Toward a Topological Classification of Integrable PDE’s

Abstract

We model Fomenko’s topological classification of 2 degree of freedom integrable stratifications in an infinite dimensional soliton system. Specifically, the analyticity of the Floquet discriminant Δ(q, ») in both of its arguments provides a transparent realization of a Bott function and of the remaining building blocks of the stratification; in this manner, Fomenko’s structure theorems are expressed through the inverse spectral transform. Thus, soliton equations are shown to provide natural representatives of the classification in the context of PDE’s.

Support is gratefully acknowledged from the National Science Foundation under grant # DNS8703397 and from the United States Air Force under grant #AFOSRU F49620-86-C0130. We wish to take this opportunity to thank R. Devaney, H. Flaschka, K. Meyer, and T. Ratiu for organizing the workshop at the Mathematical Sciences Research Institute where we first learned of Professor Fomenko’s work through his lectures at that workshop. Finally, one of us (DWM) acknowledges several extremely useful conversations with P. Deift.