一类复合方程的古典和非古典对称分类

Classical and Nonclassical Symmetry Classification of a Composite Type Equation

摘要: 本文确定了一类复合方程的古典对称分类和非古典对称分类。首先，基于微分特征列集算法确定了复合方程的古典对称分类。其次，确定了复合方程的非古典对称的分类。第一步，添加不变曲面条件与原方程组成一个新的偏微分方程组(PDEs)，利用符号计算软件Mathematica确定上面PDEs的对称对应的确定方程组(DTEs)；第二步，根据所得的DTEs进行非古典对称分类，得到复合方程中参数F(u)的具体形式。第三步，确定了非古典对称所对应的不变解以及精确解。所得的不变解和精确解无法利用古典对称得到，所以丰富了复合方程的精确解。

Abstract: In this paper, the classifications of classical and nonclassical symmetries to a composite type equation are determined. Firstly, the classification of classical symmetries to the composite equation is determined based on the differential characteristic set algorithm. Secondly, the classification of nonclassical symmetries for the composite equation is determined. First step, adding invariant surface condition and the original equation composed a new system of partial differential equations (PDEs), and the determining equations (DTEs) of symmetry to PDEs are determined by using the symbolic computation software Mathematica. Second step, the nonclassical symmetries are classified by calculating DTEs, so we can obtain the specific form of F(u) which is the parameter of the composite equation. Third step, the invariant solutions and exact solutions of the corresponding nonclassical symmetry are determined. The invariant solutions and exact solutions cannot be obtained by classical symmetry, so enrich the exact solutions of the composite equation.