This paper investigates the Kuramoto–Tsuzuki equation through the application of the Lie symmetry method. We derive three Lie algebra generators pertinent to this equation, facilitating a transformation that reduces the partial differential equation to an ordinary differential equation. One of the reduced ordinary differential equations is solved analytically using Nucci’s reduction method. The second reduced ordinary differential equation, derived from a combination of generators, is solved directly by some manipulations. Then we use a transformation to convert the Kuramoto–Tsuzuki equation into a second-order nonlinear ordinary differential equation. Utilizing a recently introduced technique, the second-degree generalized Abel equation method, we successfully obtain soliton solutions for the reduced ordinary differential equation. Additionally, we explore bifurcation phenomena and analyze phase portraits, providing deeper insights into the dynamic behavior of the system. The analysis of chaotic behavior and sensitivity of the corresponding dynamical system contributes to a better understanding of the Kuramoto–Tsuzuki equation and highlights the effectiveness of the proposed methods for analyzing complex nonlinear equations.
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