Auto-Bäcklund Transformation and Exact Solutions for a New Integrable (3+1)-dimensional KdV-CBS Equation

The Korteweg-de Vries–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation is often used in dealing with long-wave propagation interactions, and is widely used in mathematics, physics, and engineering. This paper proposes a new extended (3+1)-dimensional KdV-CBS equation, and it’s never been studied. Additionally, we verify the integrability of the equation based on the Painlevé test. By employing Hirota’s method, a bilinear auto-Bäcklund transformation, the multiple-soliton solutions, and the soliton molecules of the equation are derived. New exact solutions of the equation are constructed utilizing the power series expansion method and \((G'/G)\) $(G^{\prime }/G)$ -expansion method. These exact solutions are also presented graphically. Finally, the conservation laws of the equation are obtained. Our results are helpful for understanding nonlinear wave phenomena.