New abundant exact solutions for MCBS-nMCBS equation: Painlevé analysis and auto-Bäcklund transformation

This article considers a (2 + 1)-dimensional variable coefficients combined modified Calogero-Bogoyavlenskii-Schiff equation and a negative-order modified Calogero-Bogoyavlenskii-Schiff (MCBS-nMCBS) equation. The MCBS-nMCBS equation describes the progressive shallow-water waves and other physical phenomena and is very helpful in studying the wave patterns in the soliton theory. Firstly, in this article, the integrability of the considered equation is examined by the Painlevé analysis method. This approach gives the integrability components such as leading orders, resonances, and compatibility conditions. Furthermore, the Painlevé analysis method helps to generate the auto-Bäcklund transformations (ABT). By employing the ABT approach, two analytic solution families have been generated with some free parameters and functions. These solutions explain the various physical properties of the considered model and can be visualized by the 3D graphs. These graphs depict the kink-soliton, anti-kink–soliton, bright-soliton, and dark-soliton and periodic wave surfaces for the suitable parametric values.