*(English)*Zbl 1178.65099

The authors study the Camassa-Holm equation

on the real line with $\kappa =0$ and initial condition ${u|}_{t=0}=\overline{u}$. This equation admits two distinct classes of solutions, and the dichotomy between the two classes is associated with wave breaking, which takes place in finite time such that the ${H}^{1}$ and ${L}^{\infty}$ norms of the solution remain finite while the spatial derivative ${u}_{x}$ becomes pointwise unbounded. This equation is reformulated by means of a different set of variables from Eulerian to Lagrangian coordinates to produce a system of semilinear ordinary differential equations. The existence of solutions, short-time stability and global stability are established. The system is shown to be invariant with respect to a relabeling set ${\mathcal{G}}_{0}$. Lastly, the two-direction mappings between the Eulerian variable $u\in {H}^{1}$ and the Lagrangian variable $X\in {\mathcal{G}}_{0}$ is defined.