This proposal is comprised of six otherwise disparate projects with the unifying theme of investigating how collective large-scale behavior arises in multi-scale dynamical systems. The projects are: Zero Dispersion Limits of Nonlinear Wave Equations, Macroscopic Lattice Dynamics, the NLS Limit of the CGL Equation, Moment Closure Hierarchies for Kinetic Equations, Numerical Schemes for Hyperbolic Systems, and Shallow- Water Models. From a physical perspective this proposed effort could be viewed as a study of nonequilibrium statistical mechanics. But it is more than that, for it strives to discover and study new nonlinear phenomena that could impact the way physical, geophysical, and biological models are developed and simulated.
New technologies are entering regimes in which traditional mathematical models are inadequate. For example, quantum mechanical effects play a central role in the emerging nanoscale semiconductor technology. These effects are absent from traditional device models. Including them often leads to models with "small dispersion." Such models exhibit large-scale behaviors that are not understood and are hard to simulate because they result from collective smaller-scale phenomena. Current mathematics is not up to the task of describing or simulating such things adequately. The first and second projects of this proposal address the development of mathematics that can do the job. The other projects of the proposal address the need for other mathematical developments that would impact nanoscale semiconductor design, hypersonic flight, epidemiological models, ocean circulation models, and other technologies.
