Methods of Hamiltonian Mechanics for Nonlinear Wave Equations

Brown University, Providence RI

Investigators

Abstract

The research described in this proposal is on a class of linear and nonlinear partial differential equations (PDE)which share the property that they can be written in the form of a Hamiltonian system with infinitely many degrees of freedom. Principal examples include nonlinear wave equations, nonlinear Schroedinger equations and Euler's equations for water waves. The analogy with dynamical systems motivates a number of basic questions. (1) The existence of invariant tori for the nonlinear partial differential equations involves extensions of KAM theory to infinite dimensional settings. Implications of these results include the existence of solutions of nonlinear evolution equations which exist for all time and are recurrent. Techniques that have been developed for PDE also have bearing on several open problems in classical dynamical systems concerning the persistence of resonant tori. (2) A normal forms transformation of a Hamiltonian PDE gives details of its essential nonlinearities, and it has applications to Nekhoroshev stability results and to a proof of the Arnold condition of genuine nonlinearity. (3) Starting from a normal forms transformation, long time existence results for nonlinear PDE are important in a mathematically rigorous analysis of the classical asymptotic regimes of mathematical physics. In addition the proposer will address a number of related questions in the theory of PDE. (4) Estimates of dispersive smoothing for linear and nonlinear Schroeinger equations depend upon the scattering properties of bicharacteristics, and estimates of this form play a role in the study of the initial value problem for data of low smoothness. (5) The classical problems of standing waves in two dimensional water waves and traveling waves in three dimensional water waves are instances of problems in PDE which exhibit small divisors, and analytic results involve delicate questions of convergence. Celestial mechanics, the problem of describing the motion of planetary bodies, has always intrigued mathematicians and motivated their discoveries. At the beginning of this century, the French mathematician H. Poincar' and his American contemporaries G.D. Birkhoff and G.W. Hill, essentially revolutionized the subject with their work, bringing it conceptual clarity and introducing powerful new analytic techniques. This work gave rise to a new paradigm in its field, and contributed to the birth of many modern branches of mathematics. An important question was and remains the n-body problem; the problem of stability of the motion of n planets in their mutual gravitational attraction. Indeed in the 1960's, 50 years after the death of Poincar'e, the development of a theory by three mathematicians, A.N. Kolmogorov, V.I. Arnold and J. Moser (it is now called KAM theory) made a significant contribution to the field. At the time of its development there were physical applications to the stability of high energy particles in a cyclotron, as well as to the classical questions of the stability of our solar system or to other important systems in mechanics. The equations which describe the motion of n bodies are examples of ordinary differential equations. It was a question at the time of the development of KAM theory whether such stability results could be extended to partial differential equations. These describe the evolution in time of a continuous medium such as a fluid, a gas, and electromagnetic field or an elastic solid. The analogy between mechanics and the motion of a continuum is mathematically quite elegant. However the n-body problem is a finite dimensional one, while problems of partial differential equations viewed with this analogy in mind are inherently infinite dimensional, and this summarizes the essential mathematical difficulty of extending the methods. The principal content of this research program is the extension of some aspects of the analytic techniques of Hamiltonian mechanics to physically important partial differential equations. These results will have some significant consequences in a variety of diverse physical applications, including the study of ocean waves, the propagation of signal pulses in optical fibers, and in the