Metastability of Breather Modes of Time Dependent Potentials
Metastability of Breather Modes of Time Dependent Potentials
复制标题
时间相关势的呼吸模式的亚稳态
DOI：
10.1088/0951-7715/13/3/303
复制
发表时间：
2000
期刊：
影响因子：
--
通讯作者：
Michael I. Weinstein
中科院分区：
文献类型：
--
作者：
Peter D. Miller;A. Soffer;Michael I. Weinstein
我们研究线性薛定谔方程的解，其中势能是时间的周期函数并且在空间中充分局域化。我们认为有可能接近时间周期性且明确可解的问题。 Miller 和 Akhmediev 已经构建了一个大族的此类势，并求解了相应的薛定谔方程。精确的束缚态或呼吸模式存在于未扰动的问题中，并且被发现在存在小周期性扰动的情况下一般是亚稳态的。因此，这些状态是长久存在的，但最终会衰败。在 2 阶时间尺度上（其中 是扰动大小的度量），衰减呈指数形式，衰减率由费米黄金法则的类似物给出。对于 1 阶时间，呼吸模式发生频移。这种行为首先是由经典的多尺度展开推导出来的，然后在某些情况下，我们能够应用由 Soffer 和 Weinstein 开发并由 Kirr 和 Weinstein 扩展的严格理论来证明展开的合理性，并提供更长时间的渐近线，表明束缚态最终的色散衰变与时间上的代数行为。作为一种应用，我们使用我们的技术来研究某些光波导的引导特性的频率依赖性。我们用数值实验来补充我们的结果。
We study the solutions of linear Schr ¨ odinger equations in which the potential energy is a periodic function of time and is sufficiently localized in space. We consider the potential to be close to one that is time periodic and yet explicitly solvable. A large family of such potentials has been constructed and the corresponding Schr ¨ odinger equation solved by Miller and Akhmediev. Exact bound states, or breather modes, exist in the unperturbed problem and are found to be generically metastable in the presence of small periodic perturbations. Thus, these states are long-lived but eventually decay. On a time scale of order 2 , where is a measure of the perturbation size, the decay is exponential, with a rate of decay given by an analogue of Fermi's golden rule. For times of order 1 the breather modes are frequency shifted. This behaviour is derived first by classical multiple-scale expansions, and then in certain circumstances we are able to apply the rigorous theory developed by Soffer and Weinstein and extended by Kirr and Weinstein to justify the expansions and also provide longer-time asymptotics that indicate eventual dispersive decay of the bound states with behaviour that is algebraic in time. As an application, we use our techniques to study the frequency dependence of the guidance properties of certain optical waveguides. We supplement our results with numerical experiments.