Nonlinear Wave Interactions

Wave interactions are ubiquitous in nature, the laboratory, and technology. While much of the mathematics and science of wave interactions relies upon a basic property of small amplitude waves, known as the law of linear superposition, in which multiple wave shapes add together, in practice many waves are of large amplitude and do not conform to this linear behavior. Some examples of nonlinear waves include surface and internal gravity waves in the near-shore environment, intense laser light in laboratory fiber optic experiments, and quantum mechanical matter waves in experiments with ultracold atomic gases. From multidimensional interacting nonlinear waves to the interaction of nonlinear periodic waves and solitary waves, the investigator will study the mathematics and physics of nonlinear wave interactions using physical experiments and an approximation technique known as modulation theory. This interdisciplinary research will provide a fundamental understanding of nonlinear wave interactions with applications to fluid dynamics, nonlinear optics, and quantum fluids. The project will provide research training opportunities for undergraduate and graduate students. The investigator will mathematically and experimentally study nonlinear wave interactions. This will be achieved by constructing exact and numerical soliton-cnoidal (periodic traveling wave) breather solutions, solving generalized Riemann problems for colliding cnoidal waves, obtaining modulation equations and solutions for multidimensional and forced nonlinear waves, and performing fluid experiments. Methods to be employed include Whitham averaging, integrable system methods (Darboux transformations), conservation law methods, multiscale asymptotics, numerical computation, and experiments in viscous core-annular flows. To describe the complex interactions of solitons and cnoidal waves, the approach unites techniques from hyperbolic conservation laws and nonlinear dispersive waves. The equations to be analyzed are models of geophysical and technological significance, ranging from completely integrable, e.g., Korteweg-de Vries and Kadomtsev-Petviashvili, to non-integrable, e.g., Ostrovsky, 2D Benjamin-Ono, and conduit equations. The proposed viscous core-annular experiments will provide quantitative tests of the modeling and theoretical predictions. The developed theory will contribute to the field of nonlinear waves and to the understanding of atmospheric and oceanic internal and surface wave propagation. 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.