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:(1) 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 ) , ρ 2 ( 0 , t ) ) . Additionally, we could apply Hubbleʼs transformation c ( t ) = a ˙ ( 3 t ) a ( 3 t ) , to simplify the ordinary differential system involving ( a ( 3 t ) , b ( t ) , ρ 2 ( 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.