the original symmetric state and a new asymmetric one, as observed in Fig. \[unstable evolution\](a-e). Stronger instability causes complete symmetry breaking, replacing the symmetric state by an irregularly vibrating asymmetric mode, as seen in Fig. \[unstable evolution\](f).

It is relevant to note the difference between the oscillatory regimes generated by the instability of self-trapped and leaky modes. Indeed, while Fig. \[unstable evolution\](c) demonstrates that the shape of the oscillating mode is sharp in the former case, the shape is fuzzy in Fig. \[unstable evolution\](e) because it involves an essential radiation component in the case when the underlying unstable mode is a leaky one.

Collisions of free solitons with the quasi-double-well potential structure {#sec:CollisionResults}
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In addition to the analysis of the stationary states performed above, it is also relevant to consider collisions of free solitons with the elevated DWP structure. For this purpose, the initial soliton was created as the tilted (moving) version of the static one given by Eqs. (\[psi\]) and ([soliton]{}),$$\psi \left( x,z\right) =\sqrt{2k}\exp \left( i\left( k-c^{2}/2\right)
z+icx\right) \text{\textrm{sech}}\left( \sqrt{2k}\left( \left(
x-x_{0}\right) -cz\right) \right) ,  \label{tilt}$$where $c$ is a real tilt (velocity), and $x_{0}$ is the initial position of the soliton, chosen far enough from the localized potential 