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On the existence of full dimensional KAM torus for nonlinear Schrödinger equation

  • * Corresponding author: Yuan Wu

    * Corresponding author: Yuan Wu

H.C. is supported by the NNSFC (No. 11671066). L.M. is supported by the NNSFC (No. 11401041) and SPNSF (ZR2019MA062). Y.S. is supported by China Postdoctoral Science Foundation (No. 2018M641050). Y.W. is supported by NNSFC (No. 11790272 and No. 11421061)

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  • In this paper, we study the following nonlinear Schrödinger equation

    $ \begin{eqnarray} \sqrt{-1}u_{t}-u_{xx}+V*u+\epsilon f(x)|u|^4u = 0, \ x\in\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}, ~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)\end{eqnarray} $

    where $ V* $ is the Fourier multiplier defined by $ \widehat{(V* u})_n = V_{n}\widehat{u}_n, V_n\in[-1, 1] $ and $ f(x) $ is Gevrey smooth. It is shown that for $ 0\leq|\epsilon|\ll1 $, there is some $ (V_n)_{n\in\mathbb{Z}} $ such that, the equation (1) admits a time almost periodic solution (i.e., full dimensional KAM torus) in the Gevrey space. This extends results of Bourgain [7] and Cong-Liu-Shi-Yuan [8] to the case that the nonlinear perturbation depends explicitly on the space variable $ x $. The main difficulty here is the absence of zero momentum of the equation.

    Mathematics Subject Classification: Primary: 37K55; Secondary: 35Q56, 35K55.


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