Cancellations of Resonances and Long Time Dynamics of Cubic Schrödinger Equation on $${\mathbb {T}}$$

We prove a vanishing property of the normal form transformation 1D cubic nonlinear Schrödinger (NLS) equation with periodic boundary conditions on [0, L]. apply this to quintic resonance interactions and obtain description dynamics for time up $$T=\frac{L^2}{\epsilon ^4}$$ , if L is sufficiently large size initial data $$\epsilon $$ small enough. Since T characteristic wave turbulence, result implies absence turbulence behavior NLS. Our approach can be adapted other integrable systems without too many difficulties. In proof, we develop correspondence between Feynman diagrams terms in forms, which allows us calculate coefficients inductively.