  The dynamics of nonlinear flip-flop quantum walk with amplitude-dependent
phase shifts with pertubing potential barrier is investigated. Through the
adjustment between uniform local perturbations and a Kerrlike nonlinearity of
the medium we find a rich set of dynamic profiles. We will show the existence
of different Hadamard quantum walking regimes, including those with mobile
soliton-like structures or self-trapped states. The latter is predominant for
perturbations with amplitudes that tend to $\varphi\rightarrow \pi/2$. In this
system, the qubit shows an unusual behavior as we increase the amplitudes of
the potential barriers, and displays a monotonic decrease in the self-trapping
$\varphi_c$ with respect to the nonlinear parameter. A chaotic-like regime
becomes predominant for intermediate nonlinearity values. Furthermore, we show
that by changing the quantum coins ($\theta$) a non-trivial dynamic arises,
where coins close to Pauli-X drives the system to a regime with predominant
soliton-like structures, while the chaotic behavior are restricted to a narrow
region in the $\chi$-$\varphi$ plane. We believe that is possible to implement
and observe the proprieties of this model in a integrated photonic system.
