标题： 受限玻色-爱因斯坦凝聚体中涡旋的相对动力学 显示英文标题

标题： Relative Dynamics of Vortices in Confined Bose--Einstein Condensates

摘要： 我们考虑相对动力学——在这一特定情况下，模旋转对称性的动力学——使用二维Gross-Pitaevskii方程的有限维涡旋近似，研究受限玻色-爱因斯坦凝聚体（BEC）中$N$涡旋的相对动力学。我们通过表明它是某个特定李代数对偶上的李-泊松方程的一个实例，给出了相对动力学的哈密顿形式。正如我们在具有欧几里得对称性的涡旋动力学方面的配套工作中所展示的，相对动力学具有一个Casimir不变量，并在一个不变集中演化，从而产生一个能量-Casimir型稳定性条件。我们考虑三个相对平衡的例子——那些关于原点进行刚体旋转的解——使用该稳定性条件来研究它们的稳定性，其中涉及$N=2, 3, 4$。 显示英文摘要

摘要： We consider the relative dynamics -- the dynamics modulo rotational symmetry in this particular context -- of $N$ vortices in confined Bose--Einstein Condensates (BEC) using a finite-dimensional vortex approximation to the two-dimensional Gross--Pitaevskii equation. We give a Hamiltonian formulation of the relative dynamics by showing that it is an instance of the Lie--Poisson equation on the dual of a certain Lie algebra. Just as in our accompanying work on vortex dynamics with the Euclidean symmetry, the relative dynamics possesses a Casimir invariant and evolves in an invariant set, yielding an Energy--Casimir-type stability condition. We consider three examples of relative equilibria -- those solutions that are undergoing rigid rotations about the origin -- with $N=2, 3, 4$, and investigate their stability using the stability condition.