Asymptotic properties of excited states in the Thomas–Fermi limit

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. The existence and the 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 the excited states can be approximated by a product of dark solitons (localized waves of the defocusing nonlinear Schrödinger 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.