Soliton theory and the Fermi-Pasta-Ulam problem in the thermodynamic limit

Università degli Studi di Milano, Dipartimento di Matematica “F. Enriques” Via Saldini 50, 20133 Milano, Italy

Corresponding author: ponno@mat.unimi.it

Received: 22 May 2003
Accepted: 17 September 2003

Abstract

We reconsider the Fermi-Pasta-Ulam problem from the point of view of soliton theory along the lines of the original work of Zabusky and Kruskal, but with attention to the thermodynamic limit. For a special class of long-wavelength initial data, we show that, in such a limit, the modal energy spectrum is determined by the solitons, and is given by an explicit analytic expression in terms of the specific energy, which does not correspond to equipartition of energy. This is shown to occur for specific energies below a certain nonvanishing threshold, within a time interval that also depends on the specific energy. A short discussion concerning the case of generic long-wavelength initial data is also given.