Investigation of the soliton solutions, bifurcation analysis, and chaotic nature of the Akbota–Myrzakulov–Tolkynay– Zhaidary equation

Abstract We investigate exact soliton solutions, bifurcation structures, and chaotic behavior in the Akbota–Myrzakulov–Tolkynay–Zhaidary (AMTZ) equation, which governs nonlinear wave propagation in multi-dimensional dispersive systems. Using the improved modified Sardar sub-equation method, we develop hyperbolic, trigonometric, and rational function solutions, along with full visualizations illustrating wave stability characteristics. Bifurcation analysis yields critical equilibrium points such as saddle, center, and cuspidal configurations, and phase portraits reveal complicated nonlinear transitions. Adding external periodic perturbations reveals chaotic attractors confirmed using Poincaré sections and positive Lyapunov exponents, establishing sensitivity to initial conditions. The AMTZ equation is used extensively in the field of optical fiber communications, plasma physics, and fluid dynamics. This is the first comprehensive chaos analysis of the AMTZ equation, connecting exact soliton construction with nonlinear dynamical characterization, providing a theoretical framework for modeling and controlling nonlinear wave processes across disciplines.