Understanding the FPU state in FPU-like models

Many papers investigated, in a variety of ways, the so-called FPU state in the FermiPasta-Ulam model, namely the state, intermediate between the initial state and equipartition, that the system soon reaches if initially one or a few long-wavelength normal modes are excited. The FPU state has been observed, in particular, to obey a few characterizing scalings laws. The aim of this paper is twofold: First, reviewing and commenting the literature on the FPU state, suggesting a possible way to organize it. Second, contributing to a better understanding of the FPU state by studying the similar state in the Toda model, which provides, as is known, the closest integrable approximation to FPU. As a new tool, we analyze the dimensionality of Toda invariant tori in states corresponding to the FPU state, and observe it obeys the main scaling law characterizing the FPU state.

Automated summary

This paper investigates the FPU state in the FermiPasta-Ulam model and a similar state in the Toda model, by analyzing the dimensionality of Toda invariant tori and observing the main scaling law characterizing the FPU state. The aim is to better understand the FPU state and propose a way to organize the literature on this topic.