In this article, the dynamic behavior and solitary wave solutions of the Akbota equation were studied based on the analysis method of planar dynamic system. This method can not only analyze the dynamic behavior of a given equation, but also construct its solitary wave solution. Through traveling wave transformation, the Akbota equation can easily be transformed into an ordinary differential equation, and then into a two-dimensional dynamical system. By analyzing the two-dimensional dynamic system and its periodic disturbance system, planar phase portraits, three-dimensional phase portraits, Poincaré sections, and sensitivity analysis diagrams were drawn. Additionally, Lyapunov exponent portrait of a dynamical system with periodic disturbances was drawn using mathematical software. According to the maximum Lyapunov exponent portrait, it can be deduced whether the system is chaotic or stable. Solitary wave solutions of the Akbota equation are presented. Moreover, a visualization diagram and contour graphs of the solitary wave solutions are presented.

This article investigates the qualitative analysis and traveling wave solutions of the stochastic nonlinear Kodama equation within the Stratonovich framework. By applying a random traveling wave transformation, the equation is first converted into an ordinary differential equation. The qualitative behavior of the corresponding two-dimensional dynamical system and its perturbation is then examined using planar dynamical system analysis. Subsequently, the complete discriminant system method is employed to derive four distinct types of optical solutions. Three-dimensional plots illustrating these solutions under different parameters are also presented.