Quasi-stable Localized Excitations in the \beta-Fermi Pasta Ulam Tsingou System

TL;DR

This paper investigates the stability and lifetime of localized nonlinear excitations in the $eta$-FPUT system, revealing how nonlinear interactions and secondary frequencies influence their longevity.

Contribution

It provides a detailed analysis of LNE stability in the $eta$-FPUT system, including parameter dependencies and the effects of secondary frequencies on their lifetimes.

Findings

LNE lifetimes are significantly affected by secondary frequencies within the phonon band.

Longest LNEs occur when $ ext{A}\sqrt{eta} oughly 1.1$.

LNE lifetime in the cubic FPUT lattice follows a double log relationship with amplitude and nonlinearity.

Abstract

The lifetimes of localized nonlinear modes in both the $\beta$-Fermi-Pasta-Ulam-Tsingou ($\beta$-FPUT) chain and a cubic $\beta$-FPUT lattice are studied as functions of perturbation amplitude, and by extension, the relative strength of the nonlinear interactions compared to the linear part. We first recover the well known result that localized nonlinear excitations (LNEs) produced by a bond squeeze can be reduced to an approximate two-frequency solution and then show that the nonlinear term in the potential can lead to the production of secondary frequencies within the phonon band. This can affect the stability and lifetime of the LNE by facilitating interactions between the LNE and a low energy acoustic background which can be regarded as "noise" in the system. In the one dimensional FPUT chain, the LNE is stabilized by low energy acoustic emissions at early times; in some cases allowing for lifetimes several orders of magnitude larger than the oscillation period. The longest lived LNEs are found to satisfy the parameter dependence $\mathcal{A}\sqrt{\beta}\approx1.1$ where $\beta$ is the relative nonlinear strength and $\mathcal{A}$ is the displacement amplitude of the center particles in the LNE. In the cubic FPUT lattice, the LNE lifetime $T$ decreases rapidly with increasing amplitude $\mathcal{A}$ and is well described by the double log relationship $\log_{10}\log_{10}(T)\approx -(0.15\pm0.01)\mathcal{A}\sqrt{\beta}+(0.62\pm0.02)$.