×

Properties of some breather solutions of a nonlocal discrete NLS equation. (English) Zbl 1456.37081

Summary: 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 R. I. Ben et al. [Phys. Lett., A 379, No. 30–31, 1705–1714 (2015; Zbl 1343.35210)] 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 et al., Phys. Lett., A 379, No. 30–31, 1705–1714 (2015; Zbl 1343.35210)]. 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.

MSC:

37K60 Lattice dynamics; integrable lattice equations
39A36 Integrable difference and lattice equations; integrability tests
35Q55 NLS equations (nonlinear Schrödinger equations)
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
47J30 Variational methods involving nonlinear operators
78A60 Lasers, masers, optical bistability, nonlinear optics

Citations:

Zbl 1343.35210
PDFBibTeX XMLCite
Full Text: DOI