Wave Solutions, Bifurcation, Phase Portraits, Chaotic Behaviors, Poincaré Sections, and Numerical Analysis of the (2+1)-Dimensional Kundu–Mukherjee–Naskar Equation

Nonlinear differential equations are used for deciphering complex dynamical phenomena that cannot be captured by classical linear approximations, spanning physics, engineering, electronics, biomedicine, and computational mathematics. It is important to study nonlinear differential equations. This paper comprehensively analyzes the nonlinear dynamic characteristics of the [Formula: see text]-dimensional Kundu–Mukherjee–Naskar equation deploying the analytical method, Hamiltonian system framework, and numerical simulation. The wave solutions of the equation are derived by using the improved [Formula: see text]-expansion method, and its structural properties under different parameters are investigated. In order to better study the features of dynamics behavior of the equation, Hamiltonian system is established and nonlinear dynamics are systematically discussed with bifurcation, phase portraits, and Lyapunov exponents. Subsequently, detailed chaos analysis employing bifurcation diagrams and Poincaré maps demonstrates the complex response of the system under diverse initial conditions and parameters. In addition, we utilize the Runge–Kutta algorithm for numerical simulations to highlight the effectiveness and practical significance of these analytical solutions, and explore the attributes of the equation in depth through numerical solutions. We also compare this inclusive study with previous researches, highlighting the innovation and implication of this work. These exhaustive findings significantly enrich the application background of the [Formula: see text]-dimensional Kundu–Mukherjee–Naskar equation and provide solid theoretical assistance for the profound investigation of related nonlinear differential equations.