Stability analysis and conserved quantities of analytic nonlinear wave solutions in multi-dimensional fractional systems

The (3+1)-dimensional generalized nonlinear fractional Konopelchenko–Dubrovsky–Kaup–Kupershmidt $(𝔾𝔽𝕂𝔻𝕂𝕂)$ model represents the propagation and interaction of nonlinear waves in complex multi-dimensional physical media characterized by anomalous dispersion and dissipation phenomena. By incorporating fractional derivatives, this model introduces non-locality and memory effects into the classical $𝕂𝔻𝕂𝕂$ equations, commonly utilized in phenomena such as shallow water waves, nonlinear optics, and plasma physics. The fractional approach enhances mathematical representations, allowing for a more realistic depiction of the intricate behaviors observed in numerous modern physical systems. This study focuses on the development of accurate and efficient numerical techniques tailored for the computationally demanding $𝔾𝔽𝕂𝔻𝕂𝕂$ model, leveraging the Khater II and generalized rational approximation methods. These methodologies facilitate stable time-integration, effectively addressing the model’s stiffness and multi-dimensional nature. Through numerical analysis, insights into the stability and convergence of the algorithms are gained. Simulations conducted validate the performance of these methods against established solutions while also uncovering novel capabilities for exploring complex wave dynamics in scenarios involving complete fractional formulations. The findings underscore the potential of integrating fractional calculus into higher-dimensional nonlinear partial differential equations, offering a promising avenue for advancing the modeling and computational analysis of complex wave phenomena across a spectrum of contemporary physical disciplines.