On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions

Xavier Antoine (1, 2) , Qinglin Tang (3, 1, 2) , Yong Zhang (4, 5)

1 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization
2 IECL - Institut Élie Cartan de Lorraine
3 LMRS - Laboratoire de Mathématiques Raphaël Salem
4 WPI - Wolfgang Pauli Institute
5 IRMAR - Institut de Recherche Mathématique de Rennes

In this paper, we propose some efficient and robust numerical methods to compute the ground states anddynamics of Fractional Schrödinger Equation (FSE) with a rotation term and nonlocal nonlinear interactions.In particular, a newly developed Gaussian-sum (GauSum) solver is used for the nonlocal interactionevaluation [33]. To compute the ground states, we integrate the preconditioned Krylov subspace pseudospectralmethod [5] and the GauSum solver. For the dynamics simulation, using the rotating Lagrangiancoordinates transform [16], we first reformulate the FSE into a new equation without rotation. Then, atime-splitting pseudo-spectral scheme incorporated with the GauSum solver is proposed to simulate the newFSE. In parallel to the numerical schemes, we also prove some existence and nonexistence results for theground states. Dynamical laws of some standard quantities, including the mass, energy, angular momentumand the center of mass, are stated. The ground states properties with respect to the fractional orderand/or rotating frequencies, dynamics involving decoherence and turbulence together with some interestingphenomena are reported.

Mots clés: Gaussian-sum solver; Dynamics; Fractional Schrödinger equation; rotation; nonlocal nonlinear interaction; Rotating Lagrangian coordinates; Ground state

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Xavier Antoine, Qinglin Tang, Yong Zhang. On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions. Journal of Computational Physics, 2016, 325, pp.74-97. ⟨10.1016/j.jcp.2016.08.009⟩. ⟨hal-01244364⟩