  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.
