Emergent Directions in Asymptotics for Nonlinear Wave Models

About This Grant

Nonlinear wave formation and propagation are central to many physical phenomena across science and engineering, from ocean swells and rogue waves to plasma dynamics in fusion reactors and signal transmission in optical fibers. Despite arising in distinct systems, nonlinear waves often exhibit recognizable patterns such as rapidly oscillating wave trains, persistent localized beams, and rogue waves, under asymptotic regimes such as long-time evolution or small-dispersion. In many cases, these structures are universal, meaning they arise independently of conditions such as initial data and, in some cases, even of the governing equations. The project will develop and apply mathematical tools to identify and characterize emergent asymptotic phenomena in nonlinear wave models, with a particular focus on non-generic or anomalous behavior. The outcomes will advance the understanding of wave propagation involving large amplitudes or heavy tails, with broad applications in hydrodynamics and optical telecommunications. The project will also provide research training for graduate students and early-career scientists, broadening its impact beyond technical discovery into education and scientific workforce development. The project focuses on obtaining quantitative information on nonlinear wave behavior in asymptotic regimes not accessible through current inverse-scattering transform (IST) methods. Key goals include identifying mechanisms behind anomalously slow decay in solutions to integrable wave equations, extending IST techniques to broader classes of initial data, and establishing long-time behavior and other properties of weakly localized wave formations on modulationally unstable backgrounds using continuum Darboux transformations. To achieve these goals, the investigator will combine and further develop tools from the theory of integrable systems, complex and asymptotic analysis, and numerical methods. Anticipated outcomes include the identification of slowly decaying solutions beyond the predictions of the soliton resolution theory; novel IST methods for weakly localized data; and an expanded universality theory for the nonlinear stage of modulational instability. 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.