Around plane waves solutions of the Schrödinger-Langevin equation

Quentin Chauleur (1, 2) , Erwan Faou (1, 2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 MINGUS - Multi-scale numerical geometric schemes

We consider the logarithmic Schrödinger equations with damping, also called Schrödinger-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.

Domaines: Equations aux dérivées partielles [math.AP]

Quentin Chauleur, Erwan Faou. Around plane waves solutions of the Schrödinger-Langevin equation. SIAM Journal on Mathematical Analysis, 2022, 54 (5), pp.5103-5125. ⟨10.1137/21M1458387⟩. ⟨hal-03409804⟩