4 research outputs found
Stability of exact solutions of the defocusing nonlinear Schrodinger equation with periodic potential in two dimensions
The cubic nonlinear Schrodinger equation with repulsive nonlinearity and
elliptic function potential in two-dimensions models a repulsive dilute gas
Bose--Einstein condensate in a lattice potential. A family of exact stationary
solutions is presented and its stability is examined using analytical and
numerical methods. All stable trivial-phase solutions are off-set from the zero
level. Our results imply that a large number of condensed atoms is sufficient
to form a stable, periodic condensate.Comment: 12 pages, Latex, High resolution version available at
http://www.amath.washington.edu/~kutz/research.htm
Singular instability of exact stationary solutions of the nonlocal Gross-Pitaevskii equation
In this paper we show numerically that for nonlinear Schrodinger type systems
the presence of nonlocal perturbations can lead to a beyond-all-orders
instability of stable solutions of the local equation. For the specific case of
the nonlocal one-dimensional Gross-Pitaevskii equation with an external
standing light wave potential, we construct exact stationary solutions for an
arbitrary interaction kernel. As the nonlocal and local equations approach each
other (by letting an appropriate small parameter ), we compare
the dynamics of the respective solutions. By considering the time of onset of
instability, the singular nature of the inclusion of nonlocality is
demonstrated, independent of the form of the interaction kernel.Comment: 4 pages, 4 figure