  The augmented nonlinear Schr\"odinger equation (ANLSE), describing BEC, with
the Lee-Huang-Yang (LHY) correction has exhibited a quantum droplet state,
which has found experimental verification. In addition to the droplet, exact
kink-antikink and supersolid phases have been recently obtained in different
parameter domains. Interestingly, these solutions are associated with a
constant background, unlike the form of BEC in quasi-one dimension, where dark,
bright, and grey solitons have been experimentally obtained. Here, we connect a
wide class of solutions of the ANLSE with the Jacobi elliptic functions using a
fractional transformation method in a general scenario. The conserved energy
and momentum are obtained in this general setting which differentiates and
characterizes the different phases of the solution space. We then concentrate
on the Jacobi-elliptic $dn(x, m^2)$ function, as the same is characterized by a
non-vanishing background as compared to the other $cn$ and $sn$ functions.
