Properties of some breather solutions of a nonlocal discrete NLS equation

We present results on breather solutions of a discrete nonlinear Schrödinger equation with a cubic Hartree-type nonlinearity that models laser light propagation in waveguide arrays that use a nematic liquid crystal substratum. A recent study of that model by Ben et al [R.I. Ben, L. Cisneros Ake, A.A. Minzoni, and P. Panayotaros, Phys. Lett. A, 379:1705C-1714, 2015] showed that nonlocality leads to some novel properties such as the existence of orbitaly stable breathers with internal modes, and of shelf-like configurations with maxima at the interface. In this work we present rigorous results on these phenomena and consider some more general solutions. First, we study energy minimizing breathers, showing existence as well as symmetry and monotonicity properties. We also prove results on the spectrum of the linearization around one-peak breathers, solutions that are expected to coincide with minimizers in the regime of small linear intersite coupling. A second set of results concerns shelf-type breather solutions that may be thought of as limits of solutions examined in [R.I. Ben, L. Cisneros Ake, A.A. Minzoni, and P. Panayotaros, Phys. Lett. A, 379:1705C-1714, 2015]. We show the existence of solutions with a non-monotonic front-like shape and justify computations of the essential spectrum of the linearization around these solutions in the local and nonlocal cases.

Keywords

discrete nonlinear Schrödinger equations, nonlocal effects, localized solutions, breather solutions, linear stability

2010 Mathematics Subject Classification

35Q55, 37K60, 47J10, 47J30, 78A60

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The authors acknowledge partial support from grant “Ecuaciones diferenciales dispersivas en aplicaciones tecnológicas”, Misiones al Exterior, Ministerio de la Nación, Republica de Argentina. P.P. acknowledges partial support from grant SEP-Conacyt 177241.

Received 23 May 2016

Accepted 23 July 2017

Published 20 December 2017