Analytical solutions to a generalized Drinfel’d–Sokolov equation related to DSSH and KdV6

Analytical solutions to the generalized Drinfel’d–Sokolov (GDS) equationsut+α1uux+β1uxxx+γ(vδ)x=0andvt+α2uvx+β2vxxx=0are obtained for various values of the model parameters. In particular, we provide perturbation solutions to illustrate the strong influence of the parameters β1 and β2 on the behavior of the solutions. We then consider a Miura-type transform which reduces the gDS equations into a sixth-order nonlinear differential equation under the assumption that δ = 1. Under such a transform the GDS reduces to the sixth-order Drinfel’d–Sokolov–Satsuma–Hirota (DSSH) equation (also known as KdV6) in the very special case α1 = −α2. The method of homotopy analysis is applied in order to obtain analytical solutions to the resulting equation for arbitrary α1 and α2. An error analysis of the obtained approximate analytical solutions is provided.