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

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### Abstract

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.

### Author and article information

###### Journal
15 May 2014
2015-08-03
###### Article
1405.3892

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

###### Custom metadata
to appear in Nonlinearity
nlin.PS cond-mat.mes-hall math-ph math.AP math.MP