A KAM Theorem for Higher Dimensional Forced Nonlinear Schrödinger Equations

Abstract

In this paper, we prove an infinite dimensional KAM theorem. As an application, it is shown that there are many real-analytic small-amplitude linearly-stable quasi-periodic solutions for higher dimensional nonlinear Schrödinger equations with outer force

$$\begin{aligned} iu_t-\triangle u +M_\xi u+f(\bar{\omega }t)|u|^2u=0, \quad t\in {{\mathbb R}}, x\in {{\mathbb T}}^d \end{aligned}$$

where $M_\xi$ is a real Fourier multiplier,$f({\bar{\theta }})({\bar{\theta }}={\bar{\omega }} t)$ is real analytic and the forced frequencies $\bar{\omega }$ are fixed Diophantine vectors.

Appendix

Lemma 7.1

$$\begin{aligned} \Vert FG\Vert _{ D_\rho (r,s),{\mathcal {O}} }\le \Vert F\Vert _{ D_\rho (r,s),{\mathcal {O}} }\Vert G\Vert _{ D_\rho (r,s),{\mathcal {O}} }. \end{aligned}$$

Proof

Since $(FG)_{kl\alpha \beta }=\sum _{k',l',\alpha ',\beta '}F_{k-k',l-l',\alpha -\alpha ',\beta -\beta '} G_{k'l'\alpha '\beta '}$, we have

$$\begin{aligned} \Vert FG\Vert _{ D_\rho (r,s),{\mathcal {O}} }= & {} \sup _{D_\rho (r,s)}\sum _{k,l,\alpha ,\beta }|(FG)_{kl\alpha \beta }|_{{\mathcal {O}}}|I^{l}||z^{\alpha }| |{\bar{z}}^{\beta }|e^{|k||\mathrm{Im}\theta |}\\\le & {} \sup _{D_\rho (r,s)} \sum _{k,l,\alpha ,\beta }\sum _{k',l',\alpha ',\beta '}|F_{k-k',l-l',\alpha -\alpha ',\beta -\beta ' }G_{k'l'\alpha '\beta '}|_{{\mathcal {O}}}|I^{l}||z^{\alpha }| |{\bar{z}}^{\beta }|e^{|k||\mathrm{Im}\theta |}\\\le & {} \Vert F\Vert _{ D_\rho (r,s),{\mathcal {O}} }\Vert G\Vert _{ D_\rho (r,s),{\mathcal {O}} } \end{aligned}$$

and the proof is finished. $\square$

Lemma 7.2

(Generalized Cauchy Inequalities)

$$\begin{aligned} \Vert F_{\theta }\Vert _{D_\rho (r-\sigma ,s),{\mathcal {O}}}\le & {} \frac{c}{\sigma }\Vert F\Vert _{ D_\rho (r,s),{\mathcal {O}} },\\ \Vert F_{I}\Vert _{D_\rho (r,\frac{1}{2} s),{\mathcal {O}}}\le & {} \frac{c}{s^2}\Vert F\Vert _{ D_\rho (r,s),{\mathcal {O}} }, \end{aligned}$$

and

$$\begin{aligned}&\Vert F_{z}\Vert _{D_{\rho }(r,\frac{1}{2} s),{\mathcal {O}}}\le \frac{c}{s}\Vert F\Vert _{ D_\rho (r,s),{\mathcal {O}} },\\&\Vert F_{{\bar{z}}}\Vert _{D_{\rho }(r,\frac{1}{2} s),{\mathcal {O}}}\le \frac{c}{s}\Vert F\Vert _{ D_\rho (r,s),{\mathcal {O}} }. \end{aligned}$$

Proof

We only prove the third inequality, the others can be proved similarly. Let $w\ne 0$, then $f(t)=F(z+tw)$ is an analytic map from the complex disc $|t|<\frac{s}{\Vert w\Vert _\rho }$ in ${\mathbb {C}}$ into $D_\rho (r,s)$. Hence

$$\begin{aligned} \Vert f'(0)\Vert _{D_\rho (r,\frac{1}{2} s),{\mathcal {O}}}=\Vert F_zw\Vert _{D_\rho (r,\frac{1}{2} s),{\mathcal {O}}}\le \frac{c}{s}\Vert F\Vert _{ D_\rho (r,s),{\mathcal {O}} }\cdot \Vert w\Vert _\rho , \end{aligned}$$

by the usual Cauchy inequality. Since $w\ne 0$, so

$$\begin{aligned} \frac{\Vert F_zw\Vert _{D_\rho (r,\frac{1}{2} s),{\mathcal {O}}}}{\Vert w\Vert _\rho }\le \frac{c}{s}\Vert F\Vert _{ D_\rho (r,s),{\mathcal {O}} }, \end{aligned}$$

thus

$$\begin{aligned} \Vert F_{z}\Vert _{{D_\rho (r,\frac{1}{2} s),{\mathcal {O}}}}=\sup _{w\ne 0}\frac{\Vert F_zw\Vert _{D_\rho (r,\frac{1}{2} s),{\mathcal {O}}}}{\Vert w\Vert _\rho }\le \frac{c}{s}\Vert F\Vert _{ D_\rho (r,s),{\mathcal {O}} }. \end{aligned}$$

$\square$

Let $\{\cdot ,\cdot \}$ denote the Poisson bracket of smooth functions, i.e.,

$$\begin{aligned} \{F,G\}=\left\langle \frac{\partial F}{\partial I}, \frac{\partial G}{\partial \theta }\right\rangle -\left\langle \frac{\partial F}{ \partial \theta },\frac{\partial G}{\partial I}\right\rangle +\mathrm{i}\left( \left\langle \frac{\partial F}{\partial z},\frac{\partial G}{\partial {{\bar{z}}}}\right\rangle -\left\langle \frac{\partial F}{\partial {{\bar{z}}}},\frac{\partial G}{\partial { z}}\right\rangle \right) , \end{aligned}$$

then we have the following lemma:

Lemma 7.3

If

$$\begin{aligned} \Vert X_F\Vert _{ D_\rho (r,s),{\mathcal {O}} }< \varepsilon ',\ \Vert X_G\Vert _{ D_\rho (r,s),{\mathcal {O}} }< \varepsilon '', \end{aligned}$$

then

$$\begin{aligned} \Vert X_{\{F,G\}}\Vert _{D_\rho (r-\sigma ,\eta s),{\mathcal {O}}}<c\sigma ^{-1}\eta ^{-2}\varepsilon '\varepsilon '',\ \eta \ll 1. \end{aligned}$$

In particular, if $\eta \sim \varepsilon ^{\frac{1}{3}}$, $\varepsilon ', \varepsilon ''\sim \varepsilon$, we have $\Vert X_{\{F,G\}}\Vert _{D_\rho (r-\sigma ,\eta s),{\mathcal {O}}}\sim \varepsilon ^{\frac{4}{3}}$.

Proof

By Lemma 7.1 and Lemma 7.2,

$$\begin{aligned} \left\| \frac{\partial ^2F}{\partial I\partial I}\frac{\partial G}{\partial \theta }\right\| _{D_\rho \left( r-\sigma ,\frac{1}{2}s\right) }< & {} c\sigma ^{-1}s^{-2}\left\| \frac{\partial F}{\partial I}\right\| _{D_\rho (r,s)}\cdot \left\| \frac{\partial G}{\partial \theta }\right\| _{D_\rho (r,s)},\\ \left\| \frac{\partial ^2F}{\partial I\partial \theta }\frac{\partial G}{\partial \theta }\right\| _{D_\rho \left( r-\sigma ,\frac{1}{2}s\right) }< & {} c\sigma ^{-1}\left\| \frac{\partial F}{\partial I}\right\| _{D_\rho (r,s)}\cdot \left\| \frac{\partial G}{\partial \theta }\right\| _{D_\rho (r,s)},\\ \left\| \frac{\partial ^2F}{\partial I\partial z}\frac{\partial G}{\partial \theta }\right\| _{D_\rho \left( r-\sigma ,\frac{1}{2}s\right) }< & {} c\sigma ^{-1}s^{-1}\left\| \frac{\partial F}{\partial I}\right\| _{D_\rho (r,s)}\cdot \left\| \frac{\partial G}{\partial \theta }\right\| _{D_\rho (r,s)},\\ \left\| \frac{\partial ^2F}{\partial I\partial {\bar{z}}}\frac{\partial G}{\partial \theta }\right\| _{D_\rho \left( r-\sigma ,\frac{1}{2}s\right) }< & {} c\sigma ^{-1}s^{-1}\left\| \frac{\partial F}{\partial I}\right\| _{D_\rho (r,s)}\cdot \left\| \frac{\partial G}{\partial \theta }\right\| _{D_\rho (r,s)},\\ \left\| \frac{\partial ^2F}{\partial z\partial I}\frac{\partial G}{\partial {\bar{z}}}\right\| _{D_\rho \left( r-\sigma ,\frac{1}{2}s\right) }< & {} c\sigma ^{-1}s^{-2}\left\| \frac{\partial F}{\partial z}\right\| _{D_\rho (r,s)}\cdot \left\| \frac{\partial G}{\partial {\bar{z}}}\right\| _{D_\rho (r,s)},\\ \left\| \frac{\partial ^2F}{\partial z\partial \theta }\frac{\partial G}{\partial {\bar{z}}}\right\| _{D_\rho \left( r-\sigma ,\frac{1}{2}s\right) }< & {} c\sigma ^{-1}\left\| \frac{\partial F}{\partial z}\right\| _{D_\rho (r,s)}\cdot \left\| \frac{\partial G}{\partial {\bar{z}}}\right\| _{D_\rho (r,s)},\\ \left\| \frac{\partial ^2F}{\partial z\partial z}\frac{\partial G}{\partial {\bar{z}}}\right\| _{D_\rho \left( r-\sigma ,\frac{1}{2}s\right) }< & {} c\sigma ^{-1}s^{-1}\left\| \frac{\partial F}{\partial z}\right\| _{D_\rho (r,s)}\cdot \left\| \frac{\partial G}{\partial {\bar{z}}}\right\| _{D_\rho (r,s)},\\ \left\| \frac{\partial ^2F}{\partial z\partial {\bar{z}}}\frac{\partial G}{\partial {\bar{z}}}\right\| _{D_\rho \left( r-\sigma ,\frac{1}{2}s\right) }< & {} c\sigma ^{-1}s^{-1}\left\| \frac{\partial F}{\partial z}\right\| _{D_\rho (r,s)}\cdot \left\| \frac{\partial G}{\partial {\bar{z}}}\right\| _{D_\rho (r,s)}. \end{aligned}$$

The other cases can be obtained analogously, hence

$$\begin{aligned} \Vert X_{\{F,G\}}\Vert _{D_\rho (r-\sigma ,\eta s),{\mathcal {O}}}<c\sigma ^{-1}\eta ^{-2}\varepsilon '\varepsilon ''. \end{aligned}$$

$\square$