Auto-Bäcklund transformation, Painlevé analysis, and a variety of new multi-soliton solutions for the (2+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt system with time-dependent variable coefficients

Abstract This paper examines a (2 + 1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupersh-midt(gKDKK) system, with applications in fluid mechanics and plasma physics. Painlevé analysis, a powerful method that employs the Laurent series to explore singularity structures and evaluate the integrability of nonlinear partial differential equations(NLPDEs), is used to demonstrate that the considered equation is completely integrable. By applying the Painlevé analysis method, the Laurent series is truncated to derive an auto-Bäcklund transformation (ABT). Using ABT, three distinct families of analytical solutions are derived for the considered system. Additionally, multi-soliton solutions are obtained using the simplified Hirota method. All the obtained results are depicted in 3D graphs through numerous functions and parameter settings. These graphs provide a clear understanding of wave structures and soliton interactions of the system being studied.