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

TL;DR

This paper introduces an approximate perturbed direct homotopy reduction method for solving perturbed mKdV equations, capable of deriving similarity reductions to arbitrary orders and applicable to various wave solutions including solitons, Painlevé II, and elliptic functions.

Contribution

The paper presents a novel reduction method that effectively handles strong perturbations and derives arbitrary order similarity reductions for perturbed mKdV equations.

Findings

Method applies to single soliton, Painlevé II, and elliptic wave solutions.

Valid for strong perturbations and higher-order dispersions and dissipations.

Derives similarity reduction equations to arbitrary orders.

Abstract

An approximate perturbed direct homotopy 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 solution but also for the Painlev\'e II waves and periodic waves expressed by Jacobi elliptic functions for both fourth order dispersion and second order dissipation. The method is valid also for strong perturbations.