Wave Modulation & Stability in Applied Mathematics

This project investigates the stability and dynamics of spatially periodic solutions to nonlinear partial differential equations (PDEs) that arise across physics, engineering, and applied mathematics. The principal investigator (PI) focuses on the dynamical stability of these patterns -- their ability to persist under small perturbations – which is crucial since unstable solutions are generally not observable in practical settings except as transient phenomena. Specifically, the project aims to study the modulational stability of periodic wave patterns, examining how their fundamental wave characteristics evolve under slow spatial and temporal variations. Insights from this research have important implications for many applications, including optical signal propagation, fluid flows, and plasma physics. The project also includes opportunities for both undergraduate and graduate students to participate in advanced research training. Building on the PI’s prior success in studying modulated signals in one-dimensional media, this research extends to multi-dimensional nonlinear wave phenomena. The goal is to develop new techniques and methodologies for studying the behavior of spatially periodic patterns under small modulational perturbations. Because periodic patterns can support multiple modulated signals simultaneously, their dynamics exhibit rich, multi-scale structures that are effectively infinite-dimensional. Analyzing these systems involves addressing many nonstandard and interesting issues in dynamical systems, bifurcation and continuation theory, Whitham modulation theory, and spectral and semigroup theory for linear differential operators. The PI’s work is expected to yield new analytical tools and techniques of general interest and applicability across the scientific community. 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.