arXiv 1608.046042016-08nlin.SImath-phmath.MP

广义多组分Camassa-Holm-Novikov方程组

A generalised multicomponent system of Camassa-Holm-Novikov equations

Diego Catalano Ferraioli, Igor Leite Freire

摘要：

在本文中，我们介绍了一个两个组成部分的系统，取决于一个参数$b$，推广的Camassa-Holm（$b=1$）和Novikov方程（$b=2$）。通过研究它的经典和高阶对称的李代数到$3$阶，我们发现，对于$b\neq 2$，系统允许一个点对称的$3$维代数，显然没有高阶对称，而对于$b=2$，它有一个点对称的$6$维代数，也有高阶对称。我们还提供了所有的守恒律，一阶特征，这是承认的系统$b=1,2$。此外，对于$b=2$，我们表明，该系统是一个特殊的情况下，一个更一般的系统，其中承认一个$\mathfrak{sl}(3,\mathbb{R})$值零曲率表示。最后，我们发现系统存在峰子解，特别是对于$b=2$，存在具有非常数振幅的1-峰子解。

英文摘要：

In this paper we introduce a two-component system, depending on a parameter $b$, which generalises the Camassa-Holm ($b=1$) and Novikov equations ($b=2$). By investigating its Lie algebra of classical and higher symmetries up to order $3$, we found that for $b\neq 2$ the system admits a $3$-dimensional algebra of point symmetries and apparently no higher symmetries, whereas for $b=2$ it has a $6$-dimensional algebra of point symmetries and also higher order symmetries. Also we provide all conservation laws, with first order characteristics, which are admitted by the system for $b=1,2$. In addition, for $b=2$, we show that the system is a particular instance of a more general system which admits an $\mathfrak{sl}(3,\mathbb{R})$-valued zero-curvature representation. Finally, we found that the system admits peakon solutions and, in particular, for $b=2$ there exist 1-peakon solutions with non-constant amplitude.

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摘要：

英文摘要：

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