Factorization technique for the fourth-order nonlinear Schrödinger equation

Abstract

We consider the Cauchy problem for the cubic fourth-order nonlinear Schrodinger equation $$\left\{\begin{array}{ll}i\partial_{t}u+\frac{1}{4}\partial_{x}^{4}u = i \lambda \partial _{x}(\left| u \right| ^{2}u),&\quad t > 0,\, x \in \mathbf{R},\\ u \left( 0,x\right) = u_{0}\left( x\right) ,&\quad x \in \mathbf{R},\end{array}\right.$$ where \({\lambda \in \mathbf{R}.}\) 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.