quantum phase transitions in the DWP structure is a subject for a separate work.

The subsequent presentation is structured as follows. The model is elaborated in Section II. Results of the analysis of symmetric and spontaneously emerging asymmetric trapped and leaky modes in it are summarized in Section III. Detailed results are obtained in a numerical form, and their basic features are explained by means of an analytical approach. Both the trapped and leaky modes undergo the SSB transition with the increase of the norm, the symmetric modes getting unstable above the transition point. The nonlinear evolution of the unstable modes, which features Josephson oscillations, is studied by means of systematic simulations in Section IV. A related possibility is capture of incident solitons by the DWP structure into shuttle states. This possibility is studied in a systematic form in Section V. The paper is concluded by Section VI.

The model
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The underlying dynamical model, based on Eq. (\[NLSE\]), gives rise to stationary modes with propagation constant $k$ (in BEC, $-k$ is the chemical potential),  $$\psi \left( x,z\right) =e^{ikz}u\left( x\right) ,  \label{psi}$$where real modal functions $u(x)$ satisfy the equation

$$-ku+\frac{1}{2}u^{\prime \prime }+u^{3}=H\left( x\right) u.
\label{stationaryNLSE}$$

Solutions with $k>0$ represent self-trapped 