# American Institute of Mathematical Sciences

January  2017, 16(1): 25-68. doi: 10.3934/cpaa.2017002

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

 1 School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, P. R. China 2 School of Mathematics, Shandong University, Jinan 250100, P. R. China

Received  June 2015 Revised  August 2016 Published  November 2016

Fund Project: 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).

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.
Citation: Jie Liu, Jianguo Si. Invariant tori of a nonlinear Schrödinger equation with quasi-periodically unbounded perturbations. Communications on Pure & Applied Analysis, 2017, 16 (1) : 25-68. doi: 10.3934/cpaa.2017002
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