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We prove that the unique solution to the Yang–Yang equation arising in the context of the thermodynamics of the so-called non-linear Schrödinger 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 expansions that were argued previously in various works related to the low-temperature behaviour of integrable models.

Low-T Asymptotic Expansion of the Solution to the Yang–Yang Equation