An auto-Backlund transformation and exact solutions to a generalized KdV equation with variable coefficients and their applications

Abstract

By using the homogeneous balance principle, an auto-Backlund transformation (BT) to the generalized KdV equation with variable coefficients is derived. The auto-BT only involves one quadratic homogeneity equation to be solved. Solving the homogeneity equation by use of the ε-expansion method and using the auto-BT, generally speaking, we can obtain an exact solution containing N-solitary wave of the generalized KdV equation with variable coefficients. As an illustrative example, we obtain an exact solution containing 2 solitary wave of the equation in detail. Since the generalized KdV (GKdV) equation, cylindrical KdV equation and variable coefficient KdV equation are all the special cases of the generalized KdV equation with variable coefficients, the corresponding results of these equations are also given respectively.