  We study stability of solutions of the Cauchy problem on the line for the
Camassa-Holm equation $u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with initial
data $u_0$. In particular, we derive a new Lipschitz metric $d_\D$ with the
property that for two solutions $u$ and $v$ of the equation we have
$d_\D(u(t),v(t))\le e^{Ct} d_\D(u_0,v_0)$. The relationship between this metric
and the usual norms in $H^1$ and $L^\infty$ is clarified. The method extends to
the generalized hyperelastic-rod equation
$u_t-u_{xxt}+f(u)_x-f(u)_{xxx}+(g(u)+\frac12 f"(u)(u_x)^2)_x=0$ (for $f$
without inflection points).
