Modulations of Periodic Waves in Applied Mathematics

This project analyzes the existence and behavior of spatially periodic solutions to models for viscous fluid dynamics and optical signal propagation, with an emphasis on the stability of such patterns, that is their ability to persist when subjected to small perturbations. The study of the stability of a pattern is of practical interest since only stable solutions are expected to be observed in nature.

Results of this research will provide scientists with new analytical tools and methodologies to understand observed laboratory experiments in both fluid dynamics and optical signal propagation contexts, and will lead to experimental design principles that will be used to better explain patterns observed in experiments, to predict the presence of new structures and patterns not previously observed, and to provide insight into the experimental construction of patterns. Both undergraduate and graduate students will receive training through research involvement in the project.

The research project naturally divides into two sets of questions according to their fundamentally different physical applications. The first set seeks to develop and analyze new mathematical models relating to buoyancy driven viscous interfacial wave dynamics, such as magma rising through a porous rock. The results of this analysis is expected to provide mathematically rigorous justifications for currently unexplained laboratory observations.

The second relates to the dynamics of optics waveguides and optical resonators and will resolve several outstanding issues that are of interest to both experimentalists and theoreticians alike. Importantly, both components involve mathematical work that can be experimentally tested and compared to laboratory experiments.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.