Lax integrability and superposition wave solutions of (3+1)- dimensional generalized Calogero–Bogoyavlenskii–Schiff equation

Abstract In this article, a Generalized Calogero–Bogoyavlenskii–Schiff (CBS) equation is studied, serving as an extended shallow water wave model in higher dimensions. Firstly, utilizing the Bell polynomial method, the bilinear form of the equation, bilinear Bäcklund transformation, Lax pair and infinite conservation laws are derived, confirming the equation’s complete integrability in the context of the Lax pair. Subsequently, the nonlinear superposition formula of the equation is constructed based on the derived bilinear Bäcklund transformation and an array of infinite superposition soliton solutions of the equation are formulated using this nonlinear superposition formula. Lastly, leveraging the obtained bilinear equation, infinite superposition solutions of various functional types are constructed. Their dynamic characteristics are analyzed through illustrated solution images. It is noteworthy that this paper not only uncovers a multitude of properties through the Bell polynomial method but also derives both infinite linear and nonlinear superposition solutions, enriching the diversity of solutions, these aspects have not been previously explored in existing literature.