Type: Article

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

Hayashi, Nakao; Naumkin, Pavel I.;

Abstract:

We study the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrödinger equation with critical nonlinearity {i?t(u??2xu)+?2xu?a?4xu=?|u|2u, t>0, x?R,u(0,x)=u0(x), x?R, where a>15, ??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 L2-boundedness of the pseudodifferential operators.