Propagation of coherent states in the logarithmic Schrodinger equation

TL;DR

This paper studies how coherent states evolve in the logarithmic Schrödinger equation under semiclassical scaling with external potentials, establishing error estimates and a superposition principle for multiple states.

Contribution

It introduces the notion of criticality for the nonlinear coupling and proves error estimates and a superposition principle for multiple coherent states.

Findings

Error estimates in the critical case

Superposition principle for two Gaussian states

Improved results for Gaussian initial profiles

Abstract

We consider the logarithmic Schr{\"o}dinger equation in a semiclassical scaling, in the presence of a smooth, at most quadratic, external potential. For initial data under the form of a single coherent state, we identify the notion of criticality as far as the nonlinear coupling constant is concerned, in the semiclassical limit. In the critical case, we prove a general error estimate, and improve it in the case of initial Gaussian profiles. In this critical case, when the initial datum is the sum of two Gaussian coherent states with different centers in phase space, we prove a nonlinear superposition principle.