Modulational Instability and Discrete Localized Modes in Two Coupled Atomic Chains with Next-Nearest-Neighbor Interactions

Abstract

A pair of one dimensional atomic chains which are coupled via the Klein-Gordon potential is considered in this study, with each chain experiencing both nearest and next-nearest-neighbor interactions. The discrete nonlinear Schrödinger amplitude equation with next-nearest-neighbor interactions is thus derived from the out-phase equation of motion of the coupled chains. This is achieved by using the rotating wave approximations perturbation method, in which both the carrier wave and envelope are explicitly treated in the discrete regime. It is shown that the next-nearest-neighbor interactions greatly modifies the region of observation of modulational instability in the atomic chain. By exploring the discrete Hirota-Bilinear method, we obtain the discrete one-soliton solution which is localized around the origin and structurally stable because it conserves it form as time evolves. However when the atomic chain is purely subjected to a symmetric coupling potential, we observe a structurally unstable discrete excitation that changes into an up-and-down asymmetric localized modes; both in the presence and absence of next-nearest-neighbor interactions. Results of numerical simulations clearly depicts the long term evolution of these discrete nonlinear excitations, that evolve from symmetric to asymmetric localized modes in the atomic chain.

Nonlinear localized wave modes in optomechanical array

“…Investigation of nonlinear wave in the structures where nonlinearity and dispersion terms are competing has growing this last decade. Many mathematical model have been derived to carry out this matter [1][2][3][4][5][6][7]. In Cavity Optomechanics (COM), some relevant works have been pointed out concerning solitonic waves [8,9].…”

“…It is obvious that MI can induce unstable/stable modes as well as localized waves for a given wave vector or any other parameter of the system due to its effects. In nonlinear structure, where nonlinear and dispersion terms interact, MI is crucial for the process of generating the localized wave patterns such as dark soliton, bright soliton, rogue wave and breather [2,4,12]. MI has been carried out in several structures such as nonlinear optics [5], and fluid dynamics [6].…”

Discrete modulation instability and localized waves of the coupled semi-discrete Hirota equation

“…In [11], the authors have depicted the chaos-like motion and unstable modes incited by the self-modulated Kerr nonlinearity in optomechanical arrays. The discrete nonlinear Schrödinger equation (DNLSE) is the most well-known discrete model during MI analysis, and localized modes [1][2][3][4][5][6][7][8][9][10][11]. For example, in [25] the authors used the discrete cubic-quintic NLSE to show the effects of the nonlinear terms on the train of waves propagating in the forbidden gap (FG).…”