  This work presents asymptotic solutions to a singularly-perturbed, period-2
FPUT lattice and uses exponential asymptotics to examine `nanoptera', which are
nonlocal solitary waves with constant-amplitude, exponentially small wave
trains which appear behind the wave front. Using an exponential asymptotic
approach, this work isolates the exponentially small oscillations, and
demonstrates that they appear as special curves in the analytically-continued
solution, known as `Stokes curves' are crossed. By isolating these the
asymptotic form of these oscillations, it is shown that there are special mass
ratios which cause the oscillations to vanish, producing localized
solitary-wave solutions. The asymptotic predictions are validated through
comparison with numerical simulations.
