Modified scattering for the derivative fractional nonlinear Schrodinger equation

We study the large time asymptotic behavior of solutions to the Cauchy problem for the fractional derivative nonlinear Schrodinger equation in one space dimension {i partial derivative(t)u - 1/alpha vertical bar partial derivative(x)vertical bar(alpha) u = i lambda partial derivative(x) (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 alpha is an element of (0,1) boolean OR (1,2) boolean OR (2,3), and lambda is an element of R. The fractional derivative vertical bar partial derivative(x)vertical bar(alpha) = F-1 vertical bar xi vertical bar(alpha) F, where Fstands for the Fourier transformation (phi) over cap(xi) = 1/root 2 pi integral(R) e(-ix xi) phi f(x)dx, and F-1 is the inverse Fourier transformation. Equation (1.1) with alpha = 2is the derivative NLS equation and studied extensively. The case of alpha = 1corresponds to the so-called derivative half-wave equation. We prove that the modified scattering of solutions occurs when 0 < alpha < 3without the exceptional point alpha = 1. (c) 2023 Elsevier Inc. All rights reserved.