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

Tom Price, Dmitry L. Kovrizhin, Austen Lamacraft

SciPost Phys. 2, 005 (2017) · published 21 February 2017

### 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 Painlev\'e IV equation. Our expression for the Green function has a form analogous to the celebrated Tracy-Widom result connecting the Airy kernel with Painlev\'e 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.

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