Stability of Traveling Waves

Brigham Young University, Provo UT

Investigators

Abstract

Humpherys DMS-0607721 This work focuses on the stability theory of traveling waves, with an emphasis on front propagation arising in the continuum and kinetic theories of compressible flow. This class of problems models key real-world phenomena such as shock waves in a viscous gas or plasma, detonations in a reactive gas, and the propagation of phase boundaries in a viscous fluid. The investigator studies long-standing open questions about the stability of traveling waves in these areas by exploring the effects of key structural features, such as symmetrizability and genuine coupling. Through an exhaustive numerical study, which includes Evans function computation, he seeks a better understanding of the stability properties of traveling waves in these important problem areas, particularly in regimes where analytical methods currently provide little information. In connection with recent work with his collaborators, the investigator develops better algorithms for Evans function computation. An additional benefit of this work is the continued development of the investigator's freely available numerical Evans function toolbox, which also allows for the exploration of systems beyond our immediate interest. Moreover, the investigator uses this package as an investigative tool in student-focused undergraduate research. Traveling waves are ubiquitous in nature, occurring everywhere from population models in ecology to the propagation of tsunamis in the oceanic sciences, or from shock waves of a supersonic jet to the flickering pulses of light in a fiber optic cable. The stability properties of these traveling waves describe the degree to which they can persist in the presence of disturbances. This work focuses on the mathematics of traveling waves and develops computational methods for exploring the stability properties of waves in a myriad of areas in the pure and applied sciences.