Around plane waves solutions of the Schr{\"o}dinger-Langevin equation

TL;DR

This paper studies the stability of explicit plane wave solutions to the logarithmic Schrödinger-Langevin equation on periodic domains, demonstrating asymptotic and long-term stability under damping and non-damping conditions, supported by numerical experiments.

Contribution

It establishes the asymptotic stability of plane wave solutions for the Schrödinger-Langevin equation with damping and analyzes their stability in the non-damped case for most nonlinear parameters.

Findings

Plane wave solutions are asymptotically stable with damping.

In the absence of damping, solutions remain stable for long times.

Numerical experiments support theoretical stability results.

Abstract

We consider the logarithmic Schr{\"o}dinger equations with damping, also called Schr{\"o}dinger-Langevin equation. On a periodic domain, this equation possesses plane wave solutions that are explicit. We prove that these solutions are asymptotically stable in Sobolev regularity. In the case without damping, we prove that for almost all value of the nonlinear parameter, these solutions are stable in high Sobolev regularity for arbitrary long times when the solution is close to a plane wave. We also show and discuss numerical experiments illustrating our results.