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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.

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          Author and article information

          Journal
          15 May 2014
          2015-08-03
          Article
          1405.3892

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

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

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