Modified scattering for the critical nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equationiut+Δu=λ|u|2Nu in all dimensions N≥1, where λ∈C and ℑλ≤0. We construct a class of initial values for which the corresponding solution is global and decays as t→∞, like t−N2 if ℑλ=0 and like (tlog⁡t)−N2 if ℑλ<0. Moreover, we give an asymptotic expansion of those solutions as t→∞. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at u=0. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.