In the first part of this talk, I will revisit the dynamics of a Bose-Einstein condensate in a double-well potential, from the regime of Josephson plasma oscillations to the self-trapping regime, by means of the Bogoliubov quasiparticle projection method. For a very small population imbalance only the lowest Bogoliubov mode is significantly occupied, and the system performs plasma oscillations at the corresponding frequency. I will show that, whereas the Bogoliubov approach permits a rigorous definition of the Josephson plasma frequency, the widely employed two-mode model — commonly constructed by means of a linear combination of the low-lying states of the GP equation — only provides accurate results for weak interactions. Then, as the initial imbalance is increased, more excited modes (even if not macroscopically occupied initially) get coupled during the evolution of the system. The nonlinear mixing of Bogoliubov modes eventually drives the system into the self-trapping regime, when the population of the ground state can be transferred completely to the excited states at some time during the evolution. I the second part of the talk, I will discuss an effective scaling approach for the free expansion of a one-dimensional quantum wave packet, consisting in a self-similar evolution to be satisfied on average, i.e., by integrating over the coordinates. A direct comparison with the solution of the Gross-Pitaevskii equation shows that the effective scaling reproduces with great accuracy the exact evolution—the actual wave function is reproduced with a fidelity close to one—for arbitrary values of the interactions. This result represents a proof of concept of the effectiveness of the scaling ansatz, which has been used in different forms in the literature but never compared against the exact evolution. If time permits, I will also briefly report on the observation of quantum droplets in a 41K -87Rb bosonic mixture, in a novel experiment at LENS in Florence.