Dispersive Phenomena in Linear and Nonlinear Partial Differential Equations

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The primary aim of the proposed work is to investigate the role of dispersive phenomena in partial differential equations. While there certainly is some interest in studying dispersive phenomena for linear partial differential equations, the main motivation comes from nonlinear partial differential equations. Indeed, nonlinear effects (which can possibly lead to blowup) are stronger in regions of spatial concentration of the solutions. Hence the dispersion reduces the potential for blow-up. If nevertheless blow-up occurs, its pattern should largely be determined by the worst type of concentration allowed by the dispersion. On the other hand, nonlinear interactions can affect the dispersion. Thus one is led from the study of linear dispersion to bilinear estimates and further to the analysis of fully nonlinear interactions. In recent years this line of attack has proved to be highly successful in the study of nonlinear dispersive equations. Yet much more remains to bedone, and one has the feeling that we have only seen the tip of the iceberg. A simple way to describe dispersion is to say that waves (e.g. sound waves, elastic waves, water waves, electromagnetic waves, etc) cannot stay spatially concentrated for a long period of time; instead they must spread out and decay. In linear problems different waves cross each other without interaction. However, in nonlinear phenomena waves will always interact. The strength of this interaction depends on the strength of each wave but also on their intersection pattern. These nonlinear interactions play an essential role in both the study of the short time behavior and of the long time behavior of various physical systems. Examples include elastic waves in solids, gravitational waves in general relativity, and many others. The goal of the proposed work is to contribute to the understanding of the dynamics of nonlinear wave interactions in the context of physically motivated dispersive systems.