Soliton Dynamics for Non-Linear Wave Equations

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The natural world is governed by wave equations: the electricity on a circuit board, the light in fiber optic cables, 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 formation of the universe. The project provides research training opportunities for graduate students.The investigator will study nonlinear wave equations that admit topological solitons, which are used to model the physical phenomena described above. Technically, these are coherent solitary waves with a nontrivial topological invariant. Canonical examples include kinks in scalar field theories, harmonic maps as stationary wave maps, vortices in gauged Ginzburg-Landau theory, magnetic monopoles, Skyrmions, and Yang-Mills instantons. The goal is to understand how topological solitons influence the dynamics, and to resolve two long-standing, open questions. First, the investigator will try to prove that nonlinear waves can be uniquely continued past a singularity that develops in finite time by concentrating energy (bubbling) in a soliton. Second, the investigator seeks to show that multi-soliton collisions are necessarily inelastic for non-integrable wave equations such as the phi-4 scalar field model and the wave maps equation. Crucial parts of this program are existence and uniqueness proofs of solutions exhibiting finite time bubbling and global-in-time multi-soliton dynamics. The techniques the investigator is developing to address these problems will be useful in other related contexts.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.