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

Abstract

This paper is concerned with the derivative nonlinear Schrodinger equation with quasi-periodic forcing under periodic boundary conditions \begin{document} $ {\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}. $ \end{document} 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.