We investigate the competition between nonlinearity and inhomogeneities in discrete-time quantum walks on one-dimensional lattices and their impact on transport and localization. Nonlinear effects are introduced through a Kerr-like, intensity-dependent local phase, while spatial and temporal inhomogeneities are implemented via random variations of the quantum gate operations. By analyzing the return probability and the participation function, we identify that distinct quantum walking regimes such as the nonlinear parameter $\chi$ and the quantum gate parameter ${\theta }_0$ are varied. Spatial inhomogeneities weaken nonlinear self-trapping and constrict the region of robust localization. In this process, partially localized regimes emerge, characterized by the coexistence of a confined core and dispersive wave-packet components. In contrast, temporal inhomogeneities act as time-dependent perturbations that disrupt phase coherence and suppress nonlinear self-trapping, promoting dispersive spreading and delocalization. By using $\chi$ versus ${\theta }_0$ diagrams, we display a comprehensive characterization of how inhomogeneities modify the stability and extent of the prevailing dynamical regimes, elucidating the competition between nonlinearity and inhomogeneities in discrete-time quantum walks.

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• Quantum walks
• Solitons
• Disordered systems
• Nonlinear waves