Infinite series symmetry reduction solutions to the modified KdV--Burgers equation

From the point of view of approximate symmetry, the modifiedKorteweg--de Vries--Burgers (mKdV--Burgers) equation with weakdissipation is investigated. The symmetry of a system of thecorresponding partial differential equations which approximate theperturbed mKdV--Burgers equation is constructed and thecorresponding general approximate symmetry reduction is derived;thereby infinite series solutions and general formulae can beobtained. The obtained result shows that the zero-order similaritysolution to the mKdV--Burgers equation satisfies the Painlevé IIequation. Also, at the level of travelling wave reduction, thegeneral solution formulae are given for any travelling wave solutionof an unperturbed mKdV equation. As an illustrative example, whenthe zero-order tanh profile solution is chosen as an initialapproximate solution, physically approximate similarity solutionsare obtained recursively under the appropriate choice of parametersoccurring during computation.