Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation

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