Factorization technique for the fourth-order nonlinear Schrödinger equation

We consider the Cauchy problem for the cubic fourth-order nonlinear Schrödinger equation i∂tu+14∂x4u=iλ∂x(u2u),t>0,x∈R,u0,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}$$\left\{\begin{array}{ll}i\partial_{t}u+\frac{1}{4}\partial_{x}^{4}u = i \lambda \partial _{x}(\left| u \right| ^{2}u),&\quad t > 0,\, x \in \mathbf{R},\\ u \left( 0,x\right) = u_{0}\left( x\right) ,&\quad x \in \mathbf{R},\end{array}\right.$$\end{document}where λ∈R.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda \in \mathbf{R}.}$$\end{document} We introduce the factorization formula for the free evolution group to prove the global existence of solutions. Also we show that the large time asymptotics of solutions has a logarithmic correction in the phase comparing with the corresponding linear case.