A note on stability of Eliasson-Kuksin's KAM tori for the nonlinear Schr\"{o}dinger equation

Eliasson and Kuksin developed a KAM approach to study the persistence of the invariant tori for nonlinear Schr"{o}dinger equation on ${\mathbb{T}}^d$. In this note, we improve Eliasson and Kuksin’s KAM theorem by using Kolmogorov’s iterative scheme and obtain a local normal form for the transformed Hamiltonian. As a consequence, we are able to derive the time ${\delta }^{-1}$ stability of the obtained KAM tori.