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We show that exact time dependent single particle Green function in theImambekov-Glazman theory of nonlinear Luttinger liquids can be written, for anyvalue of the Luttinger parameter, in terms of a particular solution of thePainlev\'e IV equation. Our expression for the Green function has a formanalogous to the celebrated Tracy-Widom result connecting the Airy kernel withPainlev\'e II. The asymptotic power law of the exact solution as a function ofa single scaling variable $x/\sqrt{t}$ agrees with the mobile impurity results.The full shape of the Green function in the thermodynamic limit is recoveredwith arbitrary precision via a simple numerical integration of a nonlinear ODE.

Nonlinear Luttinger liquid: Exact result for the Green function in terms of the fourth Painlev\'e transcendent