Approximate homotopy symmetry method: Homotopy series solutions to the sixth-order Boussinesq equation

Abstract

An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation, which arises from fluid dynamics. We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders, educing the related homotopy series solutions. Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation. Higher order similarity solutions can be obtained by solving linear variable coefficients ordinary differential equations. The auxiliary parameter has an effect on the convergence of homotopy series solutions. Series solutions and similarity reduction equations from the approximate symmetry method can be retrieved from the approximate homotopy symmetry method.