# Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schr{\"o}dinger equation

## TL;DR

This paper studies the large-time behavior and semiclassical limit of solutions to the defocusing logarithmic Schr{"o}dinger equation, revealing convergence to a Gaussian profile and analyzing the associated Wigner measures.

## Contribution

It extends large-time dispersion results to the semi-classical case, provides sharp convergence rates in Kantorovich-Rubinstein metric, and links the Schr{"o}dinger equation with kinetic equations via Wigner transforms.

## Key findings

- Solutions disperse faster than usual with a logarithmic factor.
- Rescaled solution modulus converges to a Gaussian profile.
- Wigner measures exhibit the same large-time behavior as the solution modulus.

## Abstract

We consider the dispersive logarithmic Schr{\"o}dinger equation in a semi-classical scaling. We extend the results about the large time behaviour of the solution (dispersion faster than usual with an additional logarithmic factor, convergence of the rescaled modulus of the solution to a universal Gaussian profile) to the case with semi-classical constant. We also provide a sharp convergence rate to the Gaussian profile in Kantorovich-Rubinstein metric through a detailed analysis of the Fokker-Planck equation satisfied by this modulus. Moreover, we perform the semiclassical limit of this equation thanks to the Wigner Transform in order to get a (Wigner) measure. We show that those two features are compatible and the density of a Wigner Measure has the same large time behaviour as the modulus of the solution of the logarithmic Schr{\"o}dinger equation. Lastly, we discuss about the related kinetic equation (which is the Kinetic Isothermal Euler System) and its formal properties, enlightened by the previous results and a new class of explicit solutions.