Probing quasi-integrability of the Gross-Pitaevskii equation in a harmonic-oscillator potential
Probing quasi-integrability of the Gross-Pitaevskii equation in a harmonic-oscillator potential
复制标题
探讨谐波振荡器势中 Gross-Pitaevskii 方程的准可积性
DOI：
10.1088/1361-6455/aae0ba
复制
发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Bland T
中科院分区：
文献类型：
--
作者：
Bland T
先前对具有排斥非线性和谐波振荡器捕获势的一维格罗斯-皮塔耶夫斯基方程（GPE）的模拟暗示了准可积动力学的出现——在没有任何遍历性迹象的移动暗孤子的准周期演化的意义上——尽管该模型不属于可积方程列表。为了研究这个问题，我们用谐波振荡器本征模的适当截断展开（伽辽金近似）来代替完整的 GPE，它可以准确地再现完整的动态，然后分析系统的动态谱。该分析使我们能够将观察到的准可积性解释为这样一个事实：有限模式动力学总是产生准离散功率谱，没有可见的连续分量，后者的存在是遍历性的必要表现。当将强随机场分量添加到初始条件时，这个结论仍然成立。另一方面，在无限深的势箱中对 GPE 进行相同的分析会产生明显的连续功率谱，这是遍历动力学的典型特征。
Previous simulations of the one-dimensional Gross–Pitaevskii equation (GPE) with repulsive nonlinearity and a harmonic-oscillator trapping potential hint towards the emergence of quasi-integrable dynamics—in the sense of quasi-periodic evolution of a moving dark soliton without any signs of ergodicity—although this model does not belong to the list of integrable equations. To investigate this problem, we replace the full GPE by a suitably truncated expansion over harmonic-oscillator eigenmodes (the Galerkin approximation), which accurately reproduces the full dynamics, and then analyze the system's dynamical spectrum. The analysis enables us to interpret the observed quasi-integrability as the fact that the finite-mode dynamics always produces a quasi-discrete power spectrum, with no visible continuous component, the presence of the latter being a necessary manifestation of ergodicity. This conclusion remains true when a strong random-field component is added to the initial conditions. On the other hand, the same analysis for the GPE in an infinitely deep potential box leads to a clearly continuous power spectrum, typical for ergodic dynamics.