Traveling waves for some nonlocal 1D Gross-Pitaevskii equations with nonzero conditions at infinity

André de Laire (1, 2) , Pierre Mennuni (1, 2, 3)

1 MEPHYSTO-POST - Méthodes quantitatives pour les modèles aléatoires de la physique
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 Paradyse - Systèmes de particules et systèmes dynamiques

We consider a nonlocal family of Gross-Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, we show that the branch is orbitally stable. In this manner, this result generalizes known properties for the contact interaction given by a Dirac delta function. Our proof relies on the minimization of the energy at fixed momentum. As a by-product of our analysis, we provide a simple condition to ensure that the solution to the Cauchy problem is global in time.

Mots clés: orbital stability; dark soli- tons; nonzero conditions at infinity; dark solitons; Nonlocal Schrödinger equation; Gross-Pitaevskii equation; traveling waves

Domaines: Equations aux dérivées partielles [math.AP]

André de Laire, Pierre Mennuni. Traveling waves for some nonlocal 1D Gross-Pitaevskii equations with nonzero conditions at infinity. Discrete and Continuous Dynamical Systems - Series A, 2020, 40 (1), pp.635-682. ⟨10.3934/dcds.2020026⟩. ⟨hal-01962779v3⟩