Some space-time fractional bright–dark solitons and propagation manipulations for a fractional Gross–Pitaevskii equation with an external potential

Abstract

Abstract Some nonautonomous bright–dark solitons (NBDSs) and nonautonomous controllable behaviors in the conformable space-time fractional Gross–Pitaevskii (FGP) equation with some external potentials are derived. We consider the relations between the space-time FGP equation and the fractional nonlinear Schrödinger equation and analyze the properties of the obtained equation with group velocity dispersion and spatiotemporal dispersion. Then, some constraint conditions of the valid soliton solutions are given. Furthermore, we consider the effect of α and β in NBDSs of the space-time FGP equation. Some fractional spatial–temporal controlling wave prolong phenomena are considered, and some different propagation dynamics are generated via the different parameters α and β. We study novel shape bright soliton solution, novel ‘h’-shape dark soliton and some interactions of nonautonomous bright–dark solitons. The reported results of some novel interactions are considered, which can explain some models of the electrical and optical fields.