Dynamical analysis and exact solitary wave solutions of $$(2+1)$$-dimensional integro-differential Jaulent-Miodek equation using two analytical schemes

This research explores wave phenomena in the $$(2+1)$$-dimensional integro-differential Jaulent–Miodek equation associated with dispersive nonlinear systems. The primary objective of this study is to derive and analyze exact wave solutions of the governing model using the generalized Arnous approach and the modified simple equation method. By applying a suitable travelling-wave transformation, the nonlinear partial differential equation is reduced to an ordinary differential equation, which is then solved analytically using the proposed techniques. The results reveal a rich variety of nonlinear wave structures such as kink, anti-kink, bright, dark, bell-shaped, anti-bell, periodic, singular periodic, V-shaped, W-shaped, and singular wave solutions. The physical characteristics of the obtained solutions are illustrated through two-dimensional, three-dimensional, and contour plots generated using Mathematica. Furthermore, the dynamical behavior of the system is investigated through bifurcation analysis using two-dimensional phase portraits, while chaotic dynamics are examined via corresponding three-dimensional phase portraits together with time-series plots to identify transitions from periodic to quasi-periodic and chaotic behavior. In addition, sensitivity analysis is carried out using two-dimensional phase portraits to examine the dependence of the system response on initial conditions. The findings confirm the effectiveness and applicability of the proposed analytical methods for constructing exact solutions of higher-dimensional nonlinear evolution equations and provide deeper insight into nonlinear wave propagation phenomena arising in optics, plasma physics, and fluid flows.