Perturbational blowup solutions to the 2-component Camassa–Holm equations

In this article, we study the perturbational method to construct the non-radially symmetric solutions of the compressible 2-component Camassa–Holm equations. In detail, we first combine the substitutional method and the separation method to construct a new class of analytical solutions for that system. In fact, we perturb the linear velocity:equation(1)u=c(t)x+b(t),u=c(t)x+b(t), and substitute it into the system. Then, by comparing the coefficients of the polynomial, we can deduce the functional differential equations involving (c(t),b(t),ρ2(0,t))(c(t),b(t),ρ2(0,t)). Additionally, we could apply Hubbleʼs transformation c(t)=a˙(3t)a(3t), to simplify the ordinary differential system involving (a(3t),b(t),ρ2(0,t))(a(3t),b(t),ρ2(0,t)). After proving the global or local existences of the corresponding dynamical system, a new class of analytical solutions is shown. To determine that the solutions exist globally or blow up, we just use the qualitative properties about the well-known Emden equation. Our solutions obtained by the perturbational method, fully cover Yuenʼs solutions by the separation method.