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Invariant tori for the derivative nonlinear Schrödinger equation with nonlinear term depending on spatial variable

  • *Corresponding author: Shimin Wang

    *Corresponding author: Shimin Wang
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  • We focus on a class of derivative nonlinear Schrödinger equation with reversible nonlinear term depending on spatial variable $ x $:

    $ \begin{equation*} \mathrm{i} u_t+u_{xx}-\bar{u}u_{x}^2 + F(x, u, \bar{u}, u_{x}, \bar{u}_{x}) = 0, \quad x\in \mathbb{T}: = \mathbb{R}/2\pi\mathbb{Z}, \end{equation*} $

    where the nonlinear term $ F $ is an analytic function of order at least five in $ u, \bar{u}, u_{x}, \bar{u}_{x} $ and satisfies

    $ \begin{equation*} F(x, u, \bar{u}, u_{x}, \bar{u}_{x}) = \overline{F(x, \bar{u}, u, \bar{u}_{x}, u_{x})}. \end{equation*} $

    Moreover, we also assume that $ F $ satisfies the homogeneous condition (6) to overcome the degeneracy. We prove the existence of small amplitude, smooth quasi-periodic solutions for the above equation via establishing an abstract infinite dimensional Kolmogorov–Arnold–Moser (KAM) theorem for reversible systems with unbounded perturbation.

    Mathematics Subject Classification: Primary: 37K55, 37J40; Secondary: 70K43, 35Q55.


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