Modified scattering for the nonlinear nonlocal Schrödinger equation in one-dimensional case

We study the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrödinger equation with critical nonlinearity i ∂ t u - ∂ x 2 u + ∂ x 2 u - a ∂ x 4 u = λ u 2 u , t > 0 , x ∈ R , u 0 , x = u 0 x , x ∈ R , where a > 1 5 , λ ∈ R . We continue to develop the factorization techniques which was started in papers Hayashi and Naumkin (Z Angew Math Phys 59(6):1002–1028, 2008) for Klein–Gordon, Hayashi and Naumkin (J Math Phys 56(9):093502, 2015) for a fourth-order Schrödinger, Hayashi and Kaikina (Math Methods Appl Sci 40(5):1573–1597, 2017) for a third-order Schrödinger to show the modified scattering of solutions to the equation. The crucial points of our approach presented here are based on the L 2 -boundedness of the pseudodifferential operators.