoing are
deleted, while those which are not are kept, reducing the problem of reflected
(wrapped) waves. Numerical results are given, and rigorous error estimates are
described.
The TDPSF is compatible with spectral methods for solving the interior
problem. The TDPSF also fails gracefully, in the sense that the algorithm
notifies the user when the result is incorrect. We are aware of no other method
with this capability.Comment: 21 pages, 4 figure
Temporal dynamics of tunneling. Hydrodynamic approach
We use the hydrodynamic representation of the Gross -Pitaevskii/Nonlinear
Schroedinger equation in order to analyze the dynamics of macroscopic tunneling
process. We observe a tendency to a wave breaking and shock formation during
the early stages of the tunneling process. A blip in the density distribution
appears in the outskirts of the barrier and under proper conditions it may
transform into a bright soliton. Our approach, based on the theory of shock
formation in solutions of Burgers equation, allows us to find the parameters of
the ejected blip (or soliton if formed) including the velocity of its
propagation. The blip in the density is formed regardless of the value and sign
of the nonlinearity parameter. However a soliton may be formed only if this
parameter is negative (attraction) and large enough. A criterion is proposed.
An ejection of a soliton is also observed numerically. We demonstrate,
theoretically and numerically, controlled formation of soliton through
tunneling. The mass of the ejected soliton is controlled by the initial state.Comment: 11 pages, 6 figures, expanded and more detailed verions of the
previous submissio
Nonlinear effects in tunnelling escape in N-body quantum systems
We consider the problem of tunneling escape of particles from a multiparticle
system confined within a potential trap. The process is nonlinear due to the
interparticle interaction. Using the hydrodynamic representation for the
quantum equations of the multiparticle system we find the tunneling rate and
time evolutions of the number of trapped particles for different nonlinearity
values.Comment: 10 pages, 3 figure
Nonlinear Schroedinger equation with two symmetric point interactions in one dimension
We consider a time-dependent one-dimensional nonlinear Schroedinger equation
with a symmetric potential double well represented by two delta interactions.
Among our results we give an explicit formula for the integral kernel of the
unitary semigroup associated with the linear part of the Hamiltonian. Then we
establish the corresponding Strichartz-type estimate and we prove local
existence and uniqueness of the solution to the original nonlinear problem