广义浅水波方程的Lie对称性与奇异性分析
Lie symmetries and singularity analysis for generalized shallow-water equations
摘要：
利用不变点变换理论和奇异性分析方法，对广义Camassa-Holm方程和广义Benjamin-Bono-Mahoney方程进行了全面的研究.从李理论出发，我们发现这两个方程在同一个三维李代数下是不变的，这个三维李代数就是Camassa-Holm方程所承认的同一个李代数。我们确定的一维最优系统的承认李对称性，我们进行了一个完整的分类，我们研究的两个方程的相似性解决方案。利用点对称性或奇异性分析对约化方程进行了研究。最后，将奇异性分析直接应用到偏微分方程上，由此我们推断所研究的广义方程通过了奇异性检验，是可积的。
英文摘要：
We perform a complete study by using the theory of invariant point transformations and the singularity analysis for the generalized Camassa-Holm equation and the generalized Benjamin-Bono-Mahoney equation. From the Lie theory we find that the two equations are invariant under the same three-dimensional Lie algebra which is the same Lie algebra admitted by the Camassa-Holm equation. We determine the one-dimensional optimal system for the admitted Lie symmetries and we perform a complete classification of the similarity solutions for the two equations of our study. The reduced equations are studied by using the point symmetries or the singularity analysis. Finally, the singularity analysis is directly applied on the partial differential equations from where we infer that the generalized equations of our study pass the singularity test and are integrable.
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