The aim of the proposed work is to consider a wide array of challenging problems within the field of nonlinear dispersive equations. Broadly speaking, the goal is to improve our understanding of nonlinear interactions in the context of wave-like phenomena. Equations of interest include wave equations, the Schroedinger equations and KdV type equations, as well as physically relevant coupled systems such as Einstein's equations, water waves, Maxwell systems and the Chern-Simons-Schroedinger model.
While most of the work will be concerned with nonlinear problems, one cannot obtain good nonlinear results without having a good understanding of the underlying linear dispersive dynamics. The two main themes here are (i) boundary value problems and (ii) global in time dispersion on curved backgrounds. The latter topic is mainly oriented toward systems, e.g. the Maxwell system and the spin two wave equation; this is related to the study of the conjectured stability of the Schwarzchild and Kerr black holes in the context of the Einstein equations in general relativity.
On the nonlinear side, a key topic to be explored is that of large data solutions in energy critical semilinear dispersive equations. The first stage involves understanding and classifying possible obstructions (e.g. solitons, steady states, self-similar solutions) to global scattering solutions. In the absence of such obstructions the goal is to obtain global well-posedness and scattering. Another goal is to achieve a good understanding of the dynamics near these obstructions.
The study of short and long term dynamics in quasilinear dispersive equations is another main theme of this proposal. There are several models of interest, namely nonlinear wave equations (in particular Einstein's equations), nonlinear Schroedinger equations as well as some water wave models.
Although the proposed work is primarily theoretical, many of the problems have their origin in physical theories such as general relativity, many body quantum field theory, surface wave propagation and plasma physics. As such, it is hoped that the results of the research may shed some light on the corresponding physical phenomena. Two good examples in this directions are the black hole stability question in general relativity, as well as the long range propagation of water waves.
On the human resource side, the project involves a good number of graduate students (six at present), as well as postdocs (five for the first year of the project, including three NSF postdocs). It also involves many collaborators at various institutions, both domestic and abroad. The results of this research will be made widely available via publications, web based tools, presentations at conferences and summer schools.
