Lie symmetries, integrable properties and exact solutions to the variable-coefficient nonlinear evolution equations

This paper is concerned with the variable-coefficient nonlinear evolution equations, which include the cylindrical KdV types of equations. By the combination of Painlevé analysis and Lie group classification method, the integrable conditions, Bäcklund transformations and Lax pairs of the equations are obtained, all of the geometric vector fields of the equations are presented. Then, the relationship among Lie group classification, Painlevé analysis and CK transformation method is considered. Furthermore, the exact solutions generated from the Bäcklund transformations and symmetry reductions of the vc-KdV types of equations are investigated.