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An approximate perturbed direct homotopy reduction method is proposed andapplied to two perturbed modified Korteweg-de Vries (mKdV) equations withfourth order dispersion and second order dissipation. The similarity reductionequations are derived to arbitrary orders. The method is valid not only forsingle soliton solution but also for the Painlev\'e II waves and periodic wavesexpressed by Jacobi elliptic functions for both fourth order dispersion andsecond order dissipation. The method is valid also for strong perturbations.

Approximate perturbed direct homotopy reduction method: infinite series reductions to two perturbed mKdV equations