We consider the dispersive logarithmic Schrödinger equation in a semiclassical
scaling. We extend the results of Carles and Gallagher (Duke Math. J. 167:9 (2018),
1761–1801) about the large-time behavior of the solution (dispersion faster than
usual with an additional logarithmic factor and convergence of the rescaled modulus
of the solution to a universal Gaussian profile) to the case with semiclassical
constant. We also provide a sharp convergence rate to the Gaussian profile in the
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 behavior as the modulus of the solution of the
logarithmic Schrödinger 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.
