structure (we here take $x_{0}=-15$). Generic findings are produced here for incident solitons with $k=3,$ the corresponding norm being $$N=2\sqrt{2k}\approx 4.90,  \label{N}$$according to Eq. (\[sol\]), other values of $N$ giving similar results.

Figure \[ThetaVsHeightMapping\] presents a parameter chart for three different outcomes of the collisions, produced by varying the height of the outer barriers, $H_{0}$ in Eq. (\[P2\]), and tilt $c$ in Eq. (\[tilt\]). In the region designated as (1), i.e., for $c$ small and/or $H_{0}$ large enough, the incident soliton bounces back the left outer barrier, as shown in Fig. \[FundCollision1\] for $H_{0}=0.6$ and $c=0.8$. At larger $c$, in a relatively narrow region (2) of Fig. \[ThetaVsHeightMapping\], the soliton gets captured inside the potential structure, which resembles the previously explored possibility of capturing an incident Bragg-grating soliton by a cavity formed by two locally repulsive defects [@cavity]. Similar to that setting, the trapped soliton performs shuttle motion between the outer and inner potential barriers, as shown in Fig. \[FundCollision2\], for $H_{0}=0.6$ and $c=1.15$. The shuttle dynamical regime is an essential addition to the stationary states revealed by the analysis presented above. At still larger $c$, but yet staying within the boundaries of the narrow region (2), the shuttle motion 