Logarithmic Schr{\"o}dinger equation with quadratic potential

TL;DR

This paper studies the behavior of solutions to the logarithmic Schrödinger equation with quadratic potentials, demonstrating stability of solitary waves and dispersive properties depending on the potential's nature.

Contribution

It proves the orbital stability of generalized Gausson solutions and characterizes dispersive behaviors under different quadratic potentials.

Findings

Solitary waves are orbitally stable under harmonic potential.

Solutions exhibit universal dispersive behavior in partial confinement.

Dispersive rate depends on the nature of the harmonic potential.

Abstract

We analyze dynamical properties of the logarithmic Schr{\"o}dinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a universal asymptotic profile. The introduction of a harmonic potential generates solitary waves, corresponding to generalized Gaussons. We prove that they are orbitally stable, using an inequality related to relative entropy, which may be thought of as dual to the classical logarithmic Sobolev inequality. In the case of a partial confinement, we show a universal dispersive behavior for suitable marginals. For repulsive harmonic potentials, the dispersive rate is dictated by the potential, and no universal behavior must be expected.