D. Pelinovsky
Asymptotic properties of excited states in the Thomas-Fermi limit
Nonlinear Analysis 73, 2631-2643 (2010)
Abstract:
Excited states are stationary localized solutions of the
Gross-Pitaevskii equation with a harmonic potential and a
repulsive nonlinear term that have zeros on a real axis. Existence
and asymptotic properties of excited states are considered in the
semi-classical (Thomas-Fermi) limit. Using the method of
Lyapunov--Schmidt reductions and the known properties of the
ground state in the Thomas-Fermi limit, we show that excited
states can be approximated by a product of dark solitons
(localized waves of the defocusing nonlinear Schrodinger
equation with nonzero boundary conditions) and the ground state.
The dark solitons are centered at the equilibrium points where a
balance between the actions of the harmonic potential and
the tail-to-tail interaction potential is achieved.
Keywords:
Gross-Pitaevskii equation, dark solitons, existence of excited states,
Lyapunov-Schmidt reductions