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I derive a Fredholm determinant for the thermal correlator of vertexoperators with a conformally invariant density matrix and nonlinear timeevolution. Using the method of nonlinear steepest descent I find the long timeasymptotics at finite temperature. The asymptotics display exponentially smallcorrections in temperature to the finite temperature mobile impurityphenomenology.

Long time thermal asymptotics of nonlinear Luttinger liquid from inverse scattering