Modulational instability, modulated wave, and optical solitons for a generalized highly dispersive cubic-quintic-septic-nonic medium with self-frequency shift and self-steepening nonlinear terms

Abstract

In this research, we delve into a generalized highly dispersive (HD) nonlinear Schrödinger equation, enriched with cubic-quintic-septic-nonic (CQSN) nonlinearities. The core of our investigation revolves around the perturbation of plane waves, aiming to understand their stability characteristics in such a complex medium. We investigate the influence of various factors such as the amplitude of the plane wave, perturbed wave number, nonic nonlinear term, and fourth-order dispersion term. Our findings indicate that increasing the amplitude of the plane wave widens the modulation instability (MI) bands and amplifies the MI growth rate. In contrast, increasing the nonic nonlinear term has opposing effects, narrowing the MI bands and diminishing the amplitude of the MI growth rate. Increasing the fourth-order dispersion term does not affect the amplitude of the MI growth rate but narrows the MI bands. The observed pattern of increasing and then decreasing MI intensity with rising K can be attributed to the complex interplay among phase matching conditions, dispersion effects, and nonlinear saturation. Initially, higher K enhances phase matching and boosts MI growth. However, as K increases further, the combined influence of dispersion and nonlinear effects can diminish the effectiveness of phase matching, resulting in a reduction in MI intensity. A significant portion of our work is dedicated to identifying and analyzing modulated rational, polynomial Jacobi elliptic function solutions, and the emergence of optical solitons within this framework. These findings provide new insights into the nonlinear dynamics underpinning the generalized HDNLSE, enriched with CQSN nonlinearities, offering valuable contributions to the theoretical understanding of such phenomena.