A Study of Wave Patterns

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

It is familiar to everyone that waves typically organize themselves into certain identifiable patterns. For instance, a stone dropped into a pond produces concentric rings of ripples, and the wake of a ship has a characteristic V-shape. Although they may be harder to see in everyday experience, other types of waves like electromagnetic waves frequently also produce such patterns. This means that wave patterns might be understood from the mathematical study of certain underlying models of wave motion that are common to several different types of physical systems. In a similar way, understanding wave patterns from first principles is a worthwhile pursuit because such patterns occur frequently in so many physical circumstances, ranging from the very small (e.g. optics or quantum waves on the scale of atoms) to the very large (e.g. gravitational waves on the scale of the galaxy). This project is a systematic investigation of wave patterns arising in mathematical models of wave propagation, and it aims both to catalogue the various wave patterns that can occur in a given model and to determine the range of models and physical situations in which a given pattern can arise. As one example application, the outcomes of this project will contribute to our understanding of rogue waves, large disturbances of the sea surface that appear out of nowhere and disappear just as suddenly, but which can cause great damage to ships, unless they are anticipated and avoided or otherwise mitigated. Some parts of the project will serve as vehicles for the training of junior researchers such as undergraduate students, graduate students, and postdocs. This project aims to develop and apply methods from the theory of completely integrable systems to the study of wave patterns. The key mathematical methods are tools of complex, functional, and asymptotic analysis, and the results of the research will be relevant to the fields of nonlinear wave propagation, integrable systems, and special functions. Specific models to be studied include the three-wave resonant interaction equations, the focusing nonlinear Schroedinger equation, the Davey-Stewartson equations, and the family of Painleve equations. The project will study both regular wave patterns, i.e., modulated wave trains spontaneously generated from smooth initial conditions, and universal wave patterns, i.e., special structures appearing in double-scaling limits that do not depend on initial conditions (and perhaps not even on the equation of motion). Specific goals include the development of new techniques for the investigation of wave patterns via asymptotic analysis in integrable systems consisting of multiple coupled fields or in multiple space dimensions. 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.