论文标题
pearpemon方程的相似性降低：$ b $ - 家庭
Similarity reductions of peakon equations: the $b$-family
论文作者
论文摘要
$ b $ - 家庭是非进化类型的哈密顿部分偏微分方程的单参数，这是在浅水波理论中产生的。它承认了各种解决方案，包括著名的山峰源，它们是峰值孤子的弱解决方案，在峰值上具有不连续的第一衍生物，以及以精确形式和/或数值获得的其他有趣的解决方案。在每种特殊情况下，$ b = 2,3 $（分别分别是Camassa-Holm和Degasperis-Procesi方程式）该方程是完全可以集成的，从某种意义上说，它承认了lax对和通勤的本地对称性的无限层次结构，但对于其他值$ b $ it不可整合。在通过使用互惠转换讨论了流动波的讨论之后，该转化降低到这些解决方案满足的普通微分方程级别上的大节符转换后，我们将相同的技术应用于$ b $ - family的缩放缩放相似性解决方案，并显示出$ b = 2 $或$ 3 $降低的其他障碍时，涉及其他相似性的痛苦II时，杂技造影的其他障碍是相关性的，则$ b $由此产生的普通微分方程不是Painlevé类型。
The $b$-family is a one-parameter family of Hamiltonian partial differential equations of non-evolutionary type, which arises in shallow water wave theory. It admits a variety of solutions, including the celebrated peakons, which are weak solutions in the form of peaked solitons with a discontinuous first derivative at the peaks, as well as other interesting solutions that have been obtained in exact form and/or numerically. In each of the special cases $b=2,3$ (the Camassa-Holm and Degasperis-Procesi equations, respectively) the equation is completely integrable, in the sense that it admits a Lax pair and an infinite hierarchy of commuting local symmetries, but for other values of the parameter $b$ it is non-integrable. After a discussion of travelling waves via the use of a reciprocal transformation, which reduces to a hodograph transformation at the level of the ordinary differential equation satisfied by these solutions, we apply the same technique to the scaling similarity solutions of the $b$-family, and show that when $b=2$ or $3$ this similarity reduction is related by a hodograph transformation to particular cases of the Painlevé III equation, while for all other choices of $b$ the resulting ordinary differential equation is not of Painlevé type.
