Nonlinear dynamics of the generalized Hirota–Satsuma–Ito equation: Analytical solutions and graphical insights

Investigating the behavior of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation is the primary goal of this work. A detailed examination of this equation is performed by utilizing the modified Khater method. This approach yields multiple new exact solutions that systematically enhance understanding of the dynamics of the nonlinear model. The solutions include trigonometric, hyperbolic, rational, and Jacobi elliptic functions, providing a rich mathematical framework. Graphical simulations are presented to visualize the dynamical behavior of the obtained solutions, with 3D surface plots, 2D line graphs, and contour plots generated with the help of software such as MATLAB and Mathematica. The Hamiltonian function is established, and phase portraits at equilibrium points are examined. The influence of external forces on the dynamical system is also investigated to analyze chaotic behavior. Time series plots, Poincaré maps, and Lyapunov exponents are utilized for this purpose. Finally, sensitivity analysis is performed under varying initial conditions. The methods shown in this work are useful and adaptable in mathematical physics. These solutions help us understand their physical effects and behaviors, emphasizing the value of studying nonlinear wave patterns in science and engineering.