Report on 1606.08702v2
Tom Price, Dmitry L. Kovrizhin, Austen Lamacraft · 2017
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/ √ 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.