Amplitude modulated phase in a Bose-Einstein condensate: the role of non-local interactions

We consider a Gross-Pitaevskii model of BEC with non-local interactions of range of the order of the s-wave scattering length. With this model, we study the density modulated phase in 1D and 2D, which are solutions of this modified model along with the usual uniform density state. We find an exact free energy functional for our model and show that the 1D density modulated state can have lower energy than the uniform density state. Although, the density modulated state can be made to be energetically favourable, we show also that, this state is inherently dynamically unstable due to the coupling of instabilities to the spatial order.