Generalized nonpolynomial Schrödinger equations for matter waves under anisotropic transverse confinement

Abstract

Starting from the three-dimensional Gross–Pitaevskii equation we derive a 1D generalized nonpolynomial Schrödinger 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 Schrödinger 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 Schrödinger equation.