Analytical and Numerical Investigation of a Perturbed M-Fractional Kairat-X Equation with Variable Coefficients

A comprehensive analytical and numerical investigation is presented for a perturbed M-fractional Kairat-X equation with variable coefficients and dissipative effects. The model is motivated by fractional nonlinear wave phenomena arising in inhomogeneous optical, plasma, and ferromagnetic media, where spatial variability and weak perturbations play a crucial role in wave propagation and stability. In contrast to existing studies that primarily focus on constant-coefficient formulations and exact solution construction, the present work develops a rigorous analytical framework for a variable-coefficient fractional model, thereby addressing an important gap in the literature. By extending the classical fractional Kairat–X equation to include variable-coefficient potentials and perturbation terms, a more realistic and physically relevant framework is obtained. A rigorous mathematical analysis is carried out by employing a traveling-wave reduction under clearly stated assumptions on the variable coefficients, followed by the application of fixed-point theory to establish the existence and uniqueness of traveling-wave solutions. Furthermore, Lyapunov-type energy estimates are developed to derive explicit stability conditions, providing deeper insight into the influence of coefficient structure and fractional-order effects on solution behavior. In addition, a convergent numerical scheme is constructed to support the analytical results, and its stability and accuracy are demonstrated through discrete error analysis. The numerical results are presented in a systematic tabulated form, highlighting convergence properties, stability regimes, and the sensitivity of solutions with respect to the fractional order and perturbation parameters. The results demonstrate that the introduction of variable coefficients and perturbations significantly affects solution stability and parameter sensitivity, thereby extending and strengthening existing constant-coefficient fractional Kairat–X models. Overall, this study provides a unified analytical and computational framework for investigating nonlinear fractional wave equations with spatial heterogeneity, offering new insights into their long-time dynamics and stability mechanisms, with potential applications in nonlinear optics, ferromagnetic materials, and optical fiber systems.