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On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions. (English) Zbl 1380.65296
Summary: In this paper, we propose some efficient and robust numerical methods to compute the ground states and dynamics 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 interaction evaluation [L. Exl et al., ibid. 327, 629–642 (2016; Zbl 1422.65450)]. To compute the ground states, we integrate the preconditioned Krylov subspace pseudo-spectral method [the first author and R. Duboscq, ibid. 258, 509–523 (2014; Zbl 1349.82027)] and the GauSum solver. For the dynamics simulation, using the rotating Lagrangian coordinates transform [W. Bao et al., SIAM J. Sci. Comput. 35, No. 6, A2671–A2695 (2013; Zbl 1286.35213)], we first reformulate the FSE into a new equation without rotation. Then, a time-splitting pseudo-spectral scheme incorporated with the GauSum solver is proposed to simulate the new FSE. In parallel to the numerical schemes, we also prove some existence and nonexistence results for the ground states. Dynamical laws of some standard quantities, including the mass, energy, angular momentum and the center of mass, are stated. The ground states properties with respect to the fractional order and/or rotating frequencies, dynamics involving decoherence and turbulence together with some interesting phenomena are reported.

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text: DOI
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