Mathematical Sciences: Linear and Nonlinear Waves in Random Media

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This proposal concerns itself with several problems involving random media, namely the propagation of pulse waves and the focussing of singularities of solutions of the nonlinear Schroedinger equation. In particular, the principal investigator will study direct and inverse problems for pulse reflection of waves in random media, reflection and transmission of gravity waves in irregular channels, and singular solutions of the Schroedinger equation due to nonlinear focussing of light beams. Such problems arise in a host of areas, for example, seismology, plasma physics and nonlinear optics. Many phenomena involve the propagation of waves through inhomogeneous media. Think of seismic waves travelling inside the earth after an earthquake. Since the medium is not uniform or orderly, we call it random and scientists use a variety of mathematical models to help understand how local microscopic properties influence macroscopic ones. The macroscopic effects are the the ones we can measure and they usually are of most interest to us. In this proposal the principal investigator will use analytical and numerical tools to study how waves and light beams behave as they travel inside of random media.//