| Volume 10, Number 2, 2020, Pages 771-794 DOI：10.11948/20190292 |
| On the existence of full dimensional KAM torus for fractional nonlinear Schrodinger equation |
| Yuan Wu,Xiaoping Yuan |
| Keywords:KAM theory, almost periodic solution, Gevrey space, fractional nonlinear Schrodinger equation. |
| Abstract: |
| In this paper,\ we study fractional nonlinear Schrodinger equation (FNLS) with periodic boundary condition
$$
\textbf{i}u_{t}=-(-\Delta)^{s_{0}} u-V*u-\epsilon f(x)|u|^4u,\ ~~x\in \mathbb{T}, ~~t\in \mathbb{R}, ~~s_{0}\in (\frac12,1),~~~~~~~~~~~~~~~~~~~~~~~~~~~~(0.1)
$$
where $(-\Delta)^{s_{0}}$ is the Riesz fractional differentiation defined in [21] and $V*$ is the Fourier multiplier defined by $\widehat{V*u}(n)=V_n\widehat{u}(n),\ V_n\in\left[-1,1\right],$ and $f(x)$ is Gevrey smooth. We prove that for $0\leq|\epsilon|\ll1$ and appropriate $V$,\ the equation (0.1) admits a full dimensional KAM torus in the Gevrey space satisfying $ \frac12e^{-rn^{\theta}}\leq \left|q_n\right|\leq 2e^{-rn^{\theta}}, \theta\in (0,1),$
which generalizes the results given by [8-10] to fractional nonlinear Schrodinger equation. |
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