An averaging theorem for nonlinear Schrödinger equations with small nonlinearities

Consider nonlinear Schrödinger equations with small nonlinearities ddt/u+i(−△u+V(x)u)=ϵP(△u,∇u,u,x), x∈T^d. (*) Let {ζ1(x),ζ2(x),…} be the L_2-basis formed by eigenfunctions of the operator −△+V(x). For any complex function u(x), write it as u(x)=∑_(k≥1)^v_kζ_k(x) and set I_k(u)=1/2|v_k|^2. Then for any solution u(t,x) of the linear equation (∗)_(ϵ=0) we have I(u(t,⋅))=const. In this work it is proved that if (∗) is well posed on time-intervals t≲ϵ^(−1) and satisfies there some mild a-priori assumptions, then for any its solution u^ϵ(t,x), the limiting behavior of the curve I(u^ϵ(t,⋅)) on time intervals of order ϵ^(−1), as ϵ→0, can be uniquely characterized by solutions of a certain well-posed effective equation.