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Understanding the FPU state in FPU–like models

Università degli Studi di Padova, Dipartimento di Matematica “Tullio Levi-Civita”, Via Trieste 63, 35121 Padova, Italy

This contribution is part of the Special Issue: Modern methods in Hamiltonian perturbation theory
Guest Editors: Marco Sansottera; Ugo Locatelli
Link: www.aimspress.com/mine/article/5514/special-articles

Special Issues: Modern methods in Hamiltonian perturbation theory

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.
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Keywords FPU problem; Toda model; integrability; scaling laws; Toda actions

Citation: Giancarlo Benettin, Antonio Ponno. Understanding the FPU state in FPU–like models. Mathematics in Engineering, 2021, 3(3): 1-22. doi: 10.3934/mine.2021025


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