Stability analysis of Bose gas: effect of external trapping and three-body interaction with Gross–Pitaevskii equation in Tonks–Girardeau regimes

In this work, we investigate the stability aspects of quintic Gross–Pitaevskii (GP) equation with the presence and absence of external trapping potential for a Bose gas (BG) in both the Tonks–Girardeau (TG) and the super Tonks–Girardeau (sTG) regimes. For this purpose, we compute both analytically and numerically a pure quintic GP equation with the presence of the scattering lengths. Using the time-dependent variational approach, we derive the equations of motion and effective potential of the system for both cases. Through the effective potential, we discuss the stability properties of pure quintic GP equation and obtain the modulational instability condition of BECs. The variational results are verified by means of direct numerical simulations using split-step Crank–Nicolson method and the observed results are in agreement with the analytical predictions.