Understanding the FPU state in FPU-like models

Giancarlo Benettin; Antonio Ponno; Giancarlo Benettin; Antonio Ponno

doi:10.3934/mine.2021025

Mathematics in Engineering

2021, Volume 3, Issue 3: 1-22. doi: 10.3934/mine.2021025

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Understanding the FPU state in FPU-like models

Giancarlo Benettin ^,,
Antonio Ponno

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

Received: 05 March 2020 Accepted: 24 May 2020 Published: 23 July 2020

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.
- FPU problem,
- Toda model,
- integrability,
- scaling laws,
- Toda actions
Citation: Giancarlo Benettin, Antonio Ponno. Understanding the FPU state in FPU-like models[J]. Mathematics in Engineering, 2021, 3(3): 1-22. doi: 10.3934/mine.2021025

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Abstract

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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