Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations

In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \begin{equation} \left{ \begin{array} [c]{c}% \rho_{t}+u\rho_{x}+\rho u_{x}=0 m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0 \end{array} \right. \end{equation} with \begin{equation} m=u-\alpha^{2}u_{xx}. \end{equation} By the separation method, we can obtain a class of blowup or global solutions for $\sigma =1$ or $-1$. In particular, for the integrable system with $\sigma =1$, we have the global solutions:% \begin{equation} \left{ \begin{array} [c]{c}% \rho(t,x)=\left{ \begin{array} [c]{c}% \frac{f\left( \eta\right) }{a(3t)^{1/3}},$\text{ for }$\eta^{2}<\frac {\alpha^{2}}{\xi} 0,$\text{ for }$\eta^{2}\geq\frac{\alpha^{2}}{\xi}% \end{array} \right. ,u(t,x)=\frac{\overset{\cdot}{a}(3t)}{a(3t)}x \overset{\cdot\cdot}{a}(s)-\frac{\xi}{3a(s)^{1/3}}=0,$$a(0)=a_{0}%

0,$$\overset{\cdot}{a}(0)=a_{1} f(\eta)=\xi\sqrt{-\frac{1}{\xi}\eta^{2}+\left( \frac{\alpha}{\xi}\right) ^{2}}% \end{array} \right. \end{equation} where $\eta =\frac{x}{a(s{)}^{1\mathrm{/}3}}$ with $s=3t;$ $\xi >0$ and $\alpha \ge 0$ are arbitrary constants.\newline Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems.