On-Site and Off-Site Bound States of the Discrete Nonlinear Schr\"odinger Equation and the Peierls-Nabarro Barrier

We construct multiple families of solitary standing waves of the discrete cubically nonlinear Schr\"{o}dinger 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 {\it on-site 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 {\it off-site 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 {\it Peierls-Nabarro} energy barrier.