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 ρt+uρx+ρux=0, mt+2uxm+umx+σρρx=0, with m=u−α2uxx. 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, ρ(t,x)=f(η)/a(3t)1/3 for η2<α2/ξ, ρ(t,x)=0 for η2≥α2/ξ, u(t,x)=ȧ(3t)/a(3t)x, ä(s)−ξ/3a(s)1/3=0,a(0)=a0>0,ȧ(0)=a1, f(η)=ξ−1/ξη2+(α/ξ)2, where η=xa(s)1/3 with s=3t; ξ>0 and α≥0 are arbitrary constants. Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems.