Frontiers in Dispersive Wave Equations

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The atmosphere, bodies of water, optical fibers, plasmas, and Bose-Einstein condensates are all examples of nonlinear dispersive media. This means that while these media look very different from each other, waves propagating in each of them tend to display similar, characteristic behaviors. This includes the development of rapid oscillations known as dispersive shock waves, long-lasting propagating pulses known as solitons, and isolated, high-amplitude disturbances known as rogue waves. The mathematical models used to understand and describe these phenomena are similar, regardless of the media being studied. It is a major challenge to construct a cohesive mathematical framework general enough to capture common features across different models yet flexible enough to reveal details in specific situations. Drawing on the most up-to-date analytical, asymptotic, and numerical methods, the project will develop enhanced models of dispersive phenomena including shock waves, solitons, and rogue waves, with applications to areas including hydrodynamics, meteorology, and optical communications. The training and professional development of graduate students, undergraduate students, and postdoctoral researchers will be integral to all aspects of the work.The three major objectives of the project are as follows. (1) Derive a mathematical description of the basic dispersive structure, the oscillatory shock, that is robust enough to handle both shocks that decay and those that break up into trains of solitons in the KdV equation and related models. (2) Develop computationally feasible methods for computing, with error estimates, global phenomena such as the interaction of multiple dispersive shock waves in the NLS equation and related models. (3) Establish a new, universal theory of high-amplitude rogue waves applicable to the NLS, sine-Gordon, and Ablowitz-Ladik equations, among others. The work will extend and combine in new ways the latest methods from differential equations, asymptotics, scattering theory, complex analysis, potential theory, and numerical analysis.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.