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

Abstract

In this article, we study the self-similar solutions of the 2-component Degasperis-Procesi water system:% [c]{c}% ρ_{t}+k_{2}uρ_{x}+(k_{1}+k_{2})ρu_{x}=0 u_{t}-u_{xxt}+4uu_{x}-3u_{x}u_{xx}-uu_{xxx}+k_{3}ρρ_{x}=0. By the separation method, we can obtain a class of self-similar solutions,% [c]{c}% ρ(t,x)=\max(\frac{f(η)}{a(4t)^{(k_{1}+k_{2})/4}},\text{}0),\text{}u(t,x)=\frac{\overset{\cdot}{a}(4t)}{a(4t)}x \overset{\cdot\cdot}{a}(s)-\fracξ{4a(s)^κ}=0,\text{}a(0)=a_{0}% \neq0,\text{}\overset{\cdot}{a}(0)=a_{1} f(η)=\frac{k_{3}}ξ\sqrt{-\fracξ{k_{3}}η^{2}+(\fracξ{k_{3}}α) ^{2}}% where $\hat{I}\cdot =\frac{x}{a(s{)}^{1/4}}$ with $s=4t;$ $\hat{I}º=\frac{k_1}{2}$ $\hat{I}\pm \ge 0,$ $\hat{I}¾<0$, $a_0$ and $a_1$ are constants. which the local or global behavior can be determined by the corresponding Emden equation. The results are very similar to the one obtained for the 2-component Camassa-Holm equations. Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems. With the characteristic line method, blowup phenomenon for $k_3\ge 0$ is also studied.

13 Pages, Key Words: 2-Component Degasperis-Procesi, Shallow Water System, Analytical Solutions, Blowup, Global, Self-Similar, Separation Method, Construction of Solutions, Moving Boundary, 2-Component Camassa-Holm Equations