decrease of $|k|$), even if each tail’s norm vanishes in the limit of $k$$\rightarrow 0$ (when the transition to the self-trapped mode takes place, and the tails vanish). This feature is illustrated by Fig. \[tails\](c), which displays the asymmetry measure vs. $k$:$$\theta (k)\equiv \frac{\int_{\Lambda /2}^{L/2}u^{2}(x;k)dx-\int_{-\Lambda
/2}^{-L/2}u^{2}(x;k)dx}{\int_{\Lambda /2}^{L/2}u^{2}(x;k)dx+\int_{-\Lambda
/2}^{-L/2}u^{2}(x;k)dx}~.  \label{theta_tail}$$Second, the dependence of the tails’ norms on $k$ shows strong oscillations, which is explained by the commensurability-incommensurability transitions between the wavelength of the radiation tail and the total size of the free-space domains, $L/2-\Lambda /2$. Indeed, the radiation wavenumber given by the free-space dispersion relation for linearized equation ([stationaryNLSE]{}), $q=\sqrt{-2k}$, determines the the radiation half-wavelength, $\pi /q$, which, in the case of the commensurability, satisfies relation $(\pi /q)n=L/2-\Lambda /2$, with $n=1,2,3,...$ . Thus, maxima of the radiation amplitude are expected at discrete values of the propagation constant,  $$k_{n}=-2\left[ \pi n/\left( L-\Lambda \right) \right] ^{2}.  \label{k}$$As shown in Fig. \[tails\](b), Eq. (\[k\]) quite accurately predicts positions of the tail-norm peaks for $n=2,3,4,5,6,7,$ and $8$, for $%
L-\Lambda =114$, which corresponds to the present case \[$n=1$ yields $%
k_{1}\approx -1.5\times 10^{-3}$, for which the tail’s amplitudes are too small to discern the corresponding maximum, while Eq. (\[k\]) with $n=9$ predicts $k_{9}\approx -0.123$, for 