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

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In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% equation { array [c]c% ρₜ+uρₓ+ρuₓ=0 mₜ+2uₓm+umₓ+σρρₓ=0 array. equation with equation m=u-α²uxx. 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=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.