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

Low-$T$ asymptotic expansion of the solution to the Yang-Yang equation