Approximate direct reduction method: infinite series reductions to the perturbed mKdV equation

The approximate direct reduction method is applied to the perturbed mKdVequation with weak fourth order dispersion and weak dissipation. Thesimilarity reduction solutions of different orders conform to formalcoherence, accounting for infinite series reduction solutions to theoriginal equation and general formulas of similarity reductionequations. Painlevé II type equations, hyperbolic secant andJacobi elliptic function solutions are obtained for zero-ordersimilarity reduction equations. Higher order similarity reductionequations are linear variable coefficient ordinary differentialequations.