Exact wave structures, nonlinear dynamics, and stability of the conformable fractional cubic nonlinear Schrödinger equation

Abstract The conformable fractional cubic nonlinear Schrödinger equation (CNLSE) serves as a robust mathematical model for understanding nonlinear wave propagation across various media, including optical fibers and plasma. This study introduces a cohesive analytical-computational framework to explore exact wave structures, nonlinear dynamics, and stability associated with the CNLSE. By applying a conformable fractional traveling-wave transformation, the governing equation is transformed into an equivalent nonlinear ordinary differential equation. Utilizing the Modified Sardar Sub-Equation Method (MSSEM) and the Extended Generalized Riccati Equation Mapping Method (EGREMM) in conjunction leads to the derivation of extensive families of exact 21 solutions from Modified Sardar Sub-Equation Method and 27 from Extended Generalized Riccati Equation Mapping Method expanding the prior knowledge base significantly. The solutions encompass various wave structures, such as bright, dark, singular, periodic, rational, trigonometric, kink, and anti-kink forms. Notably, the integration of Hamiltonian phase-plane analysis, bifurcation theory, and dynamic stability through Hyers–Ulam stability assessment, alongside verification via the Differential Transform Method (DTM), enhances the study’s depth. The phase-plane analysis reveals critical dynamics such as equilibrium states and bifurcations, while perturbation analysis showcases transitions toward chaotic behaviors. The findings have profound implications on optical-pulse control, plasma-wave regulation, and quantum-wave transport, ultimately providing a comprehensive solution catalogue and deepening the understanding of nonlinear fractional-wave propagation mechanisms and their stability considerations.