the competition between them, crucially depend on the tunneling transparency of the central barrier and the nonlinearity strength, i.e., the barrier height, $A$, and total norm, $N$. These are two control parameters which are subject to the variation in the subsequent analysis. For the same reason, the main findings are not sensitive to a particular shape of the DWP. In particular, the rectangular form of the DWP, adopted in Eq. (\[potential\]), which may be essential for some dynamical effects, such as temporal scaling in the relaxation of perturbations [@Azbel], produces results which are essentially the same as generated by smooth DWP profiles (for the self-trapping of the leaky modes in a single potential well, this property was known before [@Carr1]). As concerns the necessity of having the elevated potential floor, it may make the experimental creation of the structure easier, as the floor" is naturally built by overlap of fringes of two potential barriers which determine the DWP (in previously reported experimental realizations of the DWP in BEC [@Markus], the bottom level of the potential had to be depressed, because those DWPs trapped condensates with the repulsive interactions, on the contrary to the case of the self-attraction, considered herein).

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