Sensitivity and Chaotic Dynamics: A Comparative Study of Optical Solitons for the Zhanbota‐IIA Equation Using Enhanced Analytical Methods

This work investigates solitary wave solutions and dynamical properties of the integrable Zhanbota‐IIA equation, which exhibits rich nonlinear dynamics and diverse soliton structures. To derive exact traveling wave solutions, two robust analytical frameworks are employed: the new extended direct algebraic method (NEDAM) and the ( G ′ / G 2 )‐expansion method. These methods generate solutions in the forms of trigonometric, hyperbolic, and rational functions, yielding diverse wave profiles, including bright and dark solitons, periodic and mixed periodic forms, parabolic waves, and shock wave profiles. A Galilean transformation is applied to reduce the equation to its associated dynamical system, enabling a comprehensive exploration of wave propagation dynamics. Sensitivity and chaotic behaviors are analyzed through phase portraits, time series, return maps, Lyapunov exponents, and parameter perturbations under elliptic forcing, revealing pronounced sensitivity to initial conditions, multistability, and complex nonlinear behavior. These findings provide valuable insights into the stability properties of the underlying physical systems and demonstrate the robustness of the proposed analytical frameworks for solving complex nonlinear partial differential equations.