Nonlinear dynamics of self-sustaining waves in anisotropic media

: This investigation undertakes an extensive rigorous and computational examination of nonlinear dispersive wave behavior within the Jaulent-Miodek ([Formula: see text]) framework, initially established by Jaulent and Miodek in 1987. The framework examines wave characteristics in two-dimensional anisotropic environments, focusing on the mechanics of localized wave packets dictated by nonlinear-dispersion interplays. Utilizing a unified strategy merging the modified Khater technique, rational analytical approximations, and Adomian decomposition, we procure exact soliton solutions. These solutions experience meticulous validation via numerical convergence examination, demonstrating substantial consonance between rigorous and computational outcomes across various self-sustaining and periodic wave arrangements. Additionally, we scrutinize the nonlinear dynamical attributes of the [Formula: see text] framework through bifurcation investigation, deploying the Hamiltonian energy function [Formula: see text] to delineate equilibrium positions. This inquiry uncovers intricate dynamical phenomena, including chaotic states and quasi-periodic pulsations, while evaluating the sensitivity features of the corresponding planar dynamical system. Our discoveries markedly elevate the comprehension of nonlinear wave mechanics and coherent formations in dispersive systems, offering significant implications for disciplines such as hydrodynamics, nonlinear optics, and plasma physics. The methodological architecture developed here provides efficacious analytical instruments for examining nonlinear dispersive wave models, creating a bridge between theoretical advancements and practical implementations across numerous fields.