  We consider a Gross-Pitaevskii model of BEC with non-local s-wave scattering
to study the density modulated state in 1D. We resort to a perturbative Taylor
series expansion for the order parameter. By perturbative calculations, we show
that under long range s-wave scattering a density modulated state is
energetically favourable as compared to the uniform density state. We obtain
density modulated state as a solution to the perturbative non-local GP
equation, rather than the conventional approach of introducing amplitude
modulations on top of the uniform density state and lowering the roton minimum.
