  Instantons are tunneling solutions that connect two vacua, and under a small
change in the potential, instantons sometimes disappear. We classify these
disappearances as smooth (decay rate goes to 0 at disappearance) or abrupt
(decay rate not equal to 0 at disappearance). Abrupt disappearances mean that a
small change in the parameters can produce a drastic change in the physics, as
some states become suddenly unreachable. The simplest abrupt disappearances are
associated with annihilation by another Euclidean solution with higher action
and one more negative mode; higher-order catastrophes can occur in cases of
enhanced symmetry. We study a few simple examples, including the 6D
Einstein-Maxwell theory, and give a unified account of instanton
disappearances.
