Considering the higher-order nonlinearities in a material can significantly change its behavior. We suggest the extended nonlinear Schr\"odinger equation to describe the propagation of ultrashort optical pulses through a dispersive medium with higher-order nonlinearities. Soliton trains are generated through the modulational instability and we point out the influence of the septic nonlinearity in the modulational instability gain. Experimental values are used for the numerical simulations and the input plane wave leads to the development of pulse trains, depending upon the sign of the septic nonlinearity.

We have investigated the modulation instability (MI) in a nonlinear optical coupler using a generalized model describing the pulse propagation of a waveguiding structure composed of two adjacent waveguides. The model consists of a nonlinear tunnel-coupled structure consisting of right- and left-handed media. The optical coupler considered here includes a local saturable nonlinear refractive index. In particular, we discuss the impact of the saturable nonlinearity for the MI of plane waves and formation of spatial solitons. The results show that MI can exist not only in the normal group velocity dispersion (GVD) regime but also in the normal GVD regime in the nonlinear positive-negative index coupler in the presence of saturable nonlinearity. The saturable nonlinearity can increase/decrease the number of sidebands or shift the existing sidebands. The maximum value of the MI gain, as well as its bandwidth, has been also affected by the saturable nonlinearity. Moreover, the saturable nonlinearity may influence considerably the number of wave trains induced by MI.

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.

We consider a higher-order complex Ginzburg-Landau equation, with the fourth-order dispersion and cubic-quintic nonlinear terms, which can describe the propagation of an ultrashort subpicosecond or femtosecond optical pulse in an optical fiber system. We investigate the modulational instability (MI) of continuous wave solution of this equation. Several types of modulational instability gains are shown to exist in both the anomalous and normal dispersion regimes. We find that depending on the sign of the fourth-order dispersion coefficient, the MI appears for normal and anomalous dispersion regime. Simulations of the full system demonstrate that the development of the MI leads to establishment of a regular or chaotic array of pulses, a chain of well-separated peaks with continuously growing or decaying amplitudes depending on the sign of the loss/gain coefficient and higher-order dispersions terms. Comparison of the calculations with reported numerical results shows a satisfactory agreement.

A three-dimensional Thomas–Fermi dense anisotropic magnetized plasma having Fermi–Dirac distributed ions and electrons as well as classical fluid negative dust impurities is considered to analyze oblique modulational instability (MI) and head-on collisions among dust-acoustic dark solitons. The Chew–Golberger–Low description is employed to define the anisotropic dust pressure. The linear analysis is investigated. It is found that for larger wavelengths, the pressure anisotropy has a strong effect on the wave frequency. Following the multiscale reductive perturbation technique, a (3 + 1)-dimensional nonlinear Schrödinger equation is derived. Also, the MI criterion is identified, and the regions of (un)stable modulated waves are determined precisely. In addition to that, (un)stable domains of the modulated structures as well as the profile of the dust-acoustic rogue waves are found to be strongly affected by dust grain density, pressure anisotropy, and the strength of the magnetic field. In the stable regions, the face-to-face dark soliton collision and their phase shifts as well as their analytical trajectories are reported by applying the extended Poincare–Lighthill–Kuo method. Numerical analysis reveals that the phase shifts increase with dust concentration but decrease with dust pressure anisotropy. The present results may be applicable in exploring the nonlinear wave dynamics and solitary wave interactions in dense astrophysical plasmas especially to white dwarfs, interiors of the neutron stars, and magnet stars.