Rogue Waves in Ring Patterns in Generalized χ2n,4n PNLSE With a Novel Conformable Derivative: Global–Local Bifurcation, Sensitivity, and Stability Analysis

The perturbed nonlinear Schrödinger equation (PNLSE) explores the interplay between self-steepening (SS), self-phase modulation (SPM), and Raman scattering (RS). The generalized PNLSE (gPNLSE) incorporates a power-law framework and extends this analysis to describe nonlinear interactions among SS, SPM, and RS. A gPNLSE with high or strong nonlinearity extends the standard cubic PNLSE by including higher-power terms, saturable, nonlocal, and derivative nonlinearities. This model equation is formulated by considering a general power-law exponent n. As the time delay plays a crucial role in the behavior of optical waves propagation in fibers, this motivated us to study the gPNLSE via introducing a novel conformable derivative, which is characterized by a periodic time delay. Exact solutions are derived through a suitable transformation and the application of the generic unified method. The findings reveal that these solutions exhibit rogue waves (RWs) propagating in optical fibers, often forming intricate ring patterns. In the classical case, however, RWs emerge alongside breathers. Additionally, various wave geometric structures are identified, including split RWs’ vectors, downward RWs’ vectors, lush flower-shaped waves, lattice waves, and wrinkle periodic waves. Notably, SS emerges as a dominant phenomenon, potentially leading to shock waves or wave breaking. A sensitivity analysis is conducted in the presence of an external periodic force. It is observed that the system is highly sensitive when n