标题： 激发态在托马斯-费米极限下的渐近性质 显示英文标题

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

摘要： 激发态是带有谐波势和排斥非线性项的Gross--Pitaevskii方程的定态局部解，在实轴上有零点。 在半经典（Thomas-Fermi）极限下考虑了激发态的存在性和渐近性质。 利用Lyapunov--Schmidt约化方法和Thomas--Fermi极限下基态已知的性质，我们证明激发态可以由暗孤子（具有非零边界条件的失焦非线性Schrödinger方程的局部波）和基态的乘积来近似。 暗孤子位于平衡点处，这些平衡点是谐波势和尾对尾相互作用势之间达到平衡的地方。 显示英文摘要

摘要： 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.