We propose an approach that permits to avoid instability phenomena for the nonlinear Schrödinger 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.
"Nonlinear Schrödinger equation and frequency saturation." Anal. PDE 5 (5) 1157 - 1173, 2012. https://doi.org/10.2140/apde.2012.5.1157