either transition may happen first, with the growth of the nonlinearity. Another essential problem addressed in the paper is a dynamical one, namely, Josephson oscillations in the DWP structure, initiated by the instability of the symmetric mode, and collisions of free solitons with the structure.

It is relevant to mention that, in terms of the BEC realizations, the present setting represents macroscopic quantum states, with the phase transitions between them being quasi-classical ones, considered in the framework of the mean-field approximation. The validity of this approximation is usually justified by the large number of atoms in the condensate [@BEC]. The consideration of a few-body state in the DWP can give rise to quantum phase transitions, such as those in the Lipkin-Meshkov-Glick model, which applies in this case [@LMG]. As suggested by a recent analysis of the three-dimensional many-body quantum gas with repulsive binary interactions, which is pulled to the center by a potential $\sim -1/r^{2}$ [@Grisha], the quantum phase transition may produce results similar to but different from their mean-field counterparts [@HS]. In particular, the GS predicted by the mean-field may be replaced by a metastable state in the quantum many-body theory [@Grisha]. In any case, the consideration of truly 