Fractional nonlinear Schrödinger equation

Abstract

We consider the Cauchy problem for the fractional nonlinear Schrodinger equation $$\begin{aligned} \left\{ \begin{array}{ll} i\partial _{t}u+\frac{2}{3}\left| \partial _{x}\right| ^{\frac{3}{2} }u=\lambda \left| u\right| ^{2}u,\,\, t>0, &{}\quad x\in \mathbb {R},\\ u\left( 1,x\right) =u_{0}\left( x\right) ,&{}\quad x\in \mathbb {R}. \end{array}\right. \end{aligned}$$ 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.