Onsite and offsite bound states of the discrete nonlinear Schrödinger equation and the Peierls–Nabarro barrier. (22nd December 2015)

• Record Type: Journal Article
• Authors: Jenkinson, M; Weinstein, M I
• Is Part Of: Nonlinearity. Volume 29:Number 1(2016:Jan.)
• Page Start: 27
• Page End: 86
• Publication Date: 2015-12-22
• Subjects: soliton -- Peierls–Nabarro energy barrier -- discrete nonlinear lattice; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/29/1/27
• Languages: English
• ISSNs: 0951-7715

Abstract

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.