Understanding the FPU state in FPU-like models. (English) Zbl 1508.37068
Summary: Many papers investigated, in a variety of ways, the so-called “FPU state” in the Fermi-Pasta-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.
MSC:
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 70H06 | Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics |
| 70K75 | Nonlinear modes |
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