Soliton Dynamics for Nonlinear Evolution Equations

The natural world is governed by wave equations: the electricity on a circuit board, the light in fiber-optic cables, the elementary particles inside atoms, and even the black hole in the center of the galaxy all propagate by wave dynamics. Though ubiquitous, wave-type equations are far from being well understood. A goal of this project is to understand how waves are affected by interference with themselves or with their environment. The research seeks to learn when and why some waves disperse, other waves persist, and still others collapse. Knowing how waves behave drives technological progress - smaller microchips, faster data transmission, and deeper insights into the fundamental physics of the universe. The project provides research training opportunities for graduate students and postdoctoral researchers. The investigator studies the long-time dynamics of solutions to nonlinear evolution equations, focusing on dispersive and heat type equations that admit topological solitons, which are used to model the physical phenomena described above. Solitons are localized solitary waves with a nontrivial topological invariant. They were introduced by Skyrme in the 1960s as candidates for particles in classical field theories. They have properties required from a particle in classical mechanics - one can define their position, momentum, and energy - and viewed from a distance, configurations of multiple solitons resemble systems of interacting particles. The investigator's work on multi-soliton dynamics has made this connection with classical mechanics explicit, reducing the dynamics of strongly interacting solitons to underlying n-body problems for their positions, momenta, scales, etc. A guiding principle in the analysis of soliton dynamics is the Soliton Resolution Conjecture, which predicts that generic solutions decompose near the final time of existence into a superposition of finitely many solitons and a term capturing the radiation, often a solution to the underlying linear equation. The investigator will work towards proving the conjecture in certain settings, going beyond it in others, and surprisingly, showing that it is false in two instances where it is widely expected to hold. The project focuses on three categories of problems: (1) the kink stability problem for the classical phi-4 equation on the line; (2) construction of explicit counterexamples to the soliton resolution conjecture for the harmonic map heat flow and the sphere-valued wave maps equation, both in two dimensions and without symmetry assumptions; and (3) the unique continuation problem for singular nonlinear waves past the blow-up time. 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. NSF Award ID: 2556011 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Andrew Lawrie | Institution: University of Maryland, College Park, COLLEGE PARK, MD | Award Amount: $200,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2556011 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2556011.html