  We study the existence and charaterization of self-trapping phenomena in
discrete-time quantum walks. By considering a Kerr-like nonlinearity, we
associate an acquisition of the intensity-dependent phase to the walker while
it propagates on the lattice. Adjusting the nonlinear parameter ($\chi$) and
the quantum gates ($\theta$), we will show the existence of different quantum
walking regimes, including those with travelling soliton-like structures or
localized by self-trapping. This latter scenario is absent for quantum gates
close enough to Pauli-X. It appears for intermediate configurations and becomes
predominant as quantum gates get closer to Pauli-Z. By using $\chi$ versus
$\theta$ diagrams, we will show that the threshold between quantum walks with
delocalized or localized regimes exhibit an unusual aspect, in which an
increment on the nonlinear strength can induce the system from localized to a
delocalized regime.
