The nonlinear Schrödinger (NLS) equation is a fundamental model equation in modern mathematical physics. This equation may serve, for example, to describe both certain water waves and the propagation of light in optical fibers. However, the nonlinear nature of this equation is an obstacle that prevents the straightforward application of most existing general-purpose mathematical tools. This project will use the special structure of this equation to examine in detail two features of its solutions, called modulational instability (MI) and supercontinuum generation (SCG). While the starting point of these investigations is based on the mathematical properties of the NLS equation, MI and SCG are important phenomena in the real world. For example, MI has been proposed as an important element in the formation of "rogue waves" in the ocean; these rare but relatively large and spontaneous waves may be quite dangerous. Furthermore, SCG is important in optical telecommunications systems and in optical coherence tomography, an imaging technique used in opthalmology to reveal detailed retinal structures. A better understanding of this mathematical phenomenon thus has the potential to influence the design and development of telecommunications systems and optical devices.
This project's investigation of MI and SCG is based on the initial-value problem for the focusing NLS equation in the small-dispersion regime. In this regime, nonlinear effects are dominant, and solutions exhibit small-scale oscillations and violent transitions (nonlinear caustics) to even more complicated oscillatory patterns. Both MI and SCG figure prominently in this behavior. Indeed, in the limit of vanishing dispersion (also known as the semiclassical limit), MI suggests that a generic plane wave solution may be expected to break immediately into some other, presumably more disordered, form. However, recent results show that there do exist distinguished perturbations which do not excite the acute modulational instabilities known to be present in the small-dispersion regime. This project will incorporate such special perturbations into the broader theory of the zero-dispersion limit of the focusing NLS equation. As part of this effort, a scheme for classifying and describing these perturbations will be developed. Once this is done, the import of such special perturbations for applications can be assessed. Another component of this research will be to extend a non-standard method for computing the Fourier Power Spectrum of solutions to a broader class of initial data; once completed, this method will give a new, direct way to measure the impact of wave breaking (nonlinear caustics) on SCG.
