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Exponential-type solutions to a generalized Drinfel'd–Sokolov equation

Exact exponential-type solutions to the generalized Drinfel'd–Sokolov (GDS) equations are obtained for the case in which α2=0, for various values of the other model parameters. A modification of the homotopy analysis method is then applied to obtain analytical solutions for nonzero values of the parameter α2, in effect extending the exact solutions. In our modification of the standard method, we employ a nonlinear auxiliary operator. In contrast to most standard perturbation methods, in which a nonlinear problem is reduced to ‘infinitely many’ linear problems, here we reduce a hard nonlinear problem to ‘infinitely many’ easier nonlinear problems. Indeed, we also provide a solution using a linear auxiliary operator and show that the convergence of obtained solutions is improved (in the sense that fewer terms are required for the approximate solutions to obtain a desired accuracy) when using the auxiliary nonlinear operator, in some cases. An error analysis of the obtained approximate analytical solutions is provided.