论文标题
对非线性波和椭圆方程式应用的代数分类方法的概括
Generalization of the algebraic method of group classification with application to nonlinear wave and elliptic equations
论文作者
论文摘要
在（1+1）维度非线性波和椭圆方程的类别上增强并基本上概括了先前的结果，我们应用了几种新技术来对该类中的可接受点变换进行分类，直到其等效组产生的等效性。这给出了对其等效群体的详尽描述。将组分类的代数分类方法扩展到微分方程的非归一化类别之后，我们解决了研究类的完整小组分类问题，直至通常和一般点等价。该解决方案包括对类的完整初步分类和构造单数外观对称扩展，这与等价代数的亚代词无关。完整的初步组分类是基于对整个无限维等价代数的适当子代数进行分类，其投影被认为是内核不变性代数的最大扩展。获得的结果可用于构建非线性波和椭圆方程的精确解。
Enhancing and essentially generalizing previous results on a class of (1+1)-dimensional nonlinear wave and elliptic equations, we apply several new techniques to classify admissible point transformations within this class up to the equivalence generated by its equivalence group. This gives an exhaustive description of its equivalence groupoid. After extending the algebraic method of group classification to non-normalized classes of differential equations, we solve the complete group classification problem for the class under study up to both usual and general point equivalences. The solution includes the complete preliminary group classification of the class and the construction of singular Lie-symmetry extensions, which are not related to subalgebras of the equivalence algebra. The complete preliminary group classification is based on classifying appropriate subalgebras of the entire infinite-dimensional equivalence algebra whose projections are qualified as maximal extensions of the kernel invariance algebra. The results obtained can be used to construct exact solutions of nonlinear wave and elliptic equations.
