Peierls-Nabarro potential profile of discrete nonlinear Schrödinger equation

We derive an analytic formula for the Peierls-Nabarro (PN) potential of a discrete nonlinear Schro¨dinger equation with cubic focusing nonlinearity. The potential is given in terms of the ratio between the strength of nonlinearity and dispersion spanning the whole range between strongly localised to smoothly moving solitons. The essential features that characterize the on-site and inter-site stationary states are obtained. The soliton's center-of-mass dynamics is also described. The derivation is performed variationaly using two trial functions with good agreement with the exact numerical calculation.