PhD, Physics, Harvard University, 1982
|Department of Mathematics
111 Cummington Street
Boston MA 02215
My research concentrates on the qualitative understanding of solutions of partial differential equations, particularly those arising in physical or biological contexts. Among my present projects are the following.
Long-time asymptotics of dissipative partial differential equations
Marrying analysis, geometry and topology in the way pioneered by Poincare has proven to be an extraordinarily fruitful way of understanding the behavior of ordinary differential equations. To date, the corresponding study of partial differential equations is far less advanced. In such systems it is often not obvious what the principal modes and mechanisms governing the stability or instability of solutions are. One of the problems is that partial differential equations describe systems which in some sense have infinitely many degrees of freedom. A main thrust of my research is to identify invariant, finite-dimensional objects in the phase space of these systems which govern the long-time behavior of solutions. Among the specific examples I have studied are dissipative partial differential equations like the Cahn-Hilliard equation which is a model of the crystallization process, or systems of reaction-diffusion equations which describe both chemical reactions and the growth of extended biological systems. I am also using these techniques to study the propagation and stability of pulses in models of neural systems.
Validity of modulation and amplitude equations
It is experimentally observed that disturbances of the ocean surface generate large, solitary waves which travel enormous distances over the sea surface without change of form. It is impossible to solve the equations which describe such waves exactly, and while a number of model equations such as the Korteweg-de Vries or Boussinesq equations which describe such traveling waves exist, the mathematical justification of such models is still largely lacking. I am currently trying to understand the circumstances which guarantee the validity of these simple approximating equations. Another interesting question which arises is that many of the approximating equations that are derived have the property of complete integrability. This means, for instance, that when two waves of the approximate equation meet, they pass through each other, emerging from the collision with no change in shape. The full water wave equations do not have this property, and I am studying how the complete integrability of the approximating equations manifests itself in the original water wave problem.
Small denominator problems in partial differential equations and other infinite dimensional dynamical systems
Resonances between two or more oscillators are a frequent occurrence in finite dimensional systems. Their appearance can drastically alter the motion of the system, sometimes leading to uncontrolled and destructive oscillations. In celestial mechanics, they were encountered very early in the attempts to use Newton's mechanics to calculate the motions of the planets, where they lead to a phenomenon known as ``small denominators''. These problems were not overcome until the work of Kolmogorov, Arnold, and Moser (KAM) in the 1950's and 1960's. While clarifying the situation for problems with finitely many degrees of freedom, the KAM theory did not apply to problems with infinitely many degrees of freedom, like nonlinear partial differential equations or infinite networks of coupled oscillators. I have recently been interested in the circumstances under which the KAM theory can be extended to such situations and what its implications for these systems is. This study has lead to some interesting and unexpected connections with the quantum mechanical behavior of random media, and the distribution of values of quadratic forms on integer lattices.
Further information is available at the CBD ongoing research pages.
Newton's Method and Periodic Solutions of Nonlinear Wave Equations. (with W. Craig) Comm. Pure and Appl. Math. vol 46, 1409-1498, (1993).
Stability of Front Solutions for Parabolic Partial Differential Equations. (with J.-P. Eckmann) Comm. Math. Phys. vol 161, 323-334, (1994).
The long wave limit for the water wave problem I. The case of zero surface tension. (with G. Schneider) to appear in Comm. Pure Appl. Math.
An Introduction to KAM Theory. Lectures in Applied Mathematics vol 31, 3-29, (1996).
Invariant manifolds for parabolic partial differential equations on unbounded domains. Arch. Rat. Mech. Anal. vol 138, 279-306, (1997).
Periodic solutions of nonlinear partial differential equations. Notices of the AMS, vol. 44, 895-901 (1997).