Bifurcation, chaotic behavior, and soliton solutions of the Kairat-II equation via two analytical methods

Abstract

This study examines the Kairat-II equation, a nonlinear partial differential equation with numerous applications in mathematical physics and engineering. We analyze the dynamical behavior of this equation and explore its phase portraits using bifurcation theory to identify critical points where the system undergoes qualitative changes. Furthermore, we investigate the chaotic and quasi-periodic behaviors of the equation when subjected to a periodic perturbation of the form $\alpha cos\left(\mu \xi \right)$. We also apply advanced techniques in soliton theory and related methods to derive exact solutions, leading to four new solution families. These solutions are thoroughly analyzed for boundedness and singularities to show their physical relevance. Finally, we present three-dimensional visualizations to illustrate the dynamics of some of the obtained solutions and discuss their physical interpretations.