Recent observations of droplets in dipolar and binary Bose–Einstein condensate (BEC) gives motivation to study the theory of droplet formation in detail. Specifically, the authors are interested in investigating the possibility of droplet formation in a quasi‐1D geometry. Recent observations have suggested that droplets are stabilized by competition between effective mean‐field and beyond mean‐field interaction. Hence, it is possible to map the effective equation of motion to a cubic‐quartic nonlinear Schrödinger equation (CQNLSE). Two analytical solutions of the modified Gross–Pitaevskii equation or CQNLSE are obtained and they are verified numerically. Based on their stability, the parameter regime for which droplets can form is investigated. The effective potential allows for conclusions about the regions of soliton domination and self‐bound droplet formations.

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We present analytical solution, which is periodic in nature, for a driven cubic-quartic nonlinear Schr\"odinger equation (DCQNLSE) placed in a bi-chromatic optical lattice. The solution indicates the creation of density wave. Since, beyond mean-field contribution in quasi one dimensional and one-dimensional geometry differs on the even exponents of the nonlinearity thus we extend our analysis towards quadratic-cubic-quartic and quadratic-cubic nonlinearities as well. Later, we study the dynamics of DCQNLSE. Our study indicates the existence of stripe phase along with considerable phase coherence. These findings allow us to comment on the possible emergence of supersolid phase in a condensate.

The study of Bose–Einstein condensation (BEC) in lower dimensions plays an important role in understanding the fundamentals of many-body physics as they can be treated theoretically with relative ease and can be verified experimentally. Recently, observation of a liquid-like state in a BEC mixture has been reported along with a theoretical prescription for its observation in the lower dimension. This observation is unique and has serious ramifications in our prevailing conception of the liquid state, which has a deep influence on the van der Waals theory. In explaining the self-bound nature of this state, quantum fluctuation and its fine balance with mean-field (MF) interaction turn out to be playing a key role. Though the experiments are performed predominantly in three dimensions, theoretical studies extend to the lower dimensions. In this brief review, we plan to summarize the recent theoretical advances in droplet research in the lower dimension and elaborate on the description of our contributions. We will mainly focus on analytical results related to this self-bound state in a one-dimension and quasi one-dimension environment. We aim to cover a few results from the family of cnoidal solutions to droplet solutions with smooth transitions between each other, finishing it by carrying a modest discussion on the supersolid phase.