Interaction of mixed localized waves in optical media with higher-order dispersion

This work focuses on the interaction of mixed localized waves in optical media with higher-order dispersions whose dynamics are governed by a modified cubic–quintic nonlinear Schrödinger equation. For proving the integrability of this model equation, we start by building a Lax pair and an infinitely many conservation laws. Applying the linear stability analysis method, the baseband modulational instability of a stationary continuous wave solution is investigated. Studying the baseband modulational instability phenomenon, we show that the optical loss influences the instability gain spectrum: the stationary continuous wave solution under consideration satisfies the condition of the baseband modulational instability only when the optical loss is neglected. According to the generalized perturbation (n,p−n)–fold Darboux transformation, the existence and properties of the parametric first-, second-, and third-order mixed localized wave solutions for the model equation are constructed when the loss term is neglected. The built solutions helping, we engineer in optical media with higher-order dispersions new nonlinear structures showing interactions between various kinds of nonlinear waves such as multi-peak bright/dark solitons, bright/dark breathers, bright/dark rogue waves, as well as periodic waves. Graphical illustrations are then used for investigating main characteristics of the mixed localized waves propagating on vanishing/nonvanishing continuous wave background. Interestingly, our study produces nonlocal breathers in which the entire optical field oscillates periodically in conjunction with the central local oscillation during transmission. Investigating the effects of various parameters on the nonlinear structures resulting from built mixed localized wave solutions of the model equation, we show that parameter of the fourth-order dispersion can be used to describe wave compression. Also, we show that the model parameters are useful for controlling the optical waves in lossless optical media with both higher-order dispersion whose dynamics are governed by the model equation under consideration. Our results are useful for investigating mixed localized waves in nonlinear metamaterials with cubic–quintic nonlinearity, detuning intermodal dispersion, self steepening and self-frequency effects, and nonlinear third- and fourth-order dispersions.