Modified scattering for higher-order nonlinear Schrödinger equation in one space dimension

We consider the large time asymptotics of solutions to the Cauchy problem for the higher-order nonlinear Schrödinger equation with critical cubic nonlinearity {i∂tu+12∂x2u-1α|∂x|αu=λ|u|2u,t>0,x∈R,u(0,x)=u0(x),x∈R,where λ∈ R and α= 4 or α≥ 5 , since while estimating pseudodifferential operators below we need that the condition such that the symbol Λ(ξ)=12ξ2+1α|ξ|α∈C5(R). We show that the modified scattering occurs in the uniform norm. We continue to develop the factorization techniques which was started in papers (Ozawa in Commun Math Phys 139(3):479–493, 1991; Hayashi and Ozawa in Ann IHP (Phys Théor) 48:17–37, 1988; Hayashi and Naumkin in Z Angew Math Phys 59(6):1002–1028, 2008; Hayashi and Kaikina in Math Methods Appl Sci 40(5):1573–1597, 2017; Hayashi and Naumkin in J Math Phys 56(9):093502, 2015). The crucial points of our approach presented here are the L²-boundedness of the pseudodifferential operators which are used to obtain estimates of nonlinear terms in weighted Sobolev space.