Collaborative Research: Nonlinear Dynamics in Dispersive Partial Differential Equations

This project advances the understanding of nonlinear dispersive partial differential equations (PDE) - mathematical models that describe waves in contexts ranging from optics and fluid dynamics to quantum gases and plasma physics. It addresses fundamental questions about the formation, stability, and interaction of coherent structures such as solitons, vortices, and singularities in contexts leading to new and richer wave behaviors. Insights into these mechanisms promise to push forward mathematical theory in analysis and PDE, improve modeling in physics and engineering (e.g., ocean wave propagation, Bose-Einstein condensates, plasma dynamics), and serve the national interest by enhancing the theoretical foundation of applied scientific disciplines. The project also supports education through training of junior researchers and fosters broader impacts via cross-institutional student mentoring. This project explores fundamental questions in the analysis of nonlinear dispersive systems, with an emphasis on understanding the formation and long-time dynamics of coherent structures such as solitons, vortices, and singularities. The focus lies on situations where classical assumptions - such as locality, vanishing boundary conditions, or scalar structure - are relaxed, leading to new challenges in the mathematical analysis. These include the study of blow-up dynamics near minimal mass thresholds, the stability and asymptotics of localized waves in higher-dimensional or nonlocal settings, and the evolution of coherent structures in systems with nontrivial background states or multiple interacting components. The research develops and applies a range of techniques from nonlinear Fourier analysis, spectral theory, and dispersive PDE, including virial-type arguments, modulation methods, and iterative profile decompositions. Anticipated contributions include new mechanisms for stable singularity formation, improved understanding of multidimensional wave interactions, and advances in the long-time behavior of coupled or constrained dispersive flows. 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.