Type: Article

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

Hayashi, N., Naumkin Pavel I.;

Abstract:

We study the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrodinger equation with critical nonlinearity {i partial derivative(t)(u - partial derivative(2)(x)u) + partial derivative(2)(x)u - a partial derivative(4)(x)u = lambda vertical bar u vertical bar(2)u, t > 0, x is an element of R, u(0, x) = u(0)(x), x is an element of R, where a >1/5, lambda is an element of 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 Schrodinger, Hayashi and Kaikina (Math Methods Appl Sci 40(5):1573-1597, 2017) for a third-order Schrodinger 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.