Normal Form for the Fractional Nonlinear Schrödinger Equation with Cubic Nonlinearity

Abstract

We study the Birkhoff normal form for the cubic Fractional NLS

$$\begin{aligned} iu_t-(|\partial _x|+\sigma )^{\frac{1}{2}}u+|u|^2u=0,\quad x\in \mathbb {T},t\in \mathbb {R}. \end{aligned}$$

It is proved that there exists some interval $\mathcal I\subset [0,1]$, such that for any $\sigma \in \mathcal I$, the equation above admits a family of small-amplitude, linearly stable quasi-periodic solutions.

Appendix

Lemma 4.1

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

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

with a constant c depending only on a.

For the proof of Lemma 4.1, see [12, 20, 22].