183 research outputs found
Fundamental gaps of the Gross-Pitaevskii equation with repulsive interaction
We study asymptotically and numerically the fundamental gaps (i.e. the
difference between the first excited state and the ground state) in energy and
chemical potential of the Gross-Pitaevskii equation (GPE) -- nonlinear
Schrodinger equation with cubic nonlinearity -- with repulsive interaction
under different trapping potentials including box potential and harmonic
potential. Based on our asymptotic and numerical results, we formulate a gap
conjecture on the fundamental gaps in energy and chemical potential of the GPE
on bounded domains with the homogeneous Dirichlet boundary condition, and in
the whole space with a convex trapping potential growing at least quadratically
in the far field. We then extend these results to the GPE on bounded domains
with either the homogeneous Neumann boundary condition or periodic boundary
condition.Comment: 26 pages, 16 figure
An explicit unconditionally stable numerical method for solving damped nonlinear Schr\"{o}dinger equations with a focusing nonlinearity
This paper introduces an extension of the time-splitting sine-spectral (TSSP)
method for solving damped focusing nonlinear Schr\"{o}dinger equations (NLS).
The method is explicit, unconditionally stable and time transversal invariant.
Moreover, it preserves the exact decay rate for the normalization of the wave
function if linear damping terms are added to the NLS. Extensive numerical
tests are presented for cubic focusing nonlinear Schr\"{o}dinger equations in
2d with a linear, cubic or a quintic damping term. Our numerical results show
that quintic or cubic damping always arrests blowup, while linear damping can
arrest blowup only when the damping parameter \dt is larger than a threshold
value \dt_{\rm th}. We note that our method can also be applied to solve the
3d Gross-Pitaevskii equation with a quintic damping term to model the dynamics
of a collapsing and exploding Bose-Einstein condensate (BEC).Comment: SIAM Journal on Numerical Analysis, to appea
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