Summary:
In this work we study a discrete nonlinear Schrodinger lattice with a parabolic
trapping potential. The model, describing, e.g., an array of repulsive
Bose-Einstein condensate droplets
confined in the wells of an optical lattice, is analytically and numerically
investigated. Starting from the linear limit of the problem, we use global
bifurcation theory to rigorously prove that - in the discrete regime - all
linear states lead to nonlinear generalizations thereof, which assume the form
of a chain of discrete dark solitons (as the density increases). The stability
of the ensuing nonlinear states is studied and
it is found that the ground state is stable, while the excited states feature
a chain of stability/instability bands. We illustrate the mechanisms under
which discrete ness destabilizes the dark-soliton configurations, which become
stable only in the continuum regime. Continuation from the anti-continuum limit
is also considered,
and a rich bifurcation structure is revealed.
Keywords:
BEC, Solitons, Πλέγμα, Non-linear, Bifurcations
