Asymptotics of solutions to the fractional nonlinear Schrödinger equation with $\alpha \gt \frac{5}{2}$

Abstract

We study the large time asymptotic behavior of solutions to the Cauchy problem for the fractional nonlinear Schrödinger equation \begin{equation*}\left\{ \begin{array}{c}i\partial _{t}u-\frac{1}{\alpha }\left\vert \partial _{x}\right\vert^{\alpha }u=\lambda \left\vert u\right\vert ^{\alpha }u,\text{ }t\gt 0,\text{ } x\in \mathbb{R}\mathbf{,} \ u\left( 0,x\right) =u_{0}\left( x\right) ,\text{ }x\in \mathbb{R}\mathbf{,}\end{array}\right.\end{equation*}where $\lambda \gt 0,$ the fractional derivative $\left\vert \partial_{x}\right\vert ^{\alpha }=\mathcal{F}^{-1}\left\vert \xi \right\vert^{\alpha }\mathcal{F},$ $\alpha \gt \frac{5}{2}.$ This paper is a sequel to our previous papers [17] for $2\lt \alpha \lt \frac{5}{2}$ and [36] for $\alpha =\frac{5}{2}$. We show that solutions decay in time at the rate $t^{-\frac{1}{\alpha }}\left( \log t\right) ^{-\frac{1}{\alpha }}$, namely that the nonlinearity acts as a dissipative term, when $\lambda \gt 0$. This phenomena does not occur for the cubic problem \begin{equation*}\left\{ \begin{array}{c}i\partial _{t}u-\frac{1}{\alpha }\left\vert \partial _{x}\right\vert^{\alpha }u=\lambda \left\vert u\right\vert ^{2}u,\text{ }t\gt 0,\text{ }x\in \mathbb{R}\mathbf{,} \ u\left( 0,x\right) =u_{0}\left( x\right) ,\text{ }x\in \mathbb{R}\mathbf{,}\end{array}\right.\end{equation*}with $0 \lt \alpha \leq 2.$