Stability of KAM tori for nonlinear Schrödinger equation

The authors prove the long time stability of KAM tori (thus quasi-periodic solutions) for nonlinear Schrödinger equation \sqrt{-1}\, u_{t}=u_{xx}-M_{\xi}u+\varepsilon|u|^2u, subject to Dirichlet boundary conditions u(t,0)=u(t,\pi)=0, where M_{\xi} is a real Fourier multiplier. More precisely, they show that, for a typical Fourier multiplier M_{\xi}, any solution with the initial datum in the \delta-neighborhood of a KAM torus still stays in the 2\delta-neighborhood of the KAM torus for a polynomial long time such as |t|\leq \delta^{-\mathcal{M}} for any given \mathcal M with 0\leq \mathcal{M}\leq C(\varepsilon), where C(\varepsilon) is a constant depending on \varepsilon and C(\varepsilon)\rightarrow\infty as \varepsilon\rightarrow0.