Approximate Homotopy Direct Reduction Method: Infinite Series Reductions to Perturbed mKdV Equations

An approximate homotopy direct reduction method is proposed and applied to two perturbed modified Korteweg-de Vries (mKdV) equations with fourth-order dispersion and second-order dissipation. The similarity reduction equations are derived to arbitrary orders. The method is valid not only for single soliton solutions but also for the Painlevé II waves and periodic waves expressed by Jacobi elliptic functions for both fourth-order dispersion and second-order dissipation. The method is also valid for strong perturbations.