Perturbational Blowup Solutions to the 2-Component Camassa-Holm Equations

TL;DR

This paper develops a perturbational method to construct new analytical solutions for the 2-component Camassa-Holm equations, including conditions for blowup or global existence, extending previous results with a novel approach.

Contribution

It introduces a perturbational approach combining substitution and separation methods to find new solutions and analyze their blowup or global existence.

Findings

Constructed a new class of analytical solutions.

Linked solution behavior to the Emden equation.

Extended previous results with a perturbational method.

Abstract

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: 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 $(c(t),b(t),\rho^{2}(0,t)).$ Additionally, we could apply the Hubble's transformation c(t)={\dot{a}(3t)}/{a(3t)}, to simplify the ordinary differential system involving $(a(3t),b(t),\rho ^{2}(0,t))$. After proving the global or local existences of the corresponding dynamical system, a new class of analytical solutions is shown. And the corresponding solutions in radial symmetry are also given. To determine that the solutions exist globally or blow up, we just use the qualitative properties about the well-known Emden equation: {array} [c]{c} {d^{2}/{dt^{2}}}a(3t)= {\xi}{a^{1/3}(3t)}, a(0)=a_{0}>0 ,\dot{a}(0)=a_{1} {array} . Our solutions obtained by the perturbational method, fully cover the previous known results in "M.W. Yuen, \textit{Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations,}J. Math. Phys., \textbf{51} (2010) 093524, 14pp." by the separation method.