Nonuniqueness and nonlinear instability of Gaussons under repulsive harmonic potential

TL;DR

This paper investigates Gaussian solitary waves in a Schr{"o}dinger equation with logarithmic nonlinearity and repulsive harmonic potential, revealing nonuniqueness, instability, and the absence of ground states.

Contribution

It demonstrates the existence of multiple Gaussian solutions and their instability, and clarifies the nonexistence of ground states in this nonlinear Schr{"o}dinger setting.

Findings

Existence of two positive stationary Gaussian solutions

All solutions are orbitally unstable

No ground states exist under natural definitions

Abstract

We consider the Schr{\"o}dinger equation with a nondispersive logarithmic nonlinearity and a repulsive harmonic potential. For a suitable range of the coefficients, there exist two positive stationary solutions, each one generating a continuous family of solitary waves. These solutions are Gaussian, and turn out to be orbitally unstable. We also discuss the notion of ground state in this setting: for any natural definition, the set of ground states is empty.