# Dark Solitons and Grey Solitons in Waveguide Arrays with Long-Range Linear Coupling Effects
## 1 Introduction  
Nonlinear discrete systems attract considerable attention in many branches of physics, and exhibit various physical characteristics [1; 2; 3]. In the fields of optics [4; 5; 6] and Bose-Einstein condensates (BECs) [7; 8; 9], the evolution of nonlinear waves in discrete systems is a popular topic.  
The basic model of a nonlinear discrete system in optics is an array of evanescently-coupled waveguides consisting of nonlinear materials. In a waveguide array, the propagation of light is primarily characterized by the coupling caused by the overlap between the fundamental modes of nearest-neighbouring waveguides. A crucial issue is to study the formation and properties of discrete solitons (DSs) in such nonlinear waveguide systems. DS formation is the result of a balance between on-site nonlinearity and the discrete diffraction induced by linear coupling among adjacent waveguides or lattice sites. DSs show strong potential for application in all-optical data processing; their most attractive feature is that DSs can enable intelligent functional operations such as routing, blocking, logic functions, and time gating in many all-optical devices [10].  
An interesting extension is to investigate the formation of solitons in the presence of nonlocal effects. [PERSON] and colleagues studied the shock waves and dark solitons in nonlocal nonlinear media [11; 12]. A next-nearest neighbour (NNN) model in which the linear coupling matrix becomes a quadruple-diagonal matrix after higher-order diffraction was studied by [PERSON] and colleagues [13]. In 2012, [PERSON] and colleagues conducted significant studies of the nonlinear dipolar field in a nanoparticle train [14; 15; 16]--which can be viewed as a discrete nonlinear system--and reported that a linear coupling effect can exist among all lattice sites because of long-range dipole-dipole interactions. This system can produce all non-zero off-diagonal elements in the linear coupling matrix.  
Long-range coupling in waveguide arrays is the coupling between the waveguides that were spaced with certain distance--it can affect the propagation dynamics of light field. Long-range coupling is different from that of the conventional waveguide arrays, which only consider the coupling between the adjacent waveguides (i.e., the short-range coupling). We have designed such a waveguide array with long-range coupling and studied the formation of bright solitons--see our previous paper in [17]. In this work, we further study the dynamics of dark and grey solitons in such a waveguide array, which was not considered before. The numerical results show that: regarding the dark solitons, the stability and width strongly depend on the mean power and the normalized decay length, which describes the effective length of the coupling effects in the waveguides; regarding the grey solitons, the stability and grey level are determined by the mean power and the normalized decay length.  
The remainder of this paper is organized as follows. We provide a brief description of the model and basic equations in Section 2. Then, we study dark solitons in Section 3 and the grey solitons in Section 4. The paper is concluded in Section 5.