Abstract:
In this talk, I begin with a brief derivation
of the nonlinear Schrodinger/Gross-Pitaevskii equations
(NLSE/GPE) from Bose-Einstein condensates (BEC) and/or
nonlinear optics. Then I will present some mathematical
results on the existence and uniqueness as well as
non-existence of the ground states of NLSE/GPE under different
external potentials and parameter regimes. Dynamical
properties of NLSE/GPE are then discussed, which include
conservation laws, soliton solutions, well-posedness and/or
finite time blowup. Efficient and accurate numerical methods
will be presented for computing numerically the ground states
and dynamics. Extension to NLSE/GPE with an angular momentum
rotation term and/or non-local dipole-dipole interaction will
be presented. Finally, applications to collapse and explosion
of BEC, quantum transport and quantized vortex interaction
will be investigated. |