Exact Solutions of Augmented GP Equation: Solitons, Droplets and Supersolid

TL;DR

This paper derives exact solutions to the augmented nonlinear Schrödinger equation, revealing various phases like solitons, droplets, and supersolids, and characterizes their properties using Jacobi elliptic functions.

Contribution

It introduces a fractional transformation method to connect solutions of the ANLSE with Jacobi elliptic functions, providing a unified framework for different phases.

Findings

Derived exact solutions using Jacobi elliptic functions.

Connected solutions with conserved energy and momentum.

Characterized phases based on background properties.

Abstract

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.