The authors prove the long time stability of KAM tori (thus quasi-periodic solutions) for nonlinear Schrodinger equation $$sqrt{-1}, u_=u_-M_u+varepsilon|u|^2u,$$ subject to Dirichlet boundary conditions $u(t,0)=u(t,pi)=0$, where $M_$ is a real Fourier multiplier. More precisely, they show that, for a typical Fourier multiplier $M_$, any solution with the initial datum in the $delta$-neighborhood of a KAM torus still stays in the $2delta$-neighborhood of the KAM torus for a polynomial long time such as $|t|leq delta^{-mathcal}$ for any given $mathcal M$ with $0leq...