symmetric double-well potential for leaky modes (quasi-bound states), defined as per Eq. (\[potential\]).[]{data-label="potential-fig"}](potential.pdf)

Obviously, in the absence of the self-attractive cubic term, potential ([potential]{}) cannot support any bound state in the respective linear model, while weakly delocalized quasi-bound states are possible. Indeed, a straightforward estimate of the tunneling coefficient for the tall barriers separating the inner and outer parts of structure (\[potential\]) yields $$T\simeq \exp \left( -\sqrt{2H_{\max }-q^{2}}W\right) \approx \allowbreak
4.4\times 10^{-4},  \label{T}$$where $H_{\max }=2$ and $W=4$ are the height and width of the potential barriers, as per Eq. (\[potential\]), and $q\equiv \pi /l=\pi /6$ is the wavenumber of the lowest quasi-bound state in the potential box of width $%
l=6 $.

For the study of collisions of moving solitons with the DWP structure, the height of the inner barrier is fixed to be $A=0.5$, while the height, $H_{0}$, and width, $W_{0}$, of the outer barriers will be varied, to allow clearer observations of different collision scenarios: $$H\left( x\right) =\left\{
\begin{array}{ll}
0.5, & ~\mathrm{at}~~|x|<0.5, \\
H_{0}, & ~\mathrm{at}~~3<|x|<3+W_{0}\equiv \Lambda /2, \\
0, & ~\mathrm{elsewhere}.%
\end{array}%
\right.  \label{P2}$$

Detailed consideration of the SSB in the leaky modes will require an explicit calculation of small-amplitude radiation tails attached to those modes outside 