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Invariant tori of a nonlinear Schrödinger equation with quasi-periodically unbounded perturbations

The first author is supported by the National Natural Science Foundation of China (Grant 11171185, 11571201) and the Research Foundation for Doctor Programme of Henan Polytechnic University (Grant B2016-58). the second author is supported by the National Natural Science Foundation of China (Grant 11171185, 11571201).
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  • This paper is concerned with the derivative nonlinear Schrödinger equation with quasi-periodic forcing under periodic boundary conditions

    $ {\rm{i}} u_t+u_{xx}+{\rm{i}} (B+\epsilon g(\beta t))(f(|u|^2)u)_x=0, \ \ x\in \mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}. $

    Assume that the frequency vector $\beta$ is co-linear with a fixed Diophantine vector $\bar{\beta}\in \mathbb{R}.{m}$, that is, $\beta=\lambda \bar{\beta}$, $\lambda \in [1/2, 3/2]$. We show that above equation possesses a Cantorian branch of invariant $n$--tori and exists many smooth quasi-periodic solutions with $(m+n)$ non-resonance frequencies $(\lambda\bar{\beta}, \omega_{\ast})$. The proof is based on a Kolmogorov--Arnold--Moser (KAM) iterative procedure for quasi-periodically unbounded vector fields and partial Birkhoff normal form.

    Mathematics Subject Classification: Primary: 35B15, 35Q41, 35Q55, 37K55.

    Citation:

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