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 Schr\"odinger 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.