Quasi-steady state propagation in the Davydov-type model with linear on-site interactions

The problem of electron transportation along a discrete deformable medium with linear on-site interactions in the Davydov approach is considered. It is found that the quasi-stationary state of the full equations of motion leads to a discrete nonlocal nonlinear Schrödinger (DNNLS) equation whose nonlocality is of the exponential type and depending on the on-site parameter. We use the variational approach to approximate discrete traveling wave solutions in the DNNLS equation. We find that the discrete solutions continued from the discrete nonlinear Schrödinger equation, corresponding to the vanishing of the on-site parameter, bifurcates in a critical on-site value. Additionally, a threshold in the velocity of propagation of the discrete structures is found.