  Recent studies have emphasized the important role that a shape deformability
of scalar-field models pertaining to the same class with the standard $\phi^4$
field, can play in controlling the production of a specific type of breathing
bound states so-called oscillons. In the context of cosmology, the built-in
mechanism of oscillons suggests that they can affect the standard picture of
scalar ultra-light dark matter. In the present work kink scatterings are
investigated in a parametrized model of bistable system admitting the classical
$\phi^4$ field as an asymptotic limit, with focus on the formation of
long-lived low-amplitude almost harmonic oscillations of the scalar field
around a vacuum. The parametrized model is characterized by a double-well
potential with a shape-deformation parameter that changes only the steepness of
the potential walls, and hence the flatness of the hump of the potential
barrier, leaving unaffected the two degenerate minima and the barrier height.
It is found that the variation of the deformability parameter promotes several
additional vibrational modes in the kink-phonon scattering potential, leading
to suppression of the two-bounce windows in kink-antikink scatterings and the
production of oscillons. Numerical results suggest that the anharmonicity of
the potential barrier, characterized by a flat barrier hump, is the main
determinant factor for the production of oscillons in double-well systems.
