Discrete Nonlinear Schrödinger Systems for Periodic Media with Nonlocal Nonlinearity: The Case of Nematic Liquid Crystals

Panayotis Panayotaros

doi:10.3390/app11104420

Abstract

We study properties of an infinite system of discrete nonlinear Schrödinger equations that is equivalent to a coupled Schrödinger-elliptic differential equation with periodic coefficients. The differential equation was derived as a model for laser beam propagation in optical waveguide arrays in a nematic liquid crystal substrate and can be relevant to related systems with nonlocal nonlinearities. The infinite system is obtained by expanding the relevant physical quantities in a Wannier function basis associated to a periodic Schrödinger operator appearing in the problem. We show that the model can describe stable beams, and we estimate the optical power at different length scales. The main result of the paper is the Hamiltonian structure of the infinite system, assuming that the Wannier functions are real. We also give an explicit construction of real Wannier functions, and examine translation invariance properties of the linear part of the system in the Wannier basis.

Highlights

We study properties of an infinite system of discrete nonlinear Schrödinger (DNLS)equations that is equivalent to a coupled Schrödinger-elliptic system of partial differential equations with periodic coefficients
We have examined some properties of a coupled Schrödinger-elliptic system modeling optical waveguide arrays in a nematic liquid crystal substrate
The system is studied by first passing to an equivalent infinite system of discrete describing the interaction of Wannier mode amplitudes of the relevant physical quantities [1]

Summary

Introduction

We study properties of an infinite system of discrete nonlinear Schrödinger (DNLS). equations that is equivalent to a coupled Schrödinger-elliptic system of partial differential equations with periodic coefficients. The Fratalocchi-Assanto equation has a nonlocal nonlinearity that leads to new effects when compared to the cubic power DNLS model studied commonly in photonics and atomic physics [7] These effects include non-monotonic amplitude profiles of static (breather) solutions, additional internal modes in the linearization around breathers [8,9], and enhanced mobility of traveling localized solutions [10]. This observation implies that the dispersion relation and the coupling between the modes can be computed with relative ease, and that the linear part of the problem is homogenized in the Wannier basis, i.e., is effectively a translation invariant [29,30] This latter property is an additional motivation for further developing Bloch-Wannier analysis in nonlinear wave equations.

Hamiltonian Structure of Periodic Nematicon Equations

Real Wannier Functions and Dispersive Properties

Findings

Discussion