The β Fermi-Pasta-Ulam-Tsingou recurrence problem

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.