# Nonlinear normal modes in the $\beta$-Fermi-Pasta-Ulam-Tsingou chain

## TL;DR

This paper derives exact nonlinear normal mode solutions for the $eta$-FPUT chain, explores mode relaxation dynamics, and identifies different solution types including localized modes and chaotic mappings.

## Contribution

It provides explicit solutions for small and large chains with nonlinear couplings and analyzes mode decay using regularized regression techniques.

## Key findings

- Exact solutions for two-particle chain with arbitrary couplings
- Identification of localized nonlinear modes and chaotic amplitude mappings
- Mode decay follows a sigmoid and relates logarithmically to perturbation strength

## Abstract

Nonlinear normal mode solutions of the $\beta$-FPUT chain with fixed boundaries are presented in terms of the Jacobi sn function. Exact solutions for the two particle chain are found for arbitrary linear and nonlinear coupling strengths. Solutions for the N-body chain are found for purely nonlinear couplings. Three distinct solution types presented: a linear analogue, a chaotic amplitude mapping, and a localized nonlinear mode. The relaxation of perturbed modes are also explored using $l_{1}$-regularized least squares regression to estimate the free energy functional near the nonlinear normal mode solution. The perturbed modes are observed to decay sigmoidally towards a quasi-equilibrium state and a logarithmic relationship between the perturbation strength and mode lifetime is found.