标题： 峰孤立子方程的相似性约化：可积三次方程 显示英文标题

标题： Similarity reductions of peakon equations: integrable cubic equations

摘要： 我们考虑两个可积的三次非线性偏微分方程（PDEs）的尺度相似解，这些方程具有尖峰孤立波（peakon）解，即修改的Camassa-Holm（mCH）方程和Novikov方程。通过使用合适的倒数变换，这些变换分别将mCH方程和Novikov方程映射到负的mKdV流和负的Sawada-Kotera流，我们表明这些每个尺度相似约化通过hodograph变换与一个Painlevé类型的方程相关：对于mCH方程，其约化是二阶二度的，而对于Novikov方程，其约化是Painlevé V的一个特例。此外，我们表明这两个不同的Painlevé类型方程与从Camassa-Holm方程和Degasperis-Procesi方程的类似相似约化中出现的Painlevé III的特例有关。对于所考虑的每个三次非线性PDE，我们也给出了其周期性行波解的显式参数形式，以椭圆函数表示。我们展示了后者的一些参数图，并且通过使用Painlevé III的显式代数解，我们对一些最简单的尺度相似解进行了同样的处理，并描述了它们的主导阶渐近行为。 显示英文摘要

摘要： We consider the scaling similarity solutions of two integrable cubically nonlinear partial differential equations (PDEs) that admit peaked soliton (peakon) solutions, namely the modified Camassa-Holm (mCH) equation and Novikov's equation. By making use of suitable reciprocal transformations, which map the mCH equation and Novikov's equation to a negative mKdV flow and a negative Sawada-Kotera flow, respectively, we show that each of these scaling similarity reductions is related via a hodograph transformation to an equation of Painlev\'e type: for the mCH equation, its reduction is of second order and second degree, while for Novikov's equation the reduction is a particular case of Painlev\'e V. Furthermore, we show that each of these two different Painlev\'e-type equations is related to the particular cases of Painlev\'e III that arise from analogous similarity reductions of the Camassa-Holm and the Degasperis-Procesi equation, respectively. For each of the cubically nonlinear PDEs considered, we also give explicit parametric forms of their periodic travelling wave solutions in terms of elliptic functions. We present some parametric plots of the latter, and, by using explicit algebraic solutions of Painlev\'e III, we do the same for some of the simplest examples of scaling similarity solutions, together with descriptions of their leading order asymptotic behaviour.