Sixth-order Dispersion Research Articles

Dynamics of switching optical soliton in fiber with sixth order dispersion and inter modal dispersion

This work pertains to the study of switching soliton in fiber for nonlinear Schrödinger (NLS) equation with the presence of higher order dispersion and inter modal dispersion (IMD). The nonlinear wave in the optical fiber are described by the NLS equation which having the second order, fourth order, fifth order, sixth order dispersion, higher order nonlinearity and IMD. The main goal of the paper is to examine the sixth order dispersion on nonlinear wave in the fiber with the conditions of IMD. Hence, we employ the Bäcklund transformation of the Riccati equation (BTRE) approach to NLS equation and obtain the soliton solution. By the use of soliton solution along with graphical snapshots, we provide the conditions for forming switching solition in optical fiber and also analyze the effect of sixth order dispersion in fiber. The stability of the solution of NLS equation is also addressed.

Effect of weak nonlocal nonlinearity on generalized sixth-order dispersion modulational instability in optical media

Effect of weak nonlocal nonlinearity on generalized sixth-order dispersion modulational instability in optical media

Period-doubling bifurcation to chaos in a pure-sextic soliton fiber laser

We numerically investigate the dynamics of period-doubling bifurcation to chaos in a pure-sextic soliton (PSS) fiber laser dominated by sixth-order dispersion (SOD). By properly setting the cavity parameters, i.e., pump power and SOD amount, it is found that the PSS experiences multiple bifurcations before entering into chaos. The imperfect balance of SOD and self-phase modulation (SPM) provides a new scenario of period-doubling bifurcation dynamics that PSS exhibits the dynamical energy redistribution between the main lobe and the oscillating tails. In particular, a critical state that the PSS evolves spontaneously between the period-doubling bifurcation and chaos has also been demonstrated. These findings will give new insights into the period-doubling bifurcation dynamics as well as chaos in dissipative optical systems.

Exorcising the ghost condensate dark energy with a sextic dispersion relation

The universe's current acceleration is a pretty recent phenomenon in cosmological time scales. This means that the modes that have left our horizon since the beginning of the contemporary acceleration phase, have not really reached the exact IR limit. Noting this observation, we reconsider the possibility of having a ghost condensate as dark energy with a sixth-order dispersion relation. Looking at the three-point function of such a theory, we obtain the constraints on the coefficient of the sixth-order dispersion relation to avoid strong coupling. We notice that such a constraint can be easily satisfied, without fine-tuning. Such a ghost condensate, if coupled to the standard model fields, induces a constant Lorentz-violating spin-dependent force, which can gravitate or anti-gravitate. The Newtonian potential is modified with an oscillatory behavior at distance scale MPl/M2 at the time scale MPl2/M3, where M2 is the expectation value of the ghost field velocity, i.e. 〈ϕ˙〉=M2.

Generalized sixth-order dispersion solitons

Encouraged by recent experimental breakthroughs, we investigate the types of bright solitons that can exist in the presence of a Kerr nonlinearity and quadratic, quartic, and sextic dispersion by examining the nature of the soliton tails. We verify the soliton shape by full numerical calculations and find analytic solutions for some parameters. We reveal the sensitivity of the interaction dynamics to the particular choice of the dispersion parameters. These results provide a framework for experiments in this regime and for exploring parameter spaces with even higher dimensions.

Effects ofCPT-odd terms of dimensions three and five on electromagnetic propagation in continuous matter

In this work we study how CPT-odd Maxwell-Carroll-Field-Jackiw (MCFJ) electrodynamics as well as a dimension-5 extension of it affect the optical activity of continuous media. The starting point is dimension-3 MCFJ electrodynamics in matter whose modified Maxwell equations, permittivity tensor, and dispersion relations are recapitulated. Corresponding refractive indices are achieved in terms of the frequency and the vector-valued background field. For a purely timelike background, the refractive indices are real. Their associated propagation modes are circularly polarized and exhibit birefringence. For a purely spacelike background, one refractive index is always real and the other can be complex. The circularly polarized propagating modes may exhibit birefringence and dichroism (associated with absorption). Subsequently, we examine a dimension-five MCFJ-type electrodynamics, previously scrutinized in the literature, in a continuous medium. Following the same procedure, we find the refractive indices from a sixth-order dispersion equation. For a purely timelike background, three distinct refractive indices are obtained, one of them being real and two being complex. They are associated with two circularly polarized propagating modes that exhibit birefringence or dichroism, depending on the frequency range. Scenarios of propagation and absorption analogous to those found in dispersive dielectrics are also observed for spacelike background configurations. We conclude by comparing the dimension-three and five results and by emphasizing the richer phenomenology of the propagating modes in the higher-derivative model. Our results are applicable in the realm of Weyl semimetals.

Higher-order dispersion and nonlinear effects of optical fibers under septic self-steepening and self-frequency shift.

We investigate the modulational instability (MI) of a continuous wave (cw) under the combined effects of higher-order dispersions, self steepening and self-frequency shift, cubic, quintic, and septic nonlinearities. Using Maxwell's theory, an extended nonlinear Schrödinger equation is derived. The linear stability analysis of the cw solution is employed to extract an expression for the MI gain, and we point out its sensitivity to both higher-order dispersions and nonlinear terms. In particular, we insist on the balance between the sixth-order dispersion and nonlinearity, septic self-steepening, and the septic self-frequency shift terms. Additionally, the linear stability analysis of cw is confronted with the stability conditions for solitons. Different combinations of the dispersion parameters are proposed that support the stability of solitons and the occurrence of MI. This is confronted with full numerical simulations where the input cw gives rise to a broad range of behaviors, mainly related to nonlinear patterns formation. Interestingly, under the activation of MI, a suitable balance between the sixth-order dispersion and the septic self-frequency shift term is found to highly influence the propagation direction of the optical wave patterns.

Modulational instability in nonlinear doped optical fiber induced by the cubic–quintic–septic complex Ginzburg–Landau equation with higher-order dispersions

Considering the theory of electromagnetic waves, from the Maxwell’s equations, we introduce an extended ( 1 + 1 ) D complex Ginzburg–Landau equation with third-, fourth-, fifth-, and sixth-order dispersions and the cubic, quintic, and septic nonlinear terms, describing the dynamics of extremely short pulses in nonlinear doped optical fibers. Furthermore, we study the modulational instability of the plane wave theoretically using the linear stability analysis, via the gain spectrum. The linear theory predicts instability for any amplitude of the primary wave, where not only the higher-order dispersion terms highly influence the occurrence of modulational instability but also the combined effect between dispersion and higher-order nonlinear terms modifies the gain spectrum. Mainly, importance is paid to the competition between sixth-order dispersion and septic non-Kerr nonlinearities. This gives rise to new spectra of instability and therefore confirms the validity of both the proposed model and subsequent modulational instability manifestation.

Influence of septic nonlinearity on modulation instability under normal and anomalous dispersion regime

In this paper, we presented a detailed investigation on modulational instability (MI) of the higher order nonlinear Schrödinger equation with non-Kerr nonlinearities up to septimal nonlinearities to describe the propagation of attosecond pulses in a dispersive medium. In this study we demonstrate the impact of septic nonlinearity and sixth order dispersion on MI under normal and anomalous dispersion regimes. Investigation shows that MI gain and bandwidth is sensitive to septic nonlinearity and sixth order dispersion in both dispersion regimes. It is observed that at quantitative level the combined effect of fourth order dispersion parameter and sixth order dispersion plays significant role in the generation of nonconventional side bands. Furthermore, the gain control by the septic nonlinearity is clearly observed under different cases of higher order dispersion parameters.

Robust propagation of optical vortex beams, necklace-ring solitons, soliton clusters and uniform-ring beams generated in the frame of the higher-order (3 + 1)-dimensional cubic–quintic–septic complex Ginzburg–Landau equation

We have demonstrated numerically that new families of spatiotemporal dissipative optical bullets, including self-trapped, necklace-ring, ring-vortex solitons, uniform-ring beams, spherical and rhombic distributions of light bullets, and fundamental and cluster solitons, are possible in the higher-order (3 + 1)-dimensional cubic–quintic–septic complex Ginzburg–Landau equation with higher-order effects such as stimulated Raman scattering, self-steepening and third-, fourth-, fifth- and sixth-order dispersion terms. These solutions remain localized and can be self-trapped over a huge propagation distance, even in the presence of random perturbations, due to the combined influence of dispersion, diffraction, gain, loss, spectral filtering, the Raman effect and cubic–quintic–septic nonlinearities.

Rogue waves in higher-order systems: Lagrangian approach

A Lagrangian approach for finding rogue wave solutions of the extended nonlinear Schrödinger equation system is developed. A list of Lagrangian components for a variety of terms in the extended equation is presented. It turns the analysis of any equation into a simple and straightforward exercise. The fact that these terms can be summed is a major point in achieving this simplicity. Importantly, the technique provides the same result, no matter whether an extension term enters the basic part of the equation or is taken as a perturbation. This significant conclusion is demonstrated by giving several examples. We give several examples of the application of this technique for particular physically-relevant extended equations. As a typical example, the Lagrangian approach is used to evaluate the effects of fourth and sixth order dispersion on rogue waves. The results are compared with solutions found by numerical simulations of the original equation. Good agreements are found, thus confirming the usefulness of the approach. A generalization to even higher-order dispersive terms shows that the phase function for the rogue wave can be written in terms of known functions, e.g. the arcsinh function. When these equations are integrable, the solutions naturally coincide with the exact rogue wave solutions. Obtaining the exact solution this way is demonstrated in the case of Kundu–Echkaus equation.

Conservation laws, modulation instability and solitons interactions for a nonlinear Schrödinger equation with the sextic operators in an optical fiber

Under investigation in this paper is a sextic nonlinear Schrodinger equation, which describes the pulses propagating along an optical fiber. Based on the symbolic computation, Lax pair and infinitely-many conservation laws are derived. Via the modiied Hirota method, bilinear forms and multi-soliton solutions are obtained. Propagation and interactions of the solitons are illustrated graphically: Initial position and velocity of the soliton are related to the coefficient of the sixth-order dispersion, while the amplitude of the soliton is not affected by it. Head-on, overtaking and oscillating interactions between the two solitons are displayed. Through the asymptotic analysis, interaction between the two solitons is proved to be elastic. Based on the linear stability analysis, the modulation instability condition for the soliton solutions is obtained.

Integrability and solitons for the higher-order nonlinear Schrödinger equation with space-dependent coefficients in an optical fiber

Under investigation in this paper is a higher-order nonlinear Schrodinger equation with space-dependent coefficients, related to an optical fiber. Based on the self-similarity transformation and Hirota method, related to the integrability, the N-th-order bright and dark soliton solutions are derived under certain constraints. It is revealed that the velocities and trajectories of the solitons are both affected by the coefficient of the sixth-order dispersion term while the amplitudes of the solitons are determined by the gain function. Amplitudes increase when the gain function is positive and decrease when the gain function is negative. Furthermore, we find that the intensities of dark solitons are presented as a superposition of the solitons and stationary waves.

Vanishing characteristic speeds and critical dispersive points in nonlinear interfacial wave problems

Criticality plays a central role in the study of reductions and stability of hydrodynamical systems. At critical points, it is often the case that nonlinear reductions with dispersion arise to govern solution behavior. By considering when such models become bidirectional and lose their initial dispersive properties, it will be shown that higher order dispersive models may be supported in hydrodynamical systems. Precisely, this equation is a two-way Boussinesq equation with sixth order dispersion. The case of two layered shallow water is considered to illustrate this, and it is reasoned why such an environment is natural for such a system to emerge. Further, it is demonstrated that the regions in the parameter space for nontrivial flow, which admit this reduction, are vast and in fact form a continuum. The reduced model is then numerically simulated to illustrate how the two-way and higher dispersive properties suggest more exotic families of solitary wave solutions can emerge in stratified flows.

Numerical Simulation of Highly-Nonlinear Dispersion-Shifted Fiber Optical Parametric Gain Spectrum with Fiber Loss and Higher-Order Dispersion Coefficients

The previous study investigated the fiber parameters on the analytical one-pump fiber optical parametric amplifier (FOPA) gain spectrum of a loss-free highly-nonlinear dispersion-shifted fiber (HNL-DSF) and got the optimum results for each parameter. However, the FOPA gain of the combination of all the optimum values of the considered parameters was not reported. Hence, this paper intends to investigate the analytical FOPA gain of the combination of all the optimum values of the considered parameters from the previous study. Later, the analytical gain was compared with the numerical gain from the fourth-order Runge-Kutta (RK4) method and Matlab built-in function, ode45. Next, the effect of fiber loss and higher order dispersion coefficients such as fourth and sixth-order dispersion coefficients were studied. It is found that RK4 gives a smaller error and the gain reduces while the bandwidth remains same in presence of fiber loss. The fourth-order dispersion coefficient broadens the bandwidth a bit while maintaining the gain and there are two narrow-band gains generated to the left and right-side of the broad-band gain. The sixth-order dispersion coefficient just shifts the two narrow-band gains toward or away from the broadband gain depending on the positive or negative signs of the sixth-order dispersion coefficient.

Solitons for a generalized sixth-order variable-coefficient nonlinear Schrödinger equation for the attosecond pulses in an optical fiber

In this paper, under investigation is a sixth-order variable-coefficient nonlinear Schrödinger equation, which could describe the attosecond pulses in an optical fiber. Based on the self-similarity transformation and Hirota method, one- and two-soliton solutions are obtained under certain constraints. Investigation shows that the velocities and shapes of the solitons and bound solitons are both affected by the sixth-order dispersion term, and the maximum intensities of the solitons and bound solitons increase when the gain function is positive and decrease when the gain function is negative, otherwise the periodicity of the bound solitons is destroyed when the gain function is not 0.

Optimizing bandwidth of fiber optical parametric amplifier with different combinations of higher-order dispersion coefficient signs

This paper investigates the influence of different combinations of second-order (β2 ), fourth-order (β4 ) and sixth-order (β6 ) dispersion coefficient signs (negativity/positivity) to the fiber optical parametric amplifier gain performance. The numerical simulation has exploited the fourth-order Runge-Kutta method to solve the coupled amplitude equations which basically represent the parametric process of four-wave mixing. In the normal regime at which β2 is positive, the fiber optical parametric amplifier exhibits a poor gain spectrum and it is well performed once the pump is presumed to be in anomalous regime i.e. β2 is negative. The effect of β4 is significant when the signal wavelength is further from the pump wavelength and a wide amplification bandwidth is produced when the fiber have positive β4 in the anomalous regime. As for β6 , unfortunately in this simulation its influences are hardly can be seen probably because of its small value regardless its sign, but, from phase-mismatch equation it is shown that its impact is prominent when the signal is positioned way far from the pump. All in all, this shown that the signs combination of the higher-order dispersion coefficients are significant in order to optimize the FOPA amplification bandwidth.

Energy Preserving Finite Difference Scheme for Sixth Order Boussinesq Equation

We study the general Boussinesq equation with a sixth order dispersion term. For the numerical solution of this problem a conservative, nonlinear finite difference scheme is constructed. The scheme is of second order of approximation. The implementation of the scheme is based on an appropriate splitting. The numerical experiments show that this scheme has second order of convergence in space and time.

Soliton and Four-Wave Mixing Effects Induced by the Third-Order Dispersion in a Photonic Crystal Fiber With Femtosecond Pulses Pumping at Normal-Dispersion Regime

With the homemade solid-core silica-based photonic crystal fiber (PCF), we observe the generation of soliton and four-wave mixing (FWM) when the femtosecond pulse is pumped at 0.822 $\mu\hbox{m} $ , belonging to the normal-dispersion regime. It is generally acknowledged that FWM is unrelated to third-order dispersion, whereas we use the fourth-order Runge–Kutta method to simulate the PCF properties by taking into account the third-order dispersion or not; the result we got is different from the conventional one. The numerical simulations are greatly consistent with the experiment results, and the FWM is dominated by negative fourth-order and negative sixth-order dispersion in the experiment. The main dynamic process is explained as the soliton, and the FWM generation can be induced by third-order dispersion. We indicate that low Raman threshold has played a very active role in this process, and we propose a conception of stimulated FWM effect in this paper.

A viscous–convective instability in laminar Keplerian thindiscs – II. Anelastic approximation

Using the anelastic approximation of linearised hydrodynamic equations, we investigate the development of axially symmetric small perturbations in thin Keplerian discs. The sixth-order dispersion equation is derived and numerically solved for different values of relevant physical parameters (viscosity, heat conductivity, disc semi-thickness and vertical structure). The analysis reveals the appearance of two overstable modes which split out from the classical Rayleigh inertial modes in a wide range of the parameters in both ionized and neutral gases. These modes have a viscous-convective nature and can serve as a seed for turbulence in astrophysical discs even in the absence of magnetic fields.