Modulational instability and sister chirped femtosecond modulated waves in a nonlinear Schrödinger equation with self-steepening and self-frequency shift

We investigate the modulational instability phenomenon and demonstrate that the competing cubic–quintic nonlinearity induces propagating solitonlike bright (dark) solitons, first-order rogue waves embedded on a continuous wave background, and two sister modulated waves in the nonlinear Schrödinger equation with self-steepening and self-frequency shift. We show that the nonlinear chirp associated with each optical pulse propagating on a continuous wave background is directly proportional to the intensity of the wave. We also show that the frequency chirp associated with each of two sister modulated waves is simultaneously directly and inversely proportional to the intensity of the wave. Our investigations show that chirping features and behavior depend on both the self-steepening term and self-frequency shift, while its amplitude can be controlled by varying the parameters of the group velocity dispersion and cubic nonlinearity. •We consider a nonlinear Schrödinger equation with self-steepening and self-frequency.•Chirped femtosecond solitons embedded on continuous wave backgrounds are derived.•The concept of two sister modulated waves with chirping is introduced and analyzed.•Effects of the equation parameters on found chirped modulated waves are investigated.