Space-Time Resonances and Asymptotics; Stability of Self-Similar Solutions

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Nonlinear dispersive partial differential equations equations are a very wide class of equations that are used to describe varied phenomena, ranging from waves at the surface of water to plasma physics and quantum mechanics. The first aim of this project is to extend our understanding of nonlinear dispersive equations by using the method of space-time resonances, which was introduced recently by the principal investigator and his collaborators, Nader Masmoudi and Jalal Shatah. The method of space-time resonances combines the standard notion of resonances, well known since at least Poincare's time, with a study of the spatial localization of solutions. Its aim is to understand the stability of equilibria. It involves delicate multilinear harmonic analysis questions, which the principal investigator plans to study systematically. The second aim of this project is to deepen our understanding of self-similar blow-up. This is a mechanism for the formation of singularities in the solutions of nonlinear dispersive equations, a topic of fundamental importance in the study of such equations.So-called nonlinear dispersive partial differential equations are a class of equations describing extremely varied physical phenomena: surface water waves, the physics of charged fluids (plasmas), or quantum mechanical effects, to name but three. The range of applications is very broad: the physics of water waves includes the study of tsunamis; as for plasma physics, it is not only very important for our understanding of the universe but is also crucial for many industrial applications. The aim of the principal investigator is to deepen our understanding of this very general class of equations at a fundamental level. He intends to study two questions in particular. First, when are the equilibria of these equations stable? Second, how do these equations develop singularities or, stated differently, how can "smooth" solutions of these equations become increasingly "jagged"? Answers to these questions would, for instance, help understand how tsunami waves propagate or how laser beams focus.