Generalized nonpolynomial Schr_dinger equations for matter waves under anisotropic transverse confinement

摘要

Starting from the three-dimensional Gross_Pitaevskii equation we derive a 1 D generalized nonpolynomial Schr_dinger equation, which describes the dynamics of Bose_Einstein condensates under the action of a generic potential in the longitudinal axial direction and of an anisotropic harmonic potential in the transverse radial direction. This equation reduces to the familiar 1D nonpolynomial Schr_dinger equation (Salasnich, Parola and Reatto 2002 Phys. Rev. A 65, 043614) in the case of isotropic transverse harmonic confinement. In addition, we show that if the longitudinal potential models a periodic optical lattice the 3D GPE can be mapped into a 1D generalized discrete nonpolynomial Schr_dinger equation.