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}% ρ_{t}+uρ_{x}+ρu_{x}=0 m_{t}+2u_{x}m+um_{x}+σρρ_{x}=0 \end{array} \right. \end{equation} with \begin{equation} m=u-α^{2}u_{xx}. \end{equation} By the separation method, we can obtain a class of blowup or global solutions for $σ=1$ or $-1$. In particular, for the integrable system with $σ=1$, we have the global solutions:% \begin{equation} \left\{ \begin{array} [c]{c}% ρ(t,x)=\left\{ \begin{array} [c]{c}% \frac{f\left( η\right) }{a(3t)^{1/3}},\text{ for }η^{2}<\frac {α^{2}}ξ 0,\text{ for }η^{2}\geq\frac{α^{2}}ξ% \end{array} \right. ,u(t,x)=\frac{\overset{\cdot}{a}(3t)}{a(3t)}x \overset{\cdot\cdot}{a}(s)-\fracξ{3a(s)^{1/3}}=0,\text{ }a(0)=a_{0}% >0,\text{ }\overset{\cdot}{a}(0)=a_{1} f(η)=ξ\sqrt{-\frac{1}ξη^{2}+\left( \fracαξ\right) ^{2}}% \end{array} \right. \end{equation} where $η=\frac{x}{a(s)^{1/3}}$ with $s=3t;$ $ξ>0$ and $α\geq0$ are arbitrary constants.\newline Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems.

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