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

PI将研究时间相关扰动下色散偏微分方程解的长时间行为。焦点将集中在线性和非线性薛定谔方程支持束缚态(时间周期，空间定域解)的区域内。这种微扰预计会在束缚态之间重新分配能量，并将部分能量转移到辐射中。将寻求对这一过程的严格描述。为此，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 Vries。薛定谔方程是一个针对各种物理现象和工程过程的成熟模型。例如，用于高比特率通信的色散管理光纤由具有不同材料特性的串联光纤组成。当光脉冲穿过它们时，它们经历了两次主要的转变。一方面，它们被阻止扩散，这是可取的。另一方面，它们将能量释放给辐射。在这两种效果之间取得适当的平衡是PI计划要解决的设计问题。第二个例子是关于通过控制环境的变化从特殊相中的粒子(玻色-爱因斯坦凝聚体)中产生物质波的新想法。这些波的稳定性和长时间行为还不是很清楚，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，与通过固定介质传播的信号相比，到达探测器的目标反射的信号往往包含更多的信息。另一方面，同样的波动往往会从环境中辐射出来，无论是直接信号还是反射信号。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.