Nonlinear Wave Interactions & Singular Behavior

点击翻译按钮获取中文摘要

9501073 Tabak The proposed work concentrates on open problems in wave propagation arising in the natural sciences. The first problem addresses the diffraction of weak shock waves, where an apparent contradiction, known as the von Neumann paradox, has been puzzling physicists and mathematicians for almost five decades. The paradox concerns the ubiquitous occurrence of three-shock intersections in a regime where, in principle, they are not allowed. It is proposed to study this problem within the context of a simplified asymptotic model. The results so far suggest that the solution to the paradox may involve an additional singularity arising at the triple-point. The second problem regards the long-scale behavior of winds, concentrating on the formation of fronts between warm and cold air masses in the atmosphere. The mathematical structure of this problem is surprisingly similar to the one associated with the question of formation of singularities in three-dimensional incompressible fluids. Both problems involve a combination of physical modeling, asymptotic analysis and analytical and numerical solution of partial differential equations. Both proposed problems have practical as well as theoretical significance. Shock diffraction occurs in a variety of scenarios, such as supersonic flight, flood waves and even medical applications, when focusing sound waves are used to destroy kidney stones. A better understanding of the diffraction process would therefore impact not only our knowledge of waves in nature, but also our capacity for predicting wave intensity amplification due to focusing and reflection in engineering devices. The formation of sharp fronts between air masses in the atmosphere, on the other hand, is an important and ubiquitous phenomenon. Understanding its mechanics would enhance our ability to predict weather and climate changes. The surprising relationship between this problem and the question of singularity for mation in incompressible fluids, on the other hand, provides a promising alternative way to study a mechanism which may play a fundamental role in turbulent intermittency.