Equivalent transformations, bifurcations and exact solutions to a class of variable-coefficient PDE

In this paper, the equivalent transformations (ETs) of the generalized variable-coefficient KdV (vc-KdV) types of equations are constructed, so these variable-coefficient partial differential equations (vc-PDEs) are transformed into constant-coefficient partial differential equations (cc-PDEs) under some conditions, the vc-KdV and vc-mKdV equations are transformed into classical KdV and mKdV equations accordingly. Furthermore, the dynamical behavior is provided by the bifurcation analysis method, the exact parametric representations are investigated. Then the explicit solutions to the variable-coefficient equations are presented in terms of the ETs and parametric representations.