which asymmetric modes do not exist).

SSB diagrams
------------

Getting back to the consideration of the SSB for the entire system, systematic results are presented by means of plots $N(k)$ and $E(N)$ \[the Hamiltonian is defined by Eq. (\[H\])\] for symmetric and asymmetric modes, which are displayed in Figs. \[bifurcation diagrams\](a-c) and (d-f) for three different values of height $A$ of the inner barrier of potential structure (\[potential\]). Note that the $N(k)$ curves obey the Vakhitov-Kolokolov criterion [@VK; @VK2], which is necessary but not sufficient for the stability of modes supported by the self-attractive nonlinearity (it does not catch the instability of the symmetric modes coexisting with asymmetric ones). It is relevant to compare these plots with their counterparts,$$N_{\mathrm{sol}}=2\sqrt{2k},~E_{\mathrm{sol}}=-(1/3)\left( 2k\right) ^{3/2}%
\text{,}  \label{sol}$$for the NLS solitons in the free space, given by Eq. (\[psi\]) with$$u_{\mathrm{sol}}(x)=\sqrt{2k}\,\mathrm{sech}\left( \sqrt{2k}x\right) ,
\label{soliton}$$which are displayed by dashed curves in Figs. \[bifurcation diagrams\](a-f).

The SSB in the families of trapped and leaky states is characterized by the asymmetry ratio,

$$\Theta \equiv N^{-1}\left[ \int_{0}^{+\infty }u^{2}(x)dx-\int_{-\infty
}^{0}u^{2}(x)dx\right]  \label{theta}$$

\[cf. a similar definition for the tails, given by Eq. (\[theta\_tail\])\], which is shown as a function of $N$ in Figs. \[bifurcation diagrams\](g-i). These plots clearly identify the SSB-onset points, 