Painlevé integrability and new soliton solutions for (2 + 1)-dimensional Bogoyavlensky–Konopelchenko equation and generalized Bogoyavlensky–Konopelchenko equation with variable coefficients in fluid mechanics

The time-dependent variable coefficients of Bogoyavlensky–Konopelchenko (BK) equation and generalized Bogoyavlensky–Konopelchenko (gBK) equation are considered in this paper. The integrability test by Painlevé analysis is being implemented on both the considered equations. An auto-Bäcklund transformation has been generated with the help of Painlevé analysis for both equations. Auto-Bäcklund transformation method has been used for obtaining the analytic solutions. By using auto-Bäcklund transformation method, three different analytic solution families have been derived for each of the considered equations. Multi-soliton solutions are also calculated for both the considered equations by using Hereman and Nuseir algorithm. All the results are expressed graphically in 3D by varying different functions and parametric values. These graphs reveal the physical significance of equations under consideration.