Applied Analysis for Emergent Nonlinear Wave Phenomena

Wave formations are observed in a variety of media ranging from the atmosphere and the hydrosphere to optical fibers and chains of molecules. Despite the physical differences in these media, waves tend to generate identifiable patterns and structures that share common characteristics. Examples of such formations include wave trains exhibiting rapid oscillations, bump-shaped beams propagating persistently over long periods of time, and so-called rogue waves, which are large disturbances of a background state that appear out of nowhere and disappear just as suddenly.

These and other wave phenomena can be modeled by certain nonlinear differential equations for which detailed analysis is feasible. This project will develop, extend, and apply mathematical techniques to study wave formation and propagation in different physical contexts to extract accurate information about the dynamics of waves, with applications to fields including hydrodynamics, atmospheric sciences, and optical telecommunications.

Outcomes of such analysis will, for example, further understanding of the wave patterns surrounding rogue waves. Parts of the project will serve as vehicles for the training of graduate students and postdoctoral researchers.

This project aims to develop and apply analytical and computational methods from the theory of integrable systems to study nonlinear wave propagation. These methods combine tools from complex, asymptotic, and numerical analysis. Part of the project will investigate new asymptotic regimes in which universal phenomena in nonlinear wave formation may occur.

The research activity includes describing and classifying the wave patterns generated by a new type of mechanism that forms large-amplitude rogue waves, applicable to various models, including the nonlinear Schrödinger (NLS) and the sine-Gordon equations. Other topics addressed include the time-evolution of initial data that gives rise to spectral singularities in the NLS equation and the formation of dispersive shockwaves and their interactions with other structures in the Korteweg-de Vries equation.

The research will contribute to the fields of integrable systems, nonlinear waves, and special functions.

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.