On the asymptotics for cubic nonlinear Schrödinger equations

We consider the Cauchy problem for the cubic nonlinear Schrödinger equation where We prove the global existence of small solutions , if the initial data u 1 belong to some analytic function space and are sufficiently small. For the coefficients λ j we assume that there exists θ 0 > 0 such that for all and also we suppose that the initial data are such that where ϵ is a small positive constant depending on the size of initial function in a suitable norm. We also find the large time asymptotic formulas for solutions. In the short range region the solution has an additional logarithmic time decay comparing with the corresponding linear case.