Spectral Theory and Nonlinear Waves

This project is devoted to the study of wave propagation in both structured and disordered media. Modern society relies extensively on engineering applications involving transmission of waves. Cell phone communications, satellite data transmissions and information on the internet - all sent along thousands of miles of glass fiber cables - are controlled by mathematics describing the motion of waves.

A striking feature of the wave propagation, and the topic of this project, is its universality. In fact, waves propagating on astronomical scales such as the gravitational waves detected by LIGO, as well as waves on a microscopic scale such as those emitted by atoms in the form of electromagnetic radiation in a laser, are governed by exactly the same mathematical theories.

It is therefore of the utmost strategic importance for the well-being and safety of the society at large to train young specialists in the PI’s area of research; this is also one of the goals of this project. Experience shows that this broader impact can only be achieved by scientists who are actively working at the frontier of knowledge.

The PI recently completed a perturbative analysis of equivariant critical wave maps into the 2-sphere, without any symmetry assumptions on the perturbations. This nonlinear analysis is based on the semi-classical representations of wave functions for Bessel-type potentials which the PI developed more than a decade ago in the context of the Price law in general relativity.

Application of this new technique is planned to other nonlinear evolution equations, which admit separation of variables and a reduction to an infinite system of coupled semi-classical wave equations. The role of the semi-classical parameter h is typically played by the reciprocal of the angular momentum. Another area in which spectral theory is now coming to the fore is the rapidly developing field of asymptotic stability of topological solitons, specifically of kink solutions for one-dimensional scalar fields.

Jointly with others, the PI recently demonstrated how to treat Klein-Gordon equations with a nongeneric potential and long-range nonlinearities. The basic scalar field equations each exhibit a nongeneric potential with a threshold resonance and are therefore of the type to be considered in this project. The PI's work on quasi-periodic localization and its ramifications will be continued.

A particularly challenging novel perspective will be offered by nonuniformly hyperbolic dynamical systems, where there are strong indications that it is possible to bring Anderson localization techniques to bear.

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.