A spatially extended system is said to be dispersive when the speed of propagation of a wave depends upon that wave's frequency. Linear dispersive effects play a fundamental role in the study of a large number of physical scenarios and there has been an explosion of results concerning semi-linear dispersive equations. Nevertheless there are situations in which the speed of propagation of a wave depends on the wave's frequency as well as its amplitude. That is to say the mechanism which generates dispersion is nonlinear. It is the purpose of this project to develop mathematically rigorous theory for such equations when the nonlinear dispersive effects are degenerate (i.e., they vanish very rapidly as the amplitude tends to zero). The degeneracy of the equations allows for the existence of classical solutions which are not smooth and there is substantial numerical and formal evidence that cases arise in which degenerate dispersion leads to catastrophic instability akin to that of a backwards heat equation. Consequently, one of the principal goals of the project is to develop the existence theory for degenerate dispersive partial differential equations.
Degenerate dispersive equations arise as models for the dynamics of chains and lattices of hard spheres. Experimental results demonstrate that such chains exhibit tightly focused pulses which are easily generated and highly tunable. These narrow pulses are expected to find uses in non-destructive testing, shock absorption, remote sensing and medical imaging. At this time, there is no theoretically rigorous explanation for the stability and robust nature of these pulses. Though there are results concerning similar physical problems where the dispersive effects are not degenerate, they rely strongly on the fact that slow moving pulses are very wide and very small in amplitude. In the case of hard spheres, the interesting pulse solutions have a fixed width, independent of speed and amplitude and it is not obvious how to extend the known results to this case. A primary goal of the project is developing mathematical tools for the study of the stability and robustness of such pulses. The inclusion and training of graduate and undergraduate students is an integral part of this project. Among other projects, undergraduate students will be involved into investigation of an important application, models for traffic flow.
