Computational study on fractional nonlinear model in long-wave and short-wave: Modulation instability, chaotic nature with defect tool, and novel optical soliton solutions

In this research work, we study the Truncated time M-fraction 2D coupled YO system to investigate bifurcation analysis, chaotic nature, and soliton solutions, which describe complex phenomena in both long-wave and short-wave regimes. Initially, we apply the modified Sardar sub-equation and unified methods to solve the proposed fractional model. These methods explore diverse traveling and solitary waves in trigonometric, hyperbolic, rational, and exponential forms. Various parametric values ensure the validity and significance of the obtained solutions. Using MATLAB 2023, we present the physical nature and dynamical evolution of some solutions in both fractional and deterministic cases through 3D, 2D, and density plots. Additionally, using bifurcation theory, we investigate the chaotic nature and perform sensitivity analysis with respect to initial conditions. To investigate the chaotic nature, we added trigonometric perturbation terms. Also, we analyzed the Return Map, basin Attraction, Pickover Fractal, bifurcation of parameters, fractal dimension, power Spectrum, Recurrence plot, periodic, Quasi-periodic, multi-stability, and chaotic nature. Finally, we study modulation instability, which is important for explaining how a continuous wave becomes unstable to small perturbations due to the interaction between nonlinearity and dispersion, and it is also a key mechanism for energy localization and the formation of coherent structures.