Lipschitz metric for the Camassa-Holm equation on the line

We study stability of solutions of the Cauchy problem on the line for the Camassa-Holm equation $u_t-u_{xxt}+3u{u}_x-2u_x{u}_{xx}-u{u}_{xxx}=0$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_{\text{D}}$ with the property that for two solutions $u$ and $v$ of the equation we have $d_{\text{D}}(u(t),v(t))\le e^{Ct}{d}_{\text{D}}(u_0,v_0)$. The relationship between this metric and the usual norms in $H^1$ and $L^{\mathrm{\infty }}$ is clarified. The method extends to the generalized hyperelastic-rod equation $u_t-u_{xxt}+f(u{)}_x-f(u{)}_{xxx}+(g(u)+\frac{1}{2}f\mathrm{"}(u)(u_x{)}^2{)}_x=0$ (for $f$ without inflection points).