Nonlinear wave propagation

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NSF Award Abstract - DMS-0072343Mathematical Sciences: Nonlinear Wave PropagationAbstract0072343 HunterThis project studies a number of problems in nonlinear wave propagation. The first problem is the reflection of weak shock waves, where there have been longstanding discrepancies between theory and experiment. There are close relationships between this problem and transonic aerodynamics. The second problem is the study of nonlinear effects on gravitational waves in the Einstein field equations of general relativity, an issue of fundamental scientific significance. The third problem is the study of nonlocal, nonlinear equations for hyperbolic surface waves in elasticity and magnetohydrodynamics. Such waves arise, for example, in surface acoustic wave devices used in signal processing. The fourth problem is the study of the interaction of high frequency vorticity waves and mean flows in incompressible fluids. This research will describe the nonlinear development of vorticity instabilities, and is relevant to the closure problem for turbulent flows. The fifth problem is the propagation of fronts in a bistable oscillatory system of reaction-diffusion equation that provides a simplified model of the collective behavior of beta-cells in the pancreas, which produce insulin. The sixth problem is the study of surfactant deposition by a spreading liquid drop. This problem has industrial applications in the use of droplets for the deposition of surface films. Waves, such as sound waves, elastic waves, and gravitational waves, are an important feature of many physical and biological systems. Large amplitude waves may behave nonlinearly, and this leads to new effects not seen in linear waves: for example, the generation of shock waves by an aircraft traveling at speeds close to or above the speed of sound. A detailed analysis of the equations that describe nonlinear waves is often very difficult. The aim of this research is an increased understanding of nonlinear waves in the context of a variety of applications that involve qualitatively interesting and poorly understood phenomena, and that are related to problems of current scientific and technological interest.