Local and global dynamics for nonlinear dispersive equations

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The proposed work is devoted to a broad range of problems within the field of nonlinear dispersive equations. The primary goal is to achieve a good understanding of the interplay between linear dispersive dynamics and nonlinear effects. On the linear side, the project will study the global-in-time dispersive properties for wave and Schroedinger equations evolving on curved backgrounds. This is also in part directed toward the conjectured stability of the Schwarzchild and Kerr black hole solutions of the Einstein equations in general relativity. On the nonlinear side, a key topic to be explored is that of multiscale analysis. This corresponds to nonlinear dispersive equations that evolve in regimes where linear and nonlinear effects are strongly interlaced. Thus while one might see coherent linear dynamics on a very short, possibly frequency dependent time scale, these are lost at larger times. The challenge is then to uncover larger patterns that remain coherent on longer and longer time scales.Broadly speaking, dispersive equations are equations whose solutions can be thought of as superpositions of waves travelling in different directions. Nonlinear effects allow these waves to interact, which may lead to modified propagation patterns (creating new waves) or possibly to the phenomenon described by mathematicians as "blow-up." Many of the equations considered in this project have their origins in physical theories such as general relativity, many-body quantum field theory, surface wave propagation, and plasma physics. For this reason, it is hoped that the results of the research may shed some light on the corresponding physical phenomena. The project involves graduate students and postdocs, as well as several domestic and international collaborators.