localized states, while $k<0$ corresponds to delocalized leaky modes, which do not vanish at $x\rightarrow
\pm \infty $. The states of these two types are characterized, respectively, by convergent and divergent norms, $N=\int_{-\infty }^{+\infty
}u^{2}(x)dx$ (proportional to the total power of the light beam in optics, or the total number of atoms in BEC), and Hamiltonian (energy),$$E=\int_{-\infty }^{+\infty }\left[ \frac{1}{2}\left( u^{\prime }\right) ^{2}-%
\frac{1}{2}u^{4}+2H(x)u^{2}\right] dx.  \label{H}$$

The DWP can be readily implemented in the experiment. In optics, waveguides with this structure are fabricated with the help of the implanting technique [@Opt-DWP], while in BEC the DWP setting can be created by means of electromagnetic fields [@BEC-DWP]. In the present work, calculations are reported for the rectangular DWP profile with the elevated floor:

$$H\left( x\right) =\left\{
\begin{array}{ll}
A, & ~\mathrm{at}~~|x|<0.5, \\
2, & ~\mathrm{at}~~3<|x|<7, \\
0, & ~\mathrm{elsewhere},%
\end{array}%
\right.   \label{potential}$$

where $A>0$ is the height of the inner potential barrier, see Fig. [potential-fig]{}, while height $H_{\max }=2$ of the outer barriers is fixed by scaling. Values of lengths adopted in Eq. (\[potential\]) adequately represent the generic situation, as demonstrated by additional numerical results (not shown here in detail). Indeed, it is shown below that the symmetry-breaking and self-trapping transitions, and 