The localized states of a BEC for either attractive or repulsive atom interactions are products of the Bloch functions and soliton envelope functions of the nonlinear Schrödinger equation that agree well with the numerical ones.
A one-band effective-mass (EFM) theory was developed to study the properties of the localized states of a Bose-Einstein condensate (BEC) in a periodic optical potential with either attractive or repulsive atom interactions. The localized states of the BEC are solitons in the gaps or gap solitons. The analytic solutions of the gap solitons are a Bloch function from periodicity, modulated by a soliton envelope function of the nonlinear Schrödinger equation. The soliton width is determined by the detuning chemical potential from the band edge. The detuning chemical potential in the nonlinear Schrödinger equation indicates that not all the atoms in the condensate are used for forming gap solitons, and that some atoms form the background of nonlinear modes. The background atoms are determined by the chemical potential at the band edge. The gap solitons form on top of the background atoms. PACS numbers: 03.75.Lm, 03.75.Kk, 05.30.Jp I.
We discuss localized ground states of Bose-Einstein condensates (BEC's) in optical lattices with attractive and repulsive three-body interactions in the framework of a quintic nonlinear Schr\"odinger equation which extends the Gross-Pitaevskii equation to the one-dimensional case. We use both a variational method and a self-consistent approach to show the existence of unstable localized excitations which are similar to Townes solitons of the cubic nonlinear Schr\"odinger equation in two dimensions. These solutions are shown to be located in the forbidden zones of the band structure, very close to the band edges, separating decaying states from stable localized ones (gap solitons) fully characterizing their delocalizing transition. In this context the usual gap solitons appear as a mechanism for arresting the collapse in low-dimensional BEC's in optical lattices with an attractive real three-body interaction. The influence of the imaginary part of the three-body interaction, leading to dissipative effects in gap solitons, and the effect of atoms feeding from the thermal cloud are also discussed. These results may be of interest for both BEC's in atomic chips and Tonks-Girardeau gas in optical lattices.
We analyze the existence and stability of spatially extended (Bloch-type) and localized states of a Bose-Einstein condensate loaded into an optical lattice. In the framework of the Gross-Pitaevskii equation with a periodic potential, we study the band-gap structure of the matter-wave spectrum in both the linear and nonlinear regimes. We demonstrate the existence of families of spatially localized matter-wave gap solitons, and analyze their stability in different band gaps, for both repulsive and attractive atomic interactions.