Modified scattering for the derivative fractional nonlinear Schrödinger equation

Abstract

We study the large time asymptotic behavior of solutions to the Cauchy problem for the fractional derivative nonlinear Schrödinger equation in one space dimension{i∂tu−1α|∂x|αu=iλ∂x(|u|2u), t>0,x∈R,u(0,x)=u0(x),x∈R,where α∈(0,1)∪(1,2)∪(2,3), and λ∈R. The fractional derivative |∂x|α=F−1|ξ|αF, where F stands for the Fourier transformation ϕˆ(ξ)=12π∫Re−ixξϕ(x)dx, and F−1 is the inverse Fourier transformation. Equation (1.1) with α=2 is the derivative NLS equation and studied extensively. The case of α=1 corresponds to the so-called derivative half-wave equation. We prove that the modified scattering of solutions occurs when 0<α<3 without the exceptional point α=1.