Effective self-similar expansion for the Gross-Pitaevskii equation

We consider 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.