Numerical and Variational Solutions of the Dipolar Gross
Pitaevskii Equation in Reduced Dimensions¹

P. Muruganandam^{a, b} and S. K. Adhikari^a

^a Instituto de Física Teórica, UNESP—Universidade Estadual Paulista,
01.140-070 São Paulo, São Paulo, Brazil

^b School of Physics, Bharathidasan University, Palkalaiperur Campus,
Tiruchirappalli 620024, Tamilnadu, India

Received November 5, 2011; in final form, November 9, 2011; published online March 6, 2012

Abstract—We suggest a simple Gaussian Lagrangian variational scheme for the reduced time-dependent
quasi-one-and quasi-two-dimensional Gross–Pitaevskii (GP) equations of a dipolar Bose–Einstein condensate
(BEC) in cigar and disk configurations, respectively. The variational approximation for stationary states and
breathing oscillation dynamics in reduced dimensions agrees well with the numerical solution of the GP equa-
tion even for moderately large short-range and dipolar nonlinearities. The Lagrangian variational scheme also
provides much physical insight about soliton formation in dipolar BEC.

10.1134/S1054660X12040135

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