On the Cauchy problem for nonlinear Schrödinger equations with rotation

Antonelli, Paolo; Marahrens, Daniel; Sparber, Christof

doi:10.3934/dcds.2012.32.703

Cited by 45 publications

(99 citation statements)

References 29 publications

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“…The aim of this paper is to propose a simple and efficient numerical method to solve the GPE in a rotating frame, which may include a dipolar interaction term. One novel idea of the proposed method consists in the use of rotating Lagrangian coordinates as in [6], in which the angular momentum rotation term vanishes. Hence, we can easily apply the methods for nonrotating BECs in [8,12,15] to solve the rotating case.…”

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confidence: 99%

A Simple and Efficient Numerical Method for Computing the Dynamics of Rotating Bose--Einstein Condensates via Rotating Lagrangian Coordinates

Bao¹,

Marahrens²,

Tang³

et al. 2013

SIAM J. Sci. Comput.

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Abstract. We propose a simple, efficient, and accurate numerical method for simulating the dynamics of rotating Bose-Einstein condensates (BECs) in a rotational frame with or without longrange dipole-dipole interaction (DDI). We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term and/or long-range DDI, state the twodimensional (2D) GPE obtained from the 3D GPE via dimension reduction under anisotropic external potential, and review some dynamical laws related to the 2D and 3D GPEs. By introducing a rotating Lagrangian coordinate system, the original GPEs are reformulated to GPEs without the angular momentum rotation, which is replaced by a time-dependent potential in the new coordinate system. We then cast the conserved quantities and dynamical laws in the new rotating Lagrangian coordinates. Based on the new formulation of the GPE for rotating BECs in the rotating Lagrangian coordinates, a time-splitting spectral method is presented for computing the dynamics of rotating BECs. The new numerical method is explicit, simple to implement, unconditionally stable, and very efficient in computation. It is spectral-order accurate in space and second-order accurate in time and conserves the mass on the discrete level. We compare our method with some representative methods in the literature to demonstrate its efficiency and accuracy. In addition, the numerical method is applied to test the dynamical laws of rotating BECs such as the dynamics of condensate width, angular momentum expectation, and center of mass, and to investigate numerically the dynamics and interaction of quantized vortex lattices in rotating BECs without or with the long-range DDI. 1. Introduction. Bose-Einstein condensation (BEC), first observed in 1995 [4,18,23], has provided a platform to study the macroscopic quantum world. Later, with the observation of quantized vortices [2,19,34,35,37,39,50], rotating BECs have been extensively studied in the laboratory. The occurrence of quantized vortices is a hallmark of the superfluid nature of BECs. In addition, condensation of bosonic atoms and molecules with significant dipole moments whose interaction is both nonlocal and anisotropic has recently been achieved experimentally in trapped 52 Cr and 164 Dy gases [1,22,27,32,33,36,48].At temperatures T much smaller than the critical temperature T c , the properties of BEC in a rotating frame with long-range dipole-dipole interaction (DDI) are well

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confidence: 99%

A Simple and Efficient Numerical Method for Computing the Dynamics of Rotating Bose--Einstein Condensates via Rotating Lagrangian Coordinates

Bao¹,

Marahrens²,

Tang³

et al. 2013

SIAM J. Sci. Comput.

Self Cite

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“…was considered in [13], where the authors proved global existence for defocusing nonlinearities (λ > 0) without restricting the rotation frequency, generalizing earlier results given in [14] and [15], and find some blowup solutions in the focusing case λ < 0. Followed by the same idea, we consider the problem (1.2) with coupled Hartree-type nonlinearities which must compete with the local nonlinearities to ensure the existence of global or blowup solutions of system (1.2).…”

Section: Introductionmentioning

confidence: 80%

“…Followed by the same idea, we consider the problem (1.2) with coupled Hartree-type nonlinearities which must compete with the local nonlinearities to ensure the existence of global or blowup solutions of system (1.2). The analyzing of the interaction between the local and nonlocal nonlinearities did not occur in [13].…”

Section: Introductionmentioning

confidence: 94%