Onsite and offsite bound states of the discrete nonlinear Schrödinger equation and the Peierls–Nabarro barrier

Abstract

We construct multiple families of solitary standing waves of the discrete cubically nonlinear Schrödinger equation (DNLS) in dimensions d = 1, 2 and 3. These states are obtained via a bifurcation analysis about the continuum (NLS) limit. One family consists of onsite symmetric (vertex-centered) states; these are spatially localized solitary standing waves which are symmetric about any fixed lattice site. The other spatially localized states are offsite symmetric. Depending on the spatial dimension, these may be bond-centered, cell-centered or face-centered. Finally, we show that the energy difference among distinct states of the same frequency is exponentially small with respect to a natural parameter. This provides a rigorous bound for the so-called Peierls–Nabarro energy barrier.