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On the existence of full dimensional KAM torus for fractional nonlinear Schr\"odinger equation.
Yuan Wu,Xiaoping Yuan
Keywords:KAM theory, Almost periodic solution, Gevrey space, Fractional nonlinear Schr\"odinger equation.
      In this paper,\ we study fractional nonlinear Schr\"{o}dinger equation (FNLS) with periodic boundary condition \begin{eqnarray}\label{maineq0} \textbf{i}u_{t}=-(-\Delta)^{s_{0}} u-V*u-\epsilon f(x)|u|^4u,\ ~~x\in \mathbb{T}, ~~t\in \mathbb{R}, ~~s_{0}\in (\frac12,1), \end{eqnarray} where $(-\Delta)^{s_{0}}$ is the Riesz fractional differentiation defined in \cite{ki2006th} and $V*$ is the Fourier multiplier defined by \begin{equation*}\widehat{V*u}(n)=V_n\widehat{u}(n),\ V_n\in\left[-1,1\right], \end{equation*} and $f(x)$ is Gevrey smooth.\ We prove that for $0\leq|\epsilon|\ll1$ and appropriate $V$,\ the equation (\ref{maineq0}) admits a full dimensional KAM torus in the Gevrey space satisfying \begin{equation*} \frac12e^{-rn^{\theta}}\leq \left|q_n\right|\leq 2e^{-rn^{\theta}}, \theta\in (0,1), \end{equation*} which generalizes the results given by \cite{BJFA2005,CLSY,CMSW} to fractional nonlinear Schr\"{o}dinger equation.