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.

5 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

References in corpus (4)

• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• On an integrable two-component Camassa-Holm shallow water system
• Analytical Solutions to the Navier-Stokes Equations
• Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars

Cited by in corpus (1)

• Self-Similar Blowup Solutions to the 2-Component Degasperis-Procesi Shallow Water System