Periodic Solutions of Nonlinear Partial Differential Equations

As the existence theory for solutions of nonlinear partial differential equations becomes better understood, one can begin to ask more detailed questions about the behavior of solutions of such equations. Given the bewildering complexity which can arise from relatively simple systems of ordinary differential equations, it is hopeless to try to describe fully the behavior which might arise from a nonlinear partial differential equation. Thus it makes sense to first consider special solutions, in the hope that through a more concrete understanding of them one may gain insight into the behavior of more general solutions. An extremely fruitful avenue of study in the theory of ordinary differential equations has been the construction of periodic orbits: in many circumstances they form a sort of skeleton on which more complicated solutions can be built. It was Poincaré who first realized this possibility, a discovery which prompted the remark quoted above. By a careful analysis of the periodic solutions that occur in the celestial mechanics problem of three gravitationally interacting planets and of the solutions asymptotic to these periodic orbits, he proved the existence of “chaotic” orbits in this system. For the past thirty years or so there has been an active search for periodic solutions of partial differential equations, employing a variety of methods and motivated, at least in part, by the important role that periodic solutions play in understanding the behavior of ordinary differential equations. My goal in what follows is to describe a new technique for constructing such solutions which both highlights the differences between ordinary and partial differential equations and which also exhibits a surprising connection with problems in quantum mechanics. However, contrary to what Poincaré’s quotation might suggest, these periodic solutions are not only of theoretical interest but also have many practical applications. As far as I am aware, the first study of periodic solutions of a nonlinear partial differential equation was in the early 1930s in the work of Vitt ([22], described in [13]), who considered these solutions in the context of problems of electrical transmission. Additional research was carried out in the ‘30s and ‘40s, often by physicists; not until the 1960s did mathematicians begin a fairly intensive study of the existence and properties of periodic solutions. (See, for example, [14, 21].) One problem that aroused particular interest was the structure of periodic standing waves on the surface of an inviscid, irrotational fluid. (See [21] and [6].) In particular, Paul Concus [7] pointed out the difference between the existence of periodic solutions for systems of ordinary differential equations and C. Eugene Wayne is professor of mathematics at The Pennsylvania State University. His e-mail address is [email protected]. After September 1, 1997, he will be professor of mathematics at Boston University.