Numerical study of the logarithmic Schrodinger equation with repulsive harmonic potential

TL;DR

This paper investigates the dynamics of the logarithmic Schrödinger equation with a repulsive harmonic potential, employing a lens transform and splitting methods, and provides numerical results on dispersive behavior and error estimates.

Contribution

It introduces a generalized lens transform to control boundary effects and analyzes the numerical splitting methods for the logarithmic Schrödinger equation with potential.

Findings

Exponential dispersive decay observed when dispersion occurs.

The lens transform effectively neutralizes boundary effects.

Error estimates for splitting methods are provided for different nonlinearities.

Abstract

We consider the Schrodinger equation with a logarithmic nonlinearity and a repulsive harmonic potential. Depending on the parameters of the equation, the solution may or may not be dispersive. When dispersion occurs, it does with an exponential rate in time. To control this, we change the unknown function through a generalized lens transform. This approach neutralizes the possible boundary effects, and could be used in the case of the Schrodinger equation without potential. We then employ standard splitting methods on the new equation via a nonuniform grid, after the logarithmic nonlinearity has been regularized. We also discuss the case of a power nonlinearity and give some results concerning the error estimates of the first-order Lie-Trotter splitting method for both cases of nonlinearities. Finally extensive numerical experiments are reported to investigate the dynamics of the equations.