A Nekhoroshev-type theorem for the nonlinear Schrödinger equations on the d -dimensional torus

We prove a Nekhoroshev type long time stability result for the following nonlinear Schrödinger (NLS) equations on the d-dimensional torus iut=−△u+V∗u+∂F(|u|2)∂u¯(u,u¯),x∈Td,t∈R,where V is a smooth convolution potential, F(z) is a real-valued polynomial function in z satisfying F(0)=F′(0)=0. More precisely, it is proved that in modified Sobolev space (see (Equation2(2) hp(Td):={u(x)=∑j∈Zdqjeij⋅x⏐‖u‖hp(Td)2=‖q‖hp2:=∑j∈Zd⌊j⌋2p|qj|2<∞},(2) )) if the norm of initial datum is smaller than ε(0<ε≪1), then the corresponding solution is bounded by 2ε over time-intervals of length ee1ε. The result generalizes the 1-dimensional case in L. Biasco, J.E. Massetti, M. Procesi. [An abstract Birkhoff normal form theorem and exponential type stability of the 1d NLS. Comm Math Phys. 2020;375(3):2089–2153] to d-dimensional NLS equations.