Solitons in multi-body interactions for a fully modulated cubic-quintic Gross-Pitaevskii equation

A phase imprint approach is applied to a cubic Gross-Pitaevskii equation (GPE) in order to obtain a modified non-autonomous derivative cubic-quintic nonlinear GPE with fully variable coefficients. This model describes the dynamics of condensates in Bose-Einstein condensates, with both two- and three-body interatomic interactions with an external potential when the coefficient of the dispersion term is constant. This model is also applicable to fiber optics media in the absence of external potential. We show that this modified GPE model has a Lax-pair when all of its coefficients depend on time. However, the external potential depends on both the time and space variables. We obtain two classes of exact analytical solutions. These classes of solutions contain four different types of solitary-like wave solutions in the form of kink, anti-kink, bright, dark solitary, and periodic wave solutions. We reveal the effects of a quintic term on wave numbers, for both the carrier and envelope waves. Stability analysis is carried out, and conditions on parameters that determine regions with linear stability are discussed. Graphical analysis of some solutions are presented.