  We study the periodic non-linear Schrodinger equations with odd integer power
nonlinearities, for initial data which are assumed to be small in some negative
order Sobolev space, but which may have large L^2 mass. These equations are
known to be illposed in H^s for all negative s, in the sense that the solution
map fails to be uniformly continuous from H^s to itself, even for short times
and small norms. Here we show that these equations are even more unstable. For
the cubic equation, the solution map is discontinuous from H^s to even the
space of distributions. For the quintic and higher order nonlinearities, there
exist pairs of solutions which are uniformly bounded in H^s, are arbitrarily
close in any C^N norm at time zero, and fail to be close in the distribution
topology at an arbitrarily small positive time.
