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% equation { array [c]c% ρ_t+uρ_x+ρu_x=0 m_t+2u_xm+um_x+σρρ_x=0 array . equation with equation m=u-α²u_xx. 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:% equation { array [c]c% ρ(t,x)={ array [c]c% f( η) a(3t)¹/3, for η²< α²ξ 0, for η²≥α²ξ% array . ,u(t,x)=·a(3t)a(3t)x ··a(s)-ξ3a(s)¹/3=0, a(0)=a₀% >0, ·a(0)=a₁ f(η)=ξ√-1ξη²+( αξ) ²% array . equation where $η=\frac{x}{a(s)^{1/3}}$