Nonlinear normal mode solutions of the β-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 the case of purely nonlinear couplings. Three distinct solution types are presented: a linear analogue, a chaotic amplitude mapping, and a localized nonlinear mode. The relaxation of perturbed modes are also explored using l1-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. © 2020 Elsevier B.V.