Exact soliton solutions, chaotic dynamical characterization, and high-precision numerical validation of the Kairat-II-X equation for nonlinear wave propagation

The Kairat-II-X (K-II-X) integrable equation, an important theoretical model for nonlinear-wave propagation, has applications in nonlinear optics, plasma physics, and fluid mechanics. The major innovations of this paper are summarized as follows: (1) By employing the modified hyperbolic tangent function method, a series of exact soliton solutions, including anti-kink solitons, periodic solitons, breather solitons, and periodic breather solitons, are systematically derived, enriching the existing solution families. Their stable propagation characteristics are illustrated through three-dimensional surfaces, two-dimensional profiles, and contour maps. (2) The embedded seventh-order Runge–Kutta scheme is utilized for high-precision numerical validation of the analytical solutions, which has not been reported in previous studies. The absolute errors are shown to reach the order of 10−16–10−12, confirming the reliability of the obtained solutions. (3) An equivalent non-autonomous periodically forced dynamical system is constructed to investigate chaotic behaviors under different periodic forcing conditions through phase portraits, bifurcation analysis, Poincaré sections, return maps, and Lyapunov-exponent spectra. The influence mechanism of periodic forcing on chaotic behaviors is revealed.