近似对称约化简述

(同伦)近似对称方法由摄动法与对称约化方法相结合产生, 用于微分方程级数解的构造. 对称约化方法应用于微分方程或者其同伦模型经扰动展开分解而成的无穷多近似子方程, 可以得出通式形式的无穷多约化解和相应的约化方程, 再通过求解约化方程进一步得出原方程的截断级数解. 截断级数解的存在性体现原方程的可解性, 通常取决于扰动项的阶数与最高阶导数项奇偶性是否一致.

Introduction to approximate symmetry reduction

By combining the methods of perturbation and symmetry reduction, approximate (homotopy) symmetry method is adopted to seek the series solutions to differential equations. Through applying symmetry reduction method to infinite number of approximate sub-equations resulted from perturbation expansion for differential equations or their homotopy models, it yields infinite number of reduction solutions and the related reduction equations with general formulae. The reduction equations can be used to further find the series truncated solutions to the original equations. The existence of series solutions reflects the solvability of original equations, which depends upon the consistency of the parity with respect to the derivative orders of perturbation terms and the terms with the highest derivative order.