Rhombus-shaped, M-shaped- solitons, 2D-lumps vector and tunneling in perturbed Gerdjikov–Ivanov equation with fourth-order dispersion: Modulation instability and global bifurcation

The Perturbed Gerdjikov–Ivanov Equation (PGIE) describes optical solitons in nonlinear fiber optics and photonic crystal fibers. It was currently studied in the literature. Here, we consider a new complex field equation, the PGIE with fourth-order dispersion. The exact solutions of the aforementioned equation are obtained by implementing the unified method. These solutions are evaluated numerically and they are shown in graphs. Our objectives, here, are to inspect the onset of self- steepening and self-phase modulation and investigate the structures of solitons produced. It is found that these phenomena are launched for high values of the relevant coefficients, and they may be overlapped. Multiple solitons shapes are observed; M-shaped solitons, rhombus shapes, solitons with tunneling, 2D-lumps vector, and solitons- cascade. Further, the analysis of modulation instability (MI) and global bifurcation are carried out. It is established that the MI triggers when the coefficient of Raman scattering exceeds a critical value. It is worth mentioning that two kinds of bifurcations can be studied, namely local and global bifurcations. Here, we are concerned with studying global bifurcation. This is based on constructing the phase portrait via Hamiltonian function.