Localized analytical solutions and numerically stabilities of generalized Gross-Pitaevskii (GP(p,q)) equation with specific external potentials

We present the generalized Gross-Pitaevskii (GP(p,q)) equation with two kinds of specific space-time modulated potentials, including the generalized Rosen-Morse potential, the combination of harmonic and Gaussian (harmonic-Gaussian) potential. Some analytical bright soliton solutions are derived from the generalized GP(p,q) equation via the complex similarity transformation and the generalized stationary nonlinear Schrödinger (NLS) equation. The soliton solutions can be reduced to Jacobi elliptic and spikon-like solutions by choosing the different parameters, and describe some physically relevant phenomenons. Furthermore, some stabilities of the obtained matter-wave solutions are addressed numerically. We find that some solutions are stable and can be observed over a broad range of parameters through analyzing the effect of the phase noise on these solutions. The obtained results may give a certain theoretical guiding significance of relative experiments in Bose-Einstein condensates.