Fractional nonlinear Schrödinger equation

We consider the Cauchy problem for the fractional nonlinear Schrödinger equation i∂tu+23∂x32u=λu2u,t>0,x∈R,u1,x=u0x,x∈R.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} i\partial _{t}u+\frac{2}{3}\left| \partial _{x}\right| ^{\frac{3}{2} }u=\lambda \left| u\right| ^{2}u,\,\, t>0, &{}\quad x\in \mathbb {R},\\ u\left( 1,x\right) =u_{0}\left( x\right) ,&{}\quad x\in \mathbb {R}. \end{array}\right. \end{aligned}$$\end{document}We develop the factorization technique to obtain the large-time asymptotic behavior of solutions which has a logarithmic phase modifications for large time comparing with the linear problem.