  We propose an approach that permits to avoid instability phenomena for the
nonlinear Schrodinger equations. We show that by approximating the solution in
a suitable way, relying on a frequency cut-off, global well-posedness is
obtained in any Sobolev space with nonnegative regularity. The error between
the exact solution and its approximation can be measured according to the
regularity of the exact solution, with different accuracy according to the
cases considered.
