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

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

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.