In this study, we examined the nonlinear dynamics of the Boussinesq equation, a foundational equation in ocean engineering to model and investigate the behavior of waves in shallow water. The novel (G'/G²)-expansion method was employed to obtain different soliton solutions, including periodic, bright, W-type, and bell-shaped soliton solutions. These solutions are illustrated through 2D, 3D, and contour plots. We discovered different dynamical behavior, including periodic, quasi-periodic, and weak chaos, depending on the choice of initial conditions and parameters. The important outcomes included the detection of multistable attractors and the presence of weak chaotic behavior supported by Lyapunov exponents. These understandings have important effects in practical uses such as energy harvesting and wave control in ocean systems, where handling and understanding system transitions and stability is crucial. These findings also give a framework for further examination of stability and control in nonlinear wave systems.

This research is aimed at finding exact soliton solutions of the nonlinear fractional Kairat-X equation, which describes soliton behavior in nonlinear media and has applications in quantum physics, materials science, signal processing, and telecommunications. We use a unified method that generalizes the tanh-function method to find new exact soliton solutions in trigonometric, hyperbolic, and plane wave forms. Computational simulations with fixed parameters are performed to produce two-dimensional and three-dimensional visualizations, e.g., contour and density plots, representing the physical properties of the derived solitons. The simulations result in the identification of several soliton types, namely kink wave solitons, dark solitons, bright solitons, and periodic wave solitons. Our results increase the knowledge of the solution properties of the Kairat-X equation and give a platform to interpret a variety of significant physical phenomena. The systematicity and stability of our methodology prove its usefulness as a device for solving other nonlinear partial differential equations in applied physics and mathematics, always returning different exact solutions.