CAREER: New Frontiers in the Dynamics of Topological Solitons

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Nonlinear waves are ubiquitous in nature, ranging from the dynamics of quantum particles to the propagation of electromagnetic radiation and gravitational waves. Mathematically, many such wave propagation phenomena can be described in terms of nonlinear dispersive equations. While waves typically spread out and decay, a striking feature of these nonlinear evolution equations is that they may admit particle-like solutions, often called solitons, whose shapes persist as time goes by. The mathematical understanding of their dynamics is still far from complete. The main research goal of this project is to investigate, in the context of classical topological field theories that arise in mathematical physics, how nonlinear waves can form particle-like structures and how these structures interact with each other. The educational component of the project seeks to enhance the training of graduate students and postdocs by organizing minicourses and workshops related to the research of the project and by providing professional development opportunities with an emphasis on presentation skills.This project focuses on soliton dynamics for several well-known classical topological field theories in mathematical physics. Three prime examples of topological solitons are at the center of the investigation: kinks, vortices, and skyrmions in one, two, and three space dimensions, respectively. Heuristically, these solitons owe their stability to their topological underpinnings. However, the mathematical justification of this intuition is still rather poorly understood and mostly open. The overarching goal of the project is to establish asymptotic stability results for these classical topological solitons, and thus to rigorously justify the heuristics for their stability. Over the course of the project the investigator also plans to move towards studying multi-soliton configurations in these and related settings. Beyond the intrinsic interest in the fundamental problems at the center of this project, their resolution will have significant impact on the analysis of strong nonlinear interactions between solitons and radiation in the context of many other nonlinear dispersive equations.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.