Wave-breaking and Resonant Phenomena

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In the first part of the project, wave-breaking will be studied in the context of the focusing nonlinear Schroedinger equation (NLS), a well-known model of amplitude-dependent beam self-focusing. NLS wavetrains may come to a sudden break at nonlinear caustic points and rearrange themselves into more complex wave structures. These structures, which may be visualized conceptually as nonlinear wave interference, may go through successive breaking; moreover, they display modulational instability. Controlling the instability and understanding mechanisms of the wave-breaking are the focus of the project. The steepest descent method for Riemann-Hilbert problems (RHP) in the semiclassical limit will be used to peel off the dominant contribution to the waveform from the RHP that linearizes NLS. The steepest descent method will itself be refined for this purpose. A numerical component is also involved in our approach. The second part of the project addresses resonant behavior in wave propagation through media that have a repetitive structure. In previous work, anomalies in transmission through slabs of such media were observed and linked to resonant phenomena through a mathematical formula describing the transmission anomaly accurately in 2D. The current project will develop formulae for the effective density of states near resonance and make a firm link between the resonance in question and the resonance introduced by Fano in a seminal 1935 quantum mechanics paper (in both cases, following a symmetry breaking perturbation, a real eigenvalue imbedded in continuous spectrum moves into the lower complex half plane as a resonance). The project will also resolve a question in physics on the pulse response of linear materials by an elaborate use of the method of steepest descent for integrals.The understanding of the propagation of waves in natural and in man-made materials is crucial to the utilization of these materials. The first project (wave-breaking) applies to wave propagation in technologically significant materials, described to a good approximation by the focusing nonlinear Schroedinger equation. Such materials are optical fibers in which the modulational instability acts adversely to fiber-optical communication. With current technology, the breaking phenomena researched are beginning to be observable at the nanoscale and will play a role in the development of a theory of propagation in nanowires. The second project (resonant behavior) refers to the propagation of light through photonic crystals, whose repetitive structure may guide, block, or even trap light. The transmission anomalies studied consist of a sharp dip followed by sharp rise of the transmission coefficient as the frequency is varied in an experiment in which light is incident nearly normally onto a slab of photonic crystal material. Both projects will involve students in interplay between mathematical analysis, numerical experiments, and physical experiments bridging the significantly different types of intuition in the mathematics and physics communities.