  We prove that the unique solution to the Yang-Yang equation arising in the
context of the thermodynamics of the so-called non-linear Schr\"{o}dinger model
admits a low-temperature expansion to all orders. Our approach provides a
rigorous justification, for a certain class of non-linear integral equations,
of the low-temperature asymptotic expansion that were argued previously in
various works related to the low-temperature behavior of integrable models.
