Mobile localized solutions for an electron in lattices with dispersive and non-dispersive phonons

•An electron interacts both with on-site and with longitudinal phonons.•We provide existence conditions for mobile localized electron excitations.•We discuss discrete and continuum approaches to lattice dynamics.•Localized states can be stable if they have a non-zero velocity.

We consider a one dimensional lattice in which an electron can interact both with on-site non-dispersive (Einstein) phonons and with longitudinal dispersive acoustic (Debye) phonons. We provide existence conditions for mobile localized electron excitations in the long wave limit. The role of both types of phonon modes on localization is also assessed, together with a discussion of differences existing between the discrete and the continuum approaches. A striking result is that, under certain conditions, localized states can only be stable if they have a non-zero velocity.