My research interest is the analysis of the partial differential equations which describe wave propagtion, and most often equations of hyperbolic type. This involves the creation of general tools and also the the study of specific problems with origin in science, for example from continuum mechanics, quantum mechanics, acoustics and optics.

There have been several long term projects at the center of my research efforts. During the seventies and early eighties, I studied the decay, scattering, observaton and control of solutions of linear hyperbolic equations, often in collaboration with M. Taylor, C. Bardos, and G. Lebeau. Then with M. Reed I studied the propagation and interaction of singularities for nonlinear hyperbolic equations. Now, with J.-L. Joly and G. Metivier we have been studying the propagation and interaction of high frequency solutions of nonlinear hyperbolic equations, a subject which goes under the name of nonlinear geometric optics. Two key words are envelope or profile equations and resonance. The familiar Burgers, Schrodinger, and KdV Equations owe their importance to the fact that they are profile equations for a wide range of different fundamental problems. We are actively studying several interesting applications to fluid mechanics and nonlinear optics (think lasers). The goal of the subject is to find approximate solutions of nonlinear wave equations whose error tends to zero as the wavelength tends to zero. This short wavelength limit is both important in applications and inaccessible by direct numerical simulation. The approximations are obtained by solving simpler equations than those at the outset, for example the profile equations mentioned above. In some cases one arrives at approximations familiar in the applied literature, where the derivations are heuristic. The qualitative and quantitative behavior of the approximate solutions give direct access to the more complicated equations in the background. Often one can go further than the heuristics in the applied literature, describing new phenomena. An example is our construction of solutions with an infinity of wave trains. There are cases where the familiar derivations do NOT give good approximations. There are also applied problems where the heuristics are inconclusive, for example for optical parametric resonators. An introduction to these ideas is presented in my series of lectures at the Park City-IAS summer school in 1995.

All of the areas have the property that the analysis behind them owes a lot to the ideas and methods of microlocal analysis.