Approximate similarity reduction for singularly perturbed Boussinesq equation via symmetry perturbation and direct method

We investigate the singularly perturbed Boussinesq equation in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions and similarity reduction equations of different orders display formal coincidence for both methods. Series reduction solutions are consequently derived. For the approximate symmetry perturbation method, similarity reduction equations of the zero order are equivalent to the Painleve IV, Painleve I, and Weierstrass elliptic equations. For the approximate direct method, similarity reduction equations of the zero order are equivalent to the Painleve IV, Painleve II, Painleve I, or Weierstrass elliptic equations. The approximate direct method yields more general approximate similarity reductions than the approximate symmetry perturbation method.

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