Bifurcation, quasi-periodic, and chaotic analysis of the conformable fractional paraxial wave equation using the rational expansion method

In this study, the conformable space–time fractional paraxial wave equation, which describes the optical pulse propagation through nonlinear media, is investigated to gain insight into intricate nonlinear phenomena. The objective of the study is to identify novel exact soliton solutions of the concerned equation with the implication of a robust analytical approach—the rational (G′/G)-expansion scheme. The suggested method offers a variety of hyperbolic, trigonometric, and rational exact solutions to the governing equation. Assorted types of solitons, including Kuznetsov-Ma breather type, periodic, flat-soliton, bell-shaped, kink, shock wave, and singular breather, are obtained in this study by selecting appropriate values for the free parameters included in analytical solutions. We utilize the Mathematica software to plot three- and two-dimensional graphs and contours of the obtained solitons. The bifurcation theory is used to analyze the dynamical behavior of the planar dynamic system produced from the considered equation via the Galilean transformation. An external periodic force is introduced to construct a perturbed nonlinear dynamical system that allows us to observe the quasi-periodic and chaotic nature of the system. The phase portraits and time series are presented to support the consistency of the results.