Abstract
In this paper, a (3+1)-dimensional sine-Gordon equation is systematically investigated. Firstly, the integrability of the equation is demonstrated by Painlevé analysis. Secondly, based on the Hirota bilinear method, the N-soliton solution of the (3+1)-dimensional sine-Gordon equation is derived. Then, by selecting and establishing conjugate relationships between parameters, the kink solutions, the breather solutions and their hybrid solutions were obtained. Finally, the lump solutions of equation are derived by selecting appropriate functions in the solution. In addition, the dynamic behavior of these solutions is systematically analyzed by their respective density profile plots and three-dimensional diagrams.
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