  The evolution of weak discontinuity is investigated on horizons in the
$n$-dimensional static solutions in the Einstein-Maxwell-scalar-$\Lambda$
system, including the Reissner-Nordstr\"om-(anti) de Sitter black hole. The
analysis is essentially local and nonlinear. We find that the Cauchy horizon is
unstable, whereas both the black-hole event horizon and the cosmological event
horizon are stable. This new instability, the so-called kink instability, of
the Cauchy horizon is completely different from the well-known
``infinite-blueshift'' instability. The kink instability makes the analytic
continuation beyond the Cauchy horizon unstable.
