Mathematical Analysis of Parametrically Excited Hamiltonian Systems with Applications in Quantum Physics, Nonlinear Optics and Wave Propagation in Random Media

PI将研究色散偏微分方程解在时间相关扰动下的长时间行为。重点将放在线性和非线性薛定谔方程的制度中，它支持束缚态（周期性的时间，在空间的解决方案）。微扰预计将重新分配束缚态之间的能量，并将其部分转移到辐射。将寻求对这一过程的严格描述。为此目的，PI将开发新的数学技术来研究辐射场和束缚态的演化。这些工具有望推广到其他色散方程，如Klein-Gordon，Sine-Gordon和Korteweg-de弗里斯。薛定谔方程是一个建立良好的模型，用于各种物理现象和工程过程。例如，在高比特率电信中使用的色散管理光纤由具有不同材料特性的级联光纤段组成。当光脉冲通过它们时，它们经历两个主要的转变。一方面，他们被阻止扩散，这是可取的。另一方面，它们将能量释放到辐射中。在这两种效果之间取得适当的平衡是PI计划做出贡献的设计问题。第二个例子是关于在一个特殊的阶段（玻色-爱因斯坦凝聚体）通过控制其环境的变化产生物质波的新想法。这些波的稳定性和长期行为还没有得到很好的理解，将在项目中进行研究。第三个例子涉及通过波动介质的雷达探测。一方面，由于随机波动，与通过静止介质传播的信号相比，由到达检测器的目标反射的信号倾向于包含更多的信息。另一方面，相同的波动倾向于将直接信号和反射信号辐射出环境。更好地定性和定量地了解第二种现象是本项目的目标之一。

The PI will study the long time behavior of solutions of dispersive partial differential equations under time dependent perturbations. The focus will be on the Linear and Nonlinear Schroedinger Equation in the regimes in which it supports bound states (periodic in time, localized in space solutions). The perturbation is expected to redistribute the energy among the bound state and transfer part of it to radiation. A rigorous description of this process will be sought. For this purpose the PI will develop novel mathematical techniques for studying the evolution of both the radiation field and the bound states. The tools are expected to generalize to other dispersive equations like Klein-Gordon, Sine-Gordon and Korteweg - de Vries.The Schroedinger Equation is a well established model for a variety of physical phenomena and engineering processes. For example, dispersion managed optical fibers used in high bit rate telecommunications consist of concatenated pieces of fiber with different material properties. As the light pulses pass through them they suffer two main transformations. On one hand they are kept from spreading out which is desirable. On the other hand they loose energy to radiation. Striking the right balance between the two effects is a design problem to which the PI plans to contribute. The second example is related to the new ideas on generating matter waves out of particles in a special phase (Bose-Einstein Condensates) by controlled variation of their environment. The stability and long time behavior of these waves is not well understood and will be investigated in the project. The third example concerns radar detection through fluctuating media. On one hand, due to the random fluctuations, the signal reflected by the target that reaches the detector tends to contain more information compared to a signal that propagates through a stationary medium. On the other hand, the same fluctuations tend to radiate out of the environment both the direct and reflected signal. A better qualitative and quantitative understanding of the second phenomena is one of the goals of this project.