“…This formula -which differs from those obtained by means of the Darboux dressing method (e.g., see [59,54,8,14]), requiring one to treat the problem of inverting an N × N matrix featuring Jost solutions as its elements -contains and allows an immediate classification of all the reflectionless solutions, irrespective of the number and the nature of the discrete eigenvalues in the spectrum, providing their direct physical interpretation (e.g., explicit expressions for the speed and precession frequency, as well as the location and time of the interactions, period of the oscillations for the breather-like solutions, separation of the maxima for the creation of entangled states, etc, see Section 4). Indeed, by choosing in a proper way the parameters featured by this formula and naturally linked to the spectral data, we are able to generate explicit expressions for all the solutions already known in the literature [50,41,57,59,54,8,14], and, in particular, general, explicit expressions for the breather-like and multipole solutions (see Section 4). As for the latter, it is important to underline here that, in principle, the existence of multipole solutions for (1.2) might be inferred from the gauge equivalence to the nonlinear Schrödinger equation as derived in [61].…”