The main objective of this project is to explore factors that vitally influence the long-time behavior of nonlinear dispersive PDE like the non-linear Schroedinger (NLS), nonlinear wave (NLW), nonlinear Klein-Gordon (NKG), and Zakharov systems. First, the effect of domain geometry is investigated in terms of a relation between the volume growth of the underlying manifold and the decay of linear solutions. The implications of this relation for the nonlinear problem are then explored in certain tractable geometries, namely quotients of Euclidean spaces. Second, the question of small-data scattering (asymptotically linear behavior) for some weakly nonlinear dispersive equations is considered. These are equations where the nonlinearity is not of high enough degree for classical methods to succeed. One focus is on inhomogeneities represented by a potential term, and how this can affect the asymptotic behavior. Finally, the so-called weakly turbulent regime is investigated by studying the long-time behavior of the cubic NLS equation on a compact domain. In this context, a new continuum equation is derived for the envelope of the discrete Fourier modes by taking a suitable large-box limit in the spirit of weak turbulence theory. This equation turns out to enjoy remarkable symmetries and even explicit solutions, and can be used (via rigorous approximation results) to better understand the long-time frequency dynamics of the original NLS system.
The nonlinear dispersive equations considered in this project arise naturally in several areas of physics (plasma physics, nonlinear optics, general relativity, etc.) where they often serve as simplified effective models. In order for these models to be useful for engineering purposes as well, one has to understand the effect of inhomogeneities (usually modeled by a non-Euclidean domain geometry or a potential term) in making connections with the real world. Such a study would help set the parameters and limitations of not only the validity of the existing simplified models, but also the faithfulness of computer simulations (based on those models) in predicting experimental phenomena. On a more theoretical note, starting a mathematically rigorous study of the all-important physical theory of weak turbulence, as suggested in the proposal, represents a cross-disciplinary collaboration between pure mathematics, applied mathematics, and physics, with rewarding applications in plasma and fluid engineering, as well as ocean and atmospheric science.
