superposition of two stationary wave functions, $\phi $, corresponding to the states trapped separately in either potential well, with their centers located at $x=\pm a$ [@Ananikian]:$$\psi \left( x,z\right) =u_{1}(z)\phi \left( x-a\right) +u_{2}(z)\phi \left(
x+a\right) .  \label{12}$$

The analysis of the SSB in BEC and similar models based on Eq. (\[NLSE\]) was initiated in Refs. [@Milburn] and [@Smerzi]. In this case, in the framework of the mean-field approximation, the symmetry breaking is the phase transition of the second kind (alias the supercritical bifurcation, which does not admit hysteresis [@Joseph]). Further, GPE (\[NLSE\]) was extended by adding an extra (free) spatial coordinate, which transforms the DWP into a two-dimensional dual-core structure [@Warsaw]. In such a setting, the self-attractive nonlinearity gives rise to matter-wave solitons, which self-trap in the free direction [@soliton]. The SSB destabilizes symmetric solitons and replaces them by asymmetric ones, provided that the norm of the wave function (which determines the effective strength of the intrinsic nonlinearity) exceeds a critical value [Warsaw]{}. In the latter case, the mean-field symmetry breaking is a phase transition of the first kind (alias a subcritical bifurcation [@Joseph]), which includes hysteresis. The subcritical transition is typical to solitons in dual-core waveguides with the Kerr self-focusing 