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

Abstract

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 $\sigma =1$ or $-1$. In particular, for the integrable system with $\sigma =1$, we have the global solutions:% \begin{equation} \left\{ \begin{array} [c]{c}% 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 $\eta =\frac{x}{a(s{)}^{1/3}}$ 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.

arXiv comment5 more figures can be found in the corresponding journal paper (J. Math. Phys. 51, 093524 (2010) ). Key Words: 2-Component Camassa-Holm Equations, Shallow Water System, Analytical Solutions, Blowup, Global, Self-Similar, Separation Method, Construction of Solutions, Moving Boundary

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Cited by in corpus (4)

• Perturbational Blowup Solutions to the 1-dimensional Compressible Euler Equations
• Some Exact Blowup Solutions to the Pressureless Euler Equations in R^N
• Self-Similar Blowup Solutions to the 2-Component Degasperis-Procesi Shallow Water System
• Perturbational Blowup Solutions to the 2-Component Camassa-Holm Equations