Supersolid behavior in one-dimensional self-trapped Bose-Einstein condensate

Supersolid is an exotic state of matter, showing crystalline order with a superfluid background, observed recently in dipolar Bose-Einstein condensate (BEC) in a trap. Here, we present exact solutions of the desired Bloch form in the self-trapped free-floating quantum fluid. Our general solutions of the amended nonlinear Schr{\"o}dinger equation, governing the mean-field and beyond mean-field dynamics, are obtained through a M{\"o}bius transform, connecting a wide class of supersolid solutions to the ubiquitous cnoidal waves. The solutions yield the constant condensate, supersolid behavior and the self-trapped droplet in different parameter domains. The lowest residual condensate is found to be exactly one-third of the constant background.