Asymptotic properties of excited states in the Thomas--Fermi limit

TL;DR

This paper investigates the asymptotic behavior of excited states in the Gross-Pitaevskii equation under the Thomas-Fermi limit, revealing their approximation by dark solitons and ground states.

Contribution

It introduces a method to approximate excited states using dark solitons and ground states in the semi-classical limit, advancing understanding of their asymptotic properties.

Findings

Excited states can be approximated by products of dark solitons and ground states.

Dark solitons are centered at equilibrium points balancing potential and interaction.

The method employs Lyapunov--Schmidt reductions to analyze asymptotic properties.

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 Schr\"{o}dinger 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.