Large time asymptotic behavior of solutions to higher order nonlinear Schrödinger equation

Abstract

We consider the Cauchy problem for the higher-order nonlinear Schrödinger equation\begin{gather}\label{A}i\partial_{t}u+\frac{1}{2}\partial_{x}^{2}u-\frac{1}{4}\partial_{x}^{4}u=u^{3},\text{ }t>0,\text{ }x\in\mathbb{R}, u\left( 0,x\right) =u_{0}\left( x\right) ,\text{ }x\in\mathbb{R.}\notag\end{gather}The aim of the present paper is prove the global existence of solutions to (0.1) if the initial data $u_{0}\in\mathbf{H}^{1}\cap\mathbf{H}^{0,1}$. Also we find the large time asymptotics of solutions.