Optical soliton solutions, bifurcation analysis, chaotic behaviors of nonlinear Schrödinger equation and modulation instability in optical fiber

Abstract In this study, we use the powerful and strong method to obtain a solution of the Sasa–Satsuma equation as ( ℋ + G ′ G 2 ) \left({\mathcal{ {\mathcal H} }}+\frac{{{\mathcal{G}}}^{^{\prime} }}{{{\mathcal{G}}}^{2}}) -expansion method. This method plays a considerable role in solving nonlinear partial differential equations (NPDEs). We investigate the modulation instability in higher-order NPDEs. Modulation instability is a phenomenon observed in certain types of nonlinear systems, such as optical fiber or plasma waves. Modulation instability is a key process in generating optical solitons, rogue waves, and interest in various fields such as nonlinear optics and plasma physics. Using a linearizing technique, we establish the modulation instability and show the influence of a higher nonlinear component in modulation instability. We examine the bifurcation analysis of the Sasa–Satsuma equation. The time histories and Poincare mapping are used to scrutinize the chaotic behaviors of the dynamical system of the Sasa–Satsuma equation excited by a parametric excitation force. To control the vibrating system, use proportional feedback control (P-Controller). Two-dimensional and three-dimensional figures are presented for singular, dark, and bright optical soliton solutions related to optical fiber. These graphs are very important and useful in describing the behavior of solutions.