W-chirped solitons and modulated waves patterns in parabolic law medium with anti-cubic nonlinearity

In this work, investigation of the W-chirped solitons and modulation instability are addressed in nonlinear optic fibers where cubic-quintic and parabolic law parameters are used. For this reason, the modified cubic-quintic complex Ginzburg–Landau equation with parabolic law nonlinearity is employed. Using the sub-ODE method, the chirp component and combined chirped soliton solutions, including rational soliton solutions, are obtained. Using specific parameters of the nonlinear structure, one can identify dark and bright solitons. For a suitable constant parameter of the chirp component, the W-chirped bright as well as the combined bright-dark solitons are displayed via the 3D plots. The linear stability analysis of the plane wave is used to derive an expression of the modulation instability spectrum that is employed to show the effects of the nonlinear parameters in normal and anomalous dispersion regimes. It is pointed out that both cubic and quintic parameters can enlarge or reduce the bandwidth of the modulation instability. Through the numerical simulation of the continuous wave, it becomes apparent that for suitable optical fiber parameter values, modulated wave objects are formed to exhibit mild oscillations. For specific wavenumber excitation, another modulated wave object is depicted to show how the unstable mode can switch to the stable mode for the long-time evolution of the continuous wave. The obtained results could certainly open a new future for signal treatment in optical fiber, where parabolic law is considered.