# Nonlinear Luttinger liquid: exact result for the Green function in terms of the fourth Painlevé transcendent

# Abstract

We show that exact time dependent single particle Green function in the Imambekov-Glazman theory of nonlinear Luttinger liquids can be written, for any value of the Luttinger parameter, in terms of a particular solution of the Painlevé IV equation. Our expression for the Green function has a form analogous to the celebrated Tracy-Widom result connecting the Airy kernel with Painlevé II. The asymptotic power law of the exact solution as a function of a single scaling variable \(x/\sqrt{t}\) agrees with the mobile impurity results. The full shape of the Green function in the thermodynamic limit is recovered with arbitrary precision via a simple numerical integration of a nonlinear ODE.