一种约束最缓上升动力学及其在玻色-爱因斯坦凝聚激发态研究中的应用
A constrained gentlest ascent dynamics and its applications to finding excited states of Bose-Einstein condensates
摘要：
本文对[W. E和X。Zhou，Nonlinearity，24（2011），pp. 1831- 1842]推广到约束最平缓上升动力学（CGAD），以寻找具有任意指定Morse指数的约束鞍点.证明了该CGAD线性稳定的定态是一个非退化约束鞍点，并具有相应的Morse指标。同时，证明了理想CGAD在非退化约束鞍点附近的指数收敛性。通过计算相应的Gros-Pitaevskii能量泛函在归一化约束下的约束鞍点，CGAD被应用于寻找单组分玻色-爱因斯坦凝聚体（BEC）的激发态，并与Morse指数成正比.此外，在线性/非线性情况下的BEC的激发态的性质的数学/数值研究。大量的数值结果表明，我们的方法的有效性和鲁棒性，并展示了一些有趣的物理。
英文摘要：
In this paper, the gentlest ascent dynamics (GAD) developed in [W. E and X. Zhou, Nonlinearity, 24 (2011), pp. 1831--1842] is extended to a constrained gentlest ascent dynamics (CGAD) to find constrained saddle points with any specified Morse indices. It is proved that the linearly stable steady state of the proposed CGAD is exactly a nondegenerate constrained saddle point with a corresponding Morse index. Meanwhile, the locally exponential convergence of an idealized CGAD near nondegenerate constrained saddle points with corresponding indices is also verified. The CGAD is then applied to find excited states of single-component Bose--Einstein condensates (BECs) in the order of their Morse indices via computing constrained saddle points of the corresponding Gross--Pitaevskii energy functional under the normalization constraint. In addition, properties of the excited states of BECs in the linear/nonlinear cases are mathematically/numerically studied. Extensive numerical results are reported to show the effectiveness and robustness of our method and demonstrate some interesting physics.
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