“…By now, the variational technique, originally proposed in a nonlinear problem [27,28], has been developed for STA in particular systems [29][30][31][32] that cannot be treated by means of other existing approaches, i.e., invariant-based engineering [33,34], counterdiabatic driving [35][36][37], and fast-forward scaling [38,39]. More specifically, since the time-dependent variational principle can find a set of Newton-like ordinary differential equations for the parameters (i.e., the width of cloud, center, and interatomic interaction), the variational control provides a promising alternative aimed at accelerating the adiabatic compression or decompression of BECs and bright solitons [29,32], beyond the harmonic approximation of the potential [31] and Thomas-Fermi limit [33,40,41]. In this scenario, the Lewis-Riesenfeld dynamical invariant and general scaling transformations [33,34] are not required in the context of inverse engineering.…”