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Behavior and breakdown of higher-order Fermi-Pasta-Ulam-Tsingou recurrences.

Chaos (Woodbury, N.Y.)·2019

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The β Fermi-Pasta-Ulam-Tsingou recurrence problem.

Salvatore D Pace¹, Kevin A Reiss¹, David K Campbell¹

¹Department of Physics, Boston University, Boston, Massachusetts 02215, USA.

Chaos (Woodbury, N.Y.)

|November 30, 2019

Summary

The Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence time in the β-FPUT chain depends on energy and nonlinearity. For large systems, recurrence time scales with energy and nonlinearity, explained by soliton interactions.

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Area of Science:

Nonlinear dynamics
Condensed matter physics
Computational physics

Background:

The Fermi-Pasta-Ulam-Tsingou (FPUT) problem investigates energy recurrence in nonlinear systems.
The β-FPUT model introduces a specific type of nonlinearity crucial for understanding energy dynamics.

Purpose of the Study:

To thoroughly investigate the first FPUT recurrence in the β-FPUT chain for both positive and negative nonlinearity parameters (β).
To determine the scaling laws and dependencies of the recurrence time (T_r) on system parameters like energy (E), nonlinearity (β), and system size (N).

Main Methods:

Numerical simulations were performed on the β-FPUT chain for various system sizes (N) and energy levels (E).
Analytical methods, including shifted-frequency perturbation theory, were employed to study the nearly linear and highly nonlinear regimes.
The continuum limit was studied to connect discrete system behavior to wave phenomena.

Main Results:

The rescaled recurrence time (T_r) depends only on the parameter S ≡ Eβ(N+1) for large N.
For small |S|, T_r is linear in S; for large |S|, T_r is proportional to |S|^-1/2, with different constants for positive and negative β.
The |S|^-1/2 scaling in the continuum limit is explained by soliton theory, with differences attributed to soliton-kink interactions in the negative β case.
The disappearance of FPUT recurrences depends only on Eβ for large N.

Conclusions:

The study provides a comprehensive understanding of FPUT recurrence in the β-FPUT model, revealing universal scaling laws.
Soliton dynamics play a key role in explaining the observed recurrence times and their dependence on nonlinearity.
Significant differences in energy mixing between positive and negative β highlight the importance of nonlinearity sign in FPUT systems.