Trigonometric and hyperbolic type solutions to a generalized Drinfel’d–Sokolov equation

A class of trigonometric and hyperbolic type solutions to the generalized Drinfel’d–Sokolov (GDS) equationsut+α1uux+β1uxxx+γ(vδ)x=0andvt+α2uvx+β2vxxx=0is obtained for the case in which α2 = 0, for various values of the other model parameters. The method of homotopy analysis is then applied to obtain local analytical solutions for nonzero values of the parameter α2, in effect extending the exact solutions. We do not assume traveling wave solution forms for the analytical solutions; that is, we solve the generalized Drinfel’d–Sokolov equations as PDEs without resorting to transforming the system to ODEs. An error analysis of the obtained approximate local analytical solutions is provided. Then, we outline a general framework by which one many construct solutions in either sine/cosine or sinh/cosh basis. We provide the general perturbation expansion via homotopy analysis, and we also discuss a method of selecting the convergence control parameter so as to minimize residual errors. Travelling solutions with time-dependent amplitude are then discussed.