Bright–dark envelope-optical solitons in space-time reverse generalized Fokas–Lenells equation: Modulated wave gain

Here, we investigate the impact of space-time reverse (STR) problems, a result of parity-time symmetry in optics and quantum mechanics, on soliton propagation in optical fibers. The STR problems are characterized by the existence of a field and its reverse. The research introduces a new classification of two scenarios: non-interactive and interactive fields and reverse fields. The solutions for the generalized Fokas–Lenells equation (gFLE) with STR and third-order dispersion are derived. To tackle this, adaptive transformations for the field and its reverse are introduced, employing a unified method. In the non-interactive scenario, both exact and approximate solutions are found. However, in the interactive case, only exact solutions are discovered. This work reveals that the presence of the field and its reverse unveils new soliton structures, including bright–dark envelope solitons and right and left envelope-solitons. In the non-interactive case, the field displays a right envelope-soliton, while the reverse field exhibits a left envelope-soliton (or vice versa). The study hypothesizes that the presence of a reverse field might impede soliton propagation in optical fibers. The research also includes an analysis of modulation instability (MI), determining that MI is initiated when the coefficient of Raman scattering exceeds a critical value. Furthermore, the study examines the modulated wave gain and explores global bifurcation through phase portrait by constructing the Hamiltonian function.