Cubic-quintic nonlinear Helmholtz equation: Modulational instability, chirped elliptic and solitary waves

Abstract

We study the formation and propagation of chirped elliptic and solitary waves in cubic-quintic nonlinear Helmholtz (CQNLH) equation. This system describes nonparaxial pulse propagation in a planar waveguide with Kerr-like and quintic nonlinearities along with spatial dispersion originating from the nonparaxial effect that becomes dominant when the conventional slowly varying envelope approximation (SVEA) fails. We first carry out the modulational instability (MI) analysis of a plane wave in this system by employing the linear stability analysis and investigate the influence of different physical parameters on the MI gain spectra. In particular, we show the nonparaxial parameter suppresses the conventional MI gain spectrum and also leads to a nontrivial monotonic increase in the gain spectrum near the tails of the conventional MI band, a qualitatively distinct behaviour from the standard nonlinear Schrödinger (NLS) system. We then study the MI dynamics by direct numerical simulations which demonstrate production of ultra-short nonparaxial pulse trains with internal oscillations and slight distortions at the wings. Following the MI dynamics, we obtain exact elliptic and solitary wave solutions using the integration method by considering physically interesting chirped traveling wave ansatz. In particular, we show that the system features intriguing chirped anti-dark, bright, gray and dark solitary waves depending upon the nature of nonlinearities. We also show that the chirping is inversely proportional to the intensity of the optical wave. Especially, the bright and dark solitary waves exhibit unusual chirping behaviour which will have applications in nonlinear pulse compression process.Nonlinear Helmholtz (NLH) equation can play a significant role in modeling a progressive miniaturization of photonic devices and plasmonics. By obeying the intrinsic higher dimensions of spatial symmetry of uniform planar waveguides, many experimental configurations could be modeled that are otherwise inaccessible in the standard SVEA. In the present work, we consider the NLH equation with cubic-quintic (CQ) nonlinearities. The implication of non-Kerr nonlinearity in the NLH system leads to a more stabilized nonparaxial pulse propagation and it can easily be realized in available materials including semiconductors and doped fibers. We study the phenomenon of MI in the CQNLH system, where we address some peculiar features of nonparaxiality. Also, we numerically demonstrate the generation of train of ultra-short nonparaxial solitary pulses. In addition to different types of Helmholtz solitons, we present the first ever study of formation and propagation of chirped elliptic and solitary waves, which are not reported for any NLH system before. In particular, we obtain various interesting chirped solitary profiles, which include bright, dark, gray and anti-dark solitary waves depending upon the nature of nonlinearities. Among them, we find chirped profiles of bright and dark solitary waves exhibit some unusual nonlinear dynamics of compr...

Modulation instability analysis of a nonautonomous $$(3+1)$$-dimensional coupled nonlinear Schrödinger equation

“…The steady-state solution becomes unstable whenever Λ(t) has an imaginary part because, in this case, perturbation grows exponentially along the fiber length. This phenomenon is called modulation instability [4,31,32] as it leads to modulation of the steady-state solution. From Eq.…”

“…In [18] a detailed study on different physical models has been conducted. Many researchers have used the MI analysis for various models such as the nonautonomous Lenells-Fokas model [19], a variable-coefficient nonlinear Schrödinger equation with fourth-order effects [20], an integrable coupled nonlinear Schrödinger system [21], 2D quantum ultracold atoms [22], a deformed Fokas-Lenells equation [23], the coupled derivative NLS equation [24], Coupled nonlinear Schrödinger equation [25], linearly coupled complex cubic quintic Ginzburg Landau equations [26,27], the linearly-coupled NLS equations [28], an inhomogeneous NLS equation including a pseudo-stimulated-Raman-scattering term [29], perturbed nonlinear Schrödinger-Hirota equation [30], Mel'nikov system [31], Cubic-quintic nonlinear Helmholtz equation [32], coupled Zakharov-Kuznetsov [33], coupled generalized NLS equations [34] and many other equations from [35]. In 2017, Mustafa et al [36] investigated the MI analysis of the (1+1)dimensional coupled NLS equation with constant coefficients.…”

Non-Lagrangian approach for coupled complex Ginzburg-Landau systems with higher order-dispersion

“…In the context of nonlinear fiber optics, there exist a few analytical methods to study the dynamics of optical pulses. If we consider the effects of optical losses, and higher order dispersion including non-paraxial limit, the non-integrable nature tend to make them hard to study analytically and one has to seek the aid of relevant numerical methods [53]. Nevertheless, a non-Lagrangian method, namely collective variable approach (CVA) can be employed to any perturbative NLSE as it does not require any conserved energy to reduce the governing equations [54][55][56].…”

Modulation Instability Analysis of a Nonautonomous (3+1)-Dimensional Coupled Nonlinear Schrödinger Equation

“…In [18] a detailed study on different physical models has been conducted. Many researchers have used the MI analysis for various models such as the nonautonomous Lenells-Fokas model [19], a variable-coefficient nonlinear Schrödinger equation with fourth-order effects [20], an integrable coupled nonlinear Schrödinger system [21], 2D quantum ultracold atoms [22], a deformed Fokas-Lenells equation [23], the coupled derivative NLS equation [24], Coupled nonlinear Schrödinger equation [25], linearly coupled complex cubic quintic Ginzburg Landau equations [26,27], the linearly-coupled NLS equations [28], an inhomogeneous NLS equation including a pseudo-stimulated-Raman-scattering term [29], perturbed nonlinear Schrödinger-Hirota equation [30], Mel'nikov system [31], Cubic-quintic nonlinear Helmholtz equation [32], coupled Zakharov-Kuznetsov [33], coupled generalized NLS equations [34] and many other equations from [35]. In 2017, Mustafa et al [36] investigated the MI analysis of the (1+1)dimensional coupled NLS equation with constant coefficients.…”