Quasi-Periodic Solutions for Fractional Nonlinear Schrödinger Equation

Abstract

We establish an infinite dimensional KAM theorem with dense normal frequency. As an application, we use this theorem to study the Fractional NLS

$$\begin{aligned} iu_t-|\partial _x|^{1\over 2}u+M_\xi u=\epsilon f(|u|^2)u,\quad x\in {\mathbb T},t\in {\mathbb R}, \end{aligned}$$

where f is real analytic in a neighborhood of $0\in {\mathbb C}$ and $\epsilon >0$ is small enough. We obtain a family of small-amplitude quasi-periodic solutions with linear stability.

A Appendix

Lemma A.1

For $p> {1\over 2}$, $\rho >0$, The space ${\ell }^{\rho , p}$ is a Banach algebra with respect to convolution of sequences, and

$$\begin{aligned}\Vert q*r \Vert _{\rho ,p }\le c \Vert q \Vert _{\rho ,p }\Vert r \Vert _{\rho ,p }\end{aligned}$$

with constant c depending only on p.

For the proof of Lemma A.1, see [5, 14, 20, 23].

Lemma A.2

(Lemma 2.1 of [25]). For any regular analytic function f, g in D(r, s) and $C^1_W$ in $\mathcal O$ with finite semi-norm 2.6, one has

$$\begin{aligned} \Vert [X_f,X_g]\Vert _{ r',s'}\le 2^{2d+1}\delta ^{-1} \Vert X_f\Vert _{ r,s}\Vert X_g\Vert _{r,s},\end{aligned}$$

$$\begin{aligned} \Vert X_{\{f,g\}}\Vert _{r',s'}\le 2^{2d+1}\delta ^{-1}\Vert X_f\Vert _{ r,s}\Vert X_g\Vert _{r,s},\quad \end{aligned}$$

where $\delta = (\frac{r'}{r})^2 \min ( s-s',1-\frac{r}{r}')$.