AN AVERAGING THEOREM FOR NONLINEAR SCHRODINGER EQUATIONS WITH SMALL NONLINEARITIES

Consider nonlinear Schrodinger equations with small nonlinearities (d)(dt)u |(-Delta u | V(x)u) = cP(Delta u, del u, u, x), x is an element of T-d. (*) dt Let {zeta(1)(x),zeta(2)(x),...} be the L-2-basis formed by eigenfunctions of the operator -Delta +V(x). For any complex function u(x), write it as u(x) = Sigma(k >= k) v(k)zeta(k) (x) and set I-k(u) = 1/2|vk|(2). Then for any solution u(t, x) of the linear equation (*)(is an element of)=0 we have I(u(t, .)) = const. In this work it is proved that if (*) is well posed on time-intervals t less than or similar to is an element of(-1) and satisfies there some mild a-priori assumptions, then for any its solution u(is an element of)(t, x), the limiting behavior of the curve I(u(is an element of)(t, .)) on time intervals of order subset of(-1), as c -> 0, can be uniquely characterized by solutions of a certain well-posed effective equation.