Nonlinear Wave Motion

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Nonlinear wave motion relates to the study of how high-power signals are transmitted and occur widely in applications. Some of the applications include surface water waves, rogue waves in the ocean, tsunamis, optical waves such as those that occur in waveguides and lasers, acoustic waves, and associated shock waves. This research effort focuses on solving a number of open problems that will substantially extend the ability of mathematicians to explain the behavior of large power wave phenomena that arise widely in applications while developing a deeper understanding of this behavior. A postdoctoral associate and undergraduate students will be trained through active participation in this research.The mathematical analysis of nonlinear wave motion presents difficulties because the underlying equations are nonlinear for which there is far less mathematical understanding than small power linear waves. One of the methods that can be used to analyze a class of nonlinear wave equations discussed in this proposal is termed the inverse scattering transform (IST). The PI has experience with this method and has found new classes of physically significant equations with interesting underlying symmetries to which IST can be applied. The method also applies to multidimensional equations and associated solutions which decay in all directions. Key properties of these latter solutions will be understood, and they can be connected to fundamental equations in quantum mechanics and optics. Research will also be conducted on a class of waves called dispersive shock waves (DSWs). DSWs are shock waves that are dispersive in nature; the underlying equations are dispersive not dissipative. Consequently, DSWs are not like atmospheric shock waves which are are strongly affected by dissipation. DSWs arise in many applications including water waves, nonlinear optics, plasma physics, and Bose-Einstein condensation amongst many others. The theory of DSWs will be extended in order to obtain improved approximations to the underlying equations in one dimension and a detailed multidimensional analysis will be developed.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.