Topological properties, exact solutions and chaotic behaviors for coupled Drinfel’d–Sokolov–Wilson equation with conformal derivative

Purpose In this paper, the dynamical properties and analytic solutions of the coupled Drinfel’d–Sokolov–Wilson equation with a conformal derivative are studied by the complete discrimination system for the polynomial method. Not only are the Hamiltonian and topological properties of this equation are presented, but also all exact traveling wave solutions are found. Design/methodology/approach The complete discrimination system for the polynomial method serves as the cornerstone of the analytical approach, facilitating both the analysis of dynamic properties and the derivation of exact solutions for the equation under study. Findings The study shows that by adjusting specific parameters, various classified solutions such as rational function solutions, solitary wave solutions and periodic function solutions can be realized in practical applications. Furthermore, numerical analysis shows that the introduction of different external perturbation terms in the coupled Drinfel’d–Sokolov–Wilson equations with conformal derivative can confirm the presence of chaotic behaviors. Originality/value The coupled Drinfel’d–Sokolov–Wilson equation is analyzed qualitatively and quantitatively using the same method. Firstly, the dynamical system is analyzed qualitatively using a third-order discriminant system to predict the type of solutions. Then, use a fourth-order discriminant system to solve the exact solutions and maintain parameter consistency. This method ensures that the results of qualitative and quantitative analysis are achieved under the same set of parameters, thereby improving the coherence and reliability of the research.