Multi-soliton Dynamics for Dispersive Partial Differential 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 well-understood. The 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 undergraduate students, graduate students, and postdoctoral researchers. The investigator studies the long-time dynamics of solutions to nonlinear wave and dispersive equations, focusing on 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 makes 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 and going beyond it in others by considering three categories of problems: (1) the soliton resolution conjecture for evolution equations without symmetry assumptions, starting with the harmonic map heat flow in two dimensions, which is a long-standing open problem; (2) the unique continuation problem for singular nonlinear waves past the blow-up time; and (3) the question of giving asymptotic descriptions of multi-soliton solutions and their collisions. 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.