ON THE EXISTENCE OF FULL DIMENSIONAL KAM TORUS FOR FRACTIONAL NONLINEAR SCHRÖDINGER EQUATION

Yuan Wu; Xiaoping Yuan

doi:10.11948/20190292

2020 Volume 10 Issue 2

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Yuan Wu, Xiaoping Yuan. ON THE EXISTENCE OF FULL DIMENSIONAL KAM TORUS FOR FRACTIONAL NONLINEAR SCHRÖDINGER EQUATION[J]. Journal of Applied Analysis & Computation, 2020, 10(2): 771-794. doi: 10.11948/20190292

Citation:

Yuan Wu, Xiaoping Yuan. ON THE EXISTENCE OF FULL DIMENSIONAL KAM TORUS FOR FRACTIONAL NONLINEAR SCHRÖDINGER EQUATION[J]. Journal of Applied Analysis & Computation, 2020, 10(2): 771-794. doi: 10.11948/20190292

ON THE EXISTENCE OF FULL DIMENSIONAL KAM TORUS FOR FRACTIONAL NONLINEAR SCHRÖDINGER EQUATION

Yuan Wu^1, ,,
Xiaoping Yuan²

1.
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2.
School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Corresponding author: Email address: 14110840003@fudan.edu.cn (Y. Wu)
Fund Project: The author was supported by National Natural Science Foundation of China (Nos. 11790272 and 11411061)

Abstract

Abstract

In this paper, we study fractional nonlinear Schrödinger equation (FNLS) with periodic boundary condition $ \begin{equation} \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{equation} $ 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 (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, 9, 10] to fractional nonlinear Schrödinger equation.
- KAM theory /
- almost periodic solution /
- Gevrey space /
- fractional nonlinear Schrödinger equation
MSC: 37K55, 37655

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References

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