Internal deep-water waves, ion-acoustic waves in a plasma, laser propagation in highly refractive materials, and collective particle behavior in very low temperature gases are all physical systems whose behavior in time is modeled by a nonlinear dispersive wave equation. The term "nonlinear" refers to a property of size-dependent response and the term "dispersive" refers to how the fluctuations of the wave influence the speed and direction of motion. At the mathematical level, one studies the behavior of general solutions and special types of solutions to these equations and seeks to provide quantitative descriptions of phenomena observed in the physical setting. In this project, the investigators explore how coherent waves travel - whether they retain their shape despite encountering obstacles, break apart and dissolve, or collapse into a singularity. Each of these possibilities hinges on both the nonlinear and the dispersive character of the equations; over the past several decades, mathematical techniques have been developed to model and measure these properties. The project aims to improve existing methods and apply the methods in new directions. The project contains educational efforts at various levels of mathematical learning. These include advising undergraduate and graduate students as well as postdoctoral scholars, with special emphasis on attracting underrepresented minorities.
The main objective of the research will be to provide analytical descriptions of the behavior of solutions to certain classes of nonlinear dispersive equations. The two main categories of equations considered are the Korteweg-de Vries (KdV) family and the nonlinear Schrödinger (NLS) family. Both classes of equations satisfy powerful dispersive estimates called local virial estimates, and the KdV family in addition satisfies a monotonicity property that controls the movement of mass. The multiple-scale method, spectral analysis, application of dispersive estimates, and monotonicity bounds are core methods that will be utilized and extended. Several focus problems are identified in which some classical feature, like scale-invariance or a property of localized influence, have been removed or weakened, providing the stimulus to develop new techniques, while at the same time, provide new descriptions of real physical phenomena. The phenomena of primary interest are the dynamics of coherent structures like solitary waves and line solitons as they interact with each other and their environment, and the description of how singular collapsing solutions arise and their asymptotic description.
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.