Dynamics of Nonlinear Wave Equations

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The world is ruled 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 propagate by wave dynamics. Though ubiquitous, wave equations are far from understood. The goal of this project is to understand how the dynamics of waves change in the face of interference with themselves or with their environment. The research seeks to learn when and why some waves disperse, while other waves persist, and still others collapse. Knowing how waves behave in critical conditions drives technological progress -- smaller microchips, faster data transmission, and deeper insights into the formation of the universe. This project will study dynamical properties of solutions to nonlinear wave and dispersive equations in geometric settings where the nonlinear structure is intimately tied to classical geometric notions such as curvature. Such equations arise in a myriad of physical models, ranging from Einstein's equations of general relativity, to the interactions of particles in nuclear physics. Particular examples of interest include the wave maps systems, the Yang-Mills system, semilinear versions of Skyrme's equation, and power-type nonlinear wave and Schrodinger equations. In many of these problems, solitons (coherent solitary waves) are thought to be the basic building blocks of global-in-time dynamics: as the solution evolves, it decomposes into a finite number of weakly interacting solitons plus a remainder term exhibiting linear dynamics. This is a loose formulation of the soliton resolution conjecture, which is of central importance in the field. Progress towards the proof of soliton resolution is the underlying motivation of several of the specific problems to be addressed in this project. The dynamics of solitons are also fundamental to singularity formation, often playing the role of the universal profiles for singularities that form via a concentration of mass or energy. The project will study the fine mechanics of singularity formation, and will seek to characterize the possible dynamical properties of the solution from the profile of the energy that has radiated away from the point of concentration.