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.