Painlevé analysis, auto-Bäcklund transformation and various new multisoliton solutions for the (3+1)-dimensional gBK equation with time-dependent variable coefficients

This paper studies the ([Formula: see text])-dimensional generalized Bogoyavlensky–Konopelchenko (gBK) equation with time-dependent variable coefficients. Integrability test with the aid of Painlevé analysis is being applied to the equation under consideration. Painlevé analysis was used to produce an auto-Bäcklund transformation for the given equation. The auto-Bäcklund transformation method was used to provide analytical solutions. By using this method, three distinct families of analytic solutions are obtained for the gBK equation. Also, Hereman and Nuseir method is employed to obtain multi-soliton solutions for the equation under consideration. All the obtained results are presented by 3D graphs using various functions and parametric values. The physical relevance of the acquired solutions has been demonstrated by these opulent graphs.