On the existence of full dimensional KAM torus for nonlinear Schr\"odinger equation

TL;DR

This paper proves the existence of full dimensional KAM tori for a nonlinear Schrödinger equation with space-dependent nonlinearities, extending previous results to more general cases with explicit space dependence.

Contribution

It demonstrates the existence of time almost periodic solutions for a nonlinear Schrödinger equation with space-dependent nonlinearities, extending prior work to cases lacking zero momentum.

Findings

Existence of full dimensional KAM tori in Gevrey space.

Extension of Bourgain's and Cong-Liu-Shi-Yuan's results.

Handling the absence of zero momentum in the analysis.

Abstract

In this paper, we study the following nonlinear Schr\"odinger equation \begin{eqnarray}\label{maineq0} \textbf{i}u_{t}-u_{xx}+V*u+\epsilon f(x)|u|^4u=0,\ x\in\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}, \end{eqnarray} where $V*$ is the Fourier multiplier defined by $\widehat{(V* u})_n=V_{n}\widehat{u}_n, V_n\in[-1,1]$ and $f(x)$ is Gevrey smooth. It is shown that for $0\leq|\epsilon|\ll1$, there is some $(V_n)_{n\in\mathbb{Z}}$ such that, the equation admits a time almost periodic solution (i.e., full dimensional KAM torus) in the Gevrey space. This extends results of Bourgain \cite{BJFA2005} and Cong-Liu-Shi-Yuan \cite{CLSY} to the case that the nonlinear perturbation depends explicitly on the space variable $x$. The main difficulty here is the absence of zero momentum of the equation.

Equations

$$\text{i}{u}_t-u_{xx}+V\ast u+ϵf(x)|u{|}^{4}u=0,x\in \mathbb{T}=\mathbb{R}/2\pi \mathbb{Z},$$

$$\sqrt{-1}{u}_t-u_{xx}+V\ast u+ϵf(x)|u{|}^{4}u=0,x\in \mathbb{T},$$

$$V\ast u=\sum _{n\in \mathbb{Z}}{V}_n{\hat{u}}_n{e}^{\text{i}nx},V_n\in [-1,1],$$

$${\dot{q}}_n=\mathbf{i}\frac{\partial H}{\partial {\overline{q}}_n}$$

$$H(q,\overline{q})=\sum _{n\in \mathbb{Z}}(n^2+V_n)|q_n{|}^2+ϵ\sum _{n\in \mathbb{Z}}\sum _{n_1-n_2+n_3-n_4+n_5-n_6=-n}\hat{f}(n)q_{n_1}{\overline{q}}_{n_2}{q}_{n_3}{\overline{q}}_{n_4}{q}_{n_5}{\overline{q}}_{n_6}.$$

$$H=N+ϵP(\theta ,I,z,\bar{z}),$$

$$N=\sum _{i=1}^d{\omega }_i{I}_i+\sum _{j\ge 1}{\Omega }_j{z}_j{\bar{z}}_j,1\le d<\infty ,$$

$${\mathcal{T}}_0={\mathbb{T}}^d\times \{I=0\}\times \{z=0\}\times \{\bar{z}=0\},$$

$$u(t,x)=\sum _{n\ge 1}{a}_n\cos {\omega }_{n}t{\varphi }_n(x)$$

$$u_{tt}-u_{xx}+V(x)u+ϵf(u)=0$$

$$\frac{1}{2}{e}^{-r|n{|}^{\theta }}\le I_n\le 2e^{-r|n{|}^{\theta }}$$

$$n_1-n_2+n_3-\cdots =0.$$

$$n_1^2-n_2^2+n_3^2-\cdots =o(1).$$

$$|\hat{f}(n)|\le C{e}^{-\mu |n{|}^{\theta }},\mu >0,\theta \in (0,1).$$

$$\Vert q{\Vert }_{r,\infty }=\underset{n\in \mathbb{Z}}{\sup }|q_n|e^{r|n{|}^{\theta }}<\infty .$$

$$\Vert \sum _{n\in \mathbb{Z}}{l}_n{\omega }_n\Vert \ge \gamma \prod _{n\in \mathbb{Z}}\frac{1}{1+l_n^2|n{|}^4}$$

$$|n|=\max \{1,n,-n\}.$$

$$\frac{1}{2}{e}^{-r|n{|}^{\theta }}\le |q_n|\le 2e^{-r|n{|}^{\theta }};$$

$$H(q,\bar{q})=N(q,\bar{q})+R(q,\bar{q}),$$

$$N(q,\bar{q})=\sum _{n\in \mathbb{Z}}(n^2+V_n)|q_n{|}^2,$$

$$R(q,\bar{q})=\sum _{a,k,k^{\prime }\in {\mathbb{N}}^{\mathbb{Z}}}{B}_{ak{k}^{\prime }}{\mathcal{M}}_{ak{k}^{\prime }}$$

$${\mathcal{M}}_{ak{k}^{\prime }}=\prod _{n\in \mathbb{Z}}{I}_n(0{)}^{a_n}{q}_n^{k_n}{\bar{q}}_n^{k_n^{\prime }},$$

$$\text{mbox}supp{\mathcal{M}}_{ak{k}^{\prime }}=\{n:2a_n+k_n+k_n^{\prime }\ne 0\},$$

$$\text{mbox}momentum{\mathcal{M}}_{ak{k}^{\prime }}:=m(k,k^{\prime })=\sum _{n\in \mathbb{Z}}(k_n-k_n^{\prime })n.$$

$$n_1^{\ast }=\max \{|n|:a_n+k_n+k_n^{\prime }\ne 0\},$$

$$m^{\ast }(k,k^{\prime })=\left|m(k,k^{\prime })\right|.$$

$$\Vert R{\Vert }_{\rho ,\mu }=\underset{a,k,k^{\prime }\in {\mathbb{N}}^{\mathbb{Z}}}{\sup }\frac{\left|B_{ak{k}^{\prime }}\right|}{e^{\rho {\sum }_n(2a_n+k_n+k_n^{\prime })|n{|}^{\theta }-2\rho (n_1^{\ast }{)}^{\theta }-\mu m^{\ast }(k,k^{\prime }{)}^{\theta }}}.$$

$$H_s\circ {\Phi }_s=N_{s+1}+R_{s+1}$$

$$R=R_0+R_1+R_2$$

$$R_0$$

Full text

[TABLE]

where $V*$ is the Fourier multiplier defined by $\widehat{(V*u})_{n}=V_{n}\widehat{u}_{n},V_{n}\in[-1,1]$ and $f(x)$ is Gevrey smooth. It is shown that for $0\leq|\epsilon|\ll 1$, there is some $(V_{n})_{n\in\mathbb{Z}}$ such that, the equation (0.1) admits a time almost periodic solution (i.e., full dimensional KAM torus) in the Gevrey space. This extends results of Bourgain [7] and Cong-Liu-Shi-Yuan [8] to the case that the nonlinear perturbation depends explicitly on the space variable $x$. The main difficulty here is the absence of zero momentum of the equation.

The first author is supported by the National Natural Science Foundation of China (No. 11671066). The third author is supported by China Postdoctoral Science Foundation Grant (No. 2018M641050).

1. Introduction and main result

In this paper, we focus on the nonlinear Schrödinger equation (NLS) with periodic boundary conditions

[TABLE]

where $\textbf{i}=\sqrt{-1}$, $V*$ is a Fourier multiplier defined by

[TABLE]

$f(x)$ is $2\pi$-periodic and real analytic in $x,y$. Written in Fourier modes $(q_{n})_{n\in\mathbb{Z}}$, then (1.1) can be rewritten as

[TABLE]

with the Hamiltonian

[TABLE]

Our aim is to show the existence of almost periodic solutions for such a family of NLS.

In the last few decades, the persistence of the invariant tori for NLS has been drawn a lot of attentions by many authors. To this end, one considers the infinite dimensional Hamiltonian of the form

[TABLE]

with the symplectic structure $\mathrm{d}\theta\wedge\mathrm{d}I+\sqrt{-1}\mathrm{d}z\wedge\mathrm{d}\bar{z}$ on $\mathbb{T}^{d}\times\mathbb{R}^{d}\times\mathcal{H}\times\mathcal{H}\ni(\theta,I,z,\bar{z})$ and

[TABLE]

where $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{d})$ is called tangent frequency vector, $(\Omega_{j})_{j\geq 1}$ is called the normal frequency vector, and $P=P(\theta,I,z,\bar{z})$ is a perturbation. The unperturbed Hamiltonian $N$ has a special invariant torus

[TABLE]

and all solutions starting on $\mathcal{T}_{0}$ are quasi-periodic with the frequency $\omega$. Under suitable assumptions on $N$ and $P$, it can be proved that for “most” frequency $\omega$, the tori $\mathcal{T}_{0}$ can be persisted for some small perturbation $\epsilon P$ (see [16, 17, 23] for example). However, the KAM theorem of this type depends heavily on the fact that the spatial dimension of the PDEs equals to $1$. Bourgain [4, 6] developed a new method initiated by Craig-Wayne [9] to deal with the KAM tori for the PDEs in high spatial dimension, based on the Newton iteration, Fröhlich-Spencer techniques, Harmonic analysis and semi-algebraic set theory (see [6]). This method is now called C-W-B method. We also mention Eliasson-Kuksin [10] where the KAM theorem is extended to deal with higher spatial dimensional nonlinear Schrödinger equation. In addition, the classical KAM theory is also developed to deal some 1D PDEs of unbounded perturbation. See, for example, [17, 15, 19, 25, 1, 2, 11] for the details. In the all above works, the obtained KAM tori are lower (finite) dimension. Naturally, the following problem is interesting: *Can the full dimensional invariant tori be expected with a suitable decay, for example, $I_{n}\sim|n|^{-S}$ with some $S>0$ as $|n|\rightarrow+\infty$ ? * The existence of the full dimensional KAM tori with polynomial decay rate $I_{n}\sim|n|^{-S}$ is still open up to now. See [18] for the details. The first result about the existence of the full dimensional tori (or almost periodic solutions) for Hamiltonian PDEs was obtained by Bourgain [3]. Precisely, using C-W-B method the almost periodic solutions (in time) of the form

[TABLE]

were constructed for 1D nonlinear wave equation (NLW)

[TABLE]

under Dirichlet boundary conditions, where $\omega_{n}\approx\sqrt{\lambda_{n}}$ and $\lambda_{n}$ is the Dirichlet spectrum of $-\partial_{xx}+V(x)$. Here, a strong decay assumption $|a_{n}|\rightarrow 0$ is needed for the amplitude $a_{n}$. Pöschel [22] proved the existence of almost periodic solutions for NLS equation by the KAM method (also see [12],[14],[20],[24]). The basic idea in these papers is to use repeatedly (infinitely many times) the KAM theorem dealing with lower dimensional KAM tori. That is why the amplitude (or action) of those almost periodic solutions decay extremely fast. In fact, the decay rate is defined implicitly and much more fast than $a_{n}\sim e^{-|n|^{C}},C>1$. See more comments in [5]. Recently, the invariant tori of full dimensions for second KdV equations with the external parameters were constructed by Geng-Hong [13], where noting that the nonlinear term contains the derivatives.

Another way is due to Bourgain in [7] where 1D NLS with periodic boundary condition was investigated (see also [21] by Pöschel where infinite dimensional Hamiltonian systems with short range was considered). It was shown in [7] that 1D NLS has a full dimensional KAM torus of prescribed frequencies where the actions of the torus obey the estimates

[TABLE]

with $\theta=\frac{1}{2}$. This is up to now only existence result about the full dimensional KAM tori with a slower decay rate than $I_{n}\sim e^{-|n|^{S}},S>1$. In a different way, Bourgain constructed the full dimensional tori directly, where a more complicated small divisor problem has to be dealt with. An important observation by Bourgain is the following: Let $(n_{i})$ be a finite set of modes satisfy $|n_{1}|\geq|n_{2}|\geq\cdots$ and

[TABLE]

In the case of a ‘near’ resonance, there is also a relation

[TABLE]

Unless $n_{1}=n_{2}$, one may then control $|n_{1}|+|n_{2}|$ from (1.7), (1.8) by $\sum_{j\geq 3}|n_{j}|.$ More recently, Cong-Liu-Shi-Yuan [8] extended Bourgain’s results to the any $\theta\in(0,1)$.

Note that the condition (1.7) is no longer valid for the Hamiltonian (1.3). But if the function $f(x)$ is Gevrey smooth with $\mu>0$, then one has

[TABLE]

Thus we use the property (1.9) to guarantee $|n_{1}|+|n_{2}|$ can be controlled by $\sum_{j\geq 3}|n_{j}|+|n|$.

To state our result precisely, we will give some definitions firstly.

Definition 1.1**.**

Given $0<\theta<1$ and $r>0$, we define the Banach space $\mathfrak{H}_{r,\infty}$ consisting of all complex sequences $q=(q_{n})_{n\in\mathbb{Z}}$ with

[TABLE]

Definition 1.2**.**

Denote $\|x\|=dist(x,\mathbb{Z})$. A vector $\omega=(\omega_{n})_{n\in\mathbb{Z}}$ is called to be Diophantine if there exists a real number $\gamma>0$ such that the following resonance issues

[TABLE]

hold, where $0\neq l=(l_{n})_{n\in\mathbb{Z}}$ is a finitely supported sequence of integers and

[TABLE]

Theorem 1.3**.**

*Given $r>0$, $0<\theta<1$ and a Diophantine vector $\omega=(\omega_{n})_{n\in\mathbb{Z}}$ satisfying $\sup_{n}|\omega_{n}|<1$, then for any $\mu>2r$, sufficiently small $\epsilon>0$ and some appropriate $V$, (1.1) has a full dimensional invariant torus $\mathcal{E}$ with amplitude in $\mathfrak{H}_{r,\infty}$ satisfying:

• (1)

the amplitude of $\mathcal{E}$ is restricted as

[TABLE]

• (2)

the frequency on $\mathcal{E}$ was prescribed to be $(n^{2}+\omega_{n})_{n\in\mathbb{Z}}$;

• (3)

the invariant torus $\mathcal{E}$ is linearly stable.

2. KAM Iteration

2.1. Some notations and the norm of the Hamiltonian

Let $q=(q_{n})_{n\in\mathbb{Z}}$ and its complex conjugate $\bar{q}=(\bar{q}_{n})_{n\in\mathbb{Z}}$. Introduce $I_{n}=|q_{n}|^{2}$ and $J_{n}=I_{n}-I_{n}(0)$ as notations but not as new variables, where $I_{n}(0)$ will be considered as the initial data. Then the Hamiltonian (1.1) has the form of

[TABLE]

where

[TABLE]

[TABLE]

with

[TABLE]

and $B_{akk^{\prime}}$ are the coefficients.

Define by

[TABLE]

and define the momentum of $\mathcal{M}_{akk^{\prime}}$ by

[TABLE]

Moreover, denote by

[TABLE]

and

[TABLE]

Now we define the norm of the Hamiltonian as follows

Definition 2.1**.**

For any given $\rho>0,\mu>0$ and $0<\theta<1$, define the norm of the Hamiltonian $R$ by

[TABLE]

2.2. Derivation of homological equations

The proof of Theorem 1.3 employs the rapidly converging iteration scheme of Newton type to deal with small divisor problems introduced by Kolmogorov, involving the infinite sequence of coordinate transformations. At the $s$-th step of the scheme, a Hamiltonian $H_{s}=N_{s}+R_{s}$ is considered, as a small perturbation of some normal form $N_{s}$. A transformation $\Phi_{s}$ is set up so that

[TABLE]

with another normal form $N_{s+1}$ and a much smaller perturbation $R_{s+1}$. We drop the index $s$ of $H_{s},N_{s},R_{s},\Phi_{s}$ and shorten the index $s+1$ as $+$.

Rewrite $R$ as

[TABLE]

where

[TABLE]

We desire to eliminate the terms $R_{0},R_{1}$ in (2.4) by the coordinate transformation $\Phi$, which is obtained as the time-1 map $X_{F}^{t}|_{t=1}$ of a Hamiltonian vector field $X_{F}$ with $F=F_{0}+F_{1}$. Let ${F}_{0}$ (resp. ${F}_{1}$) has the form of ${R}_{0}$ (resp. ${R}_{1}$), that is

[TABLE]

and the homological equations become

[TABLE]

where

[TABLE]

and

[TABLE]

The solutions of the homological equations (2.7) are given by

[TABLE]

and

[TABLE]

The new Hamiltonian ${H}_{+}$ has the form

[TABLE]

where

[TABLE]

and

[TABLE]

2.3. The solvability of the homological equations (2.7)

In this subsection, we will estimate the solutions of the homological equations (2.7). To this end, we define the new norm for the Hamiltonian ${R}$ of the form as follows:

[TABLE]

where

[TABLE]

Moreover, one has the following estimates:

Lemma 2.2**.**

Given any $\mu>\delta>0,\rho>0$, one has

[TABLE]

and

[TABLE]

where $C(\theta)$ is a positive constant depending on $\theta$ only.

Proof.

The details of the proof will be given in the Appendix. ∎

Lemma 2.3**.**

Let $(\widetilde{V}_{n})_{n\in\mathbb{Z}}$ be Diophantine with $\gamma>0$ (see (1.11)). Then for any $\rho>0,0<\delta\ll 1$ (depending only on $\theta$), the solutions of the homological equations (2.7), which are given by (2.10) and (2.11), satisfy

[TABLE]

where $i=0,1$ and $C(\theta)$ is a positive constant depending on $\theta$ only.

Proof.

The details of the proof will be given in the Appendix. ∎

2.4. The new perturbation $R_{+}$ and the new normal form $N_{+}$

Firstly, we will prove two lemmas.

Lemma 2.4**.**

(Poisson Bracket) Let $\theta\in(0,1),\rho,\mu>0$ and $0<\delta_{1},\delta_{2}\ll 1$ (depending on $\theta,\rho,\mu$). Then one has

[TABLE]

where $C(\theta)$ is a positive constant depending on $\theta$ only.

Proof.

The details of proof will be left in the Appendix. ∎

Lemma 2.5**.**

Let $\theta\in(0,1),\rho>0$ and $0<\delta_{1},\delta_{2}\ll 1$ (depending on $\theta,\rho$). Assume further

[TABLE]

where $C(\theta)$ is the constant given in (2.22) in Lemma 2.4. Then for any Hamiltonian function $H$, we get

[TABLE]

where $C_{1}(\theta)$ is a positive constant depending only on $\theta$.

Proof.

Firstly, we expand $H\circ\Phi_{F}$ into the Taylor series

[TABLE]

where $H^{(n)}=\{H^{(n-1)},F\}$ and $H^{(0)}=H$.

We will estimate $||H^{(n)}||_{\rho,\mu}$ by using Lemma 2.4 again and again:

[TABLE]

Hence in view of (2.26), one has

[TABLE]

where $C_{1}(\theta)$ is a positive constant depending on $\theta$ only. ∎

Recall the new term $R_{+}$ is given by (2.14) and write

[TABLE]

Following the proof of CLSY, one has

[TABLE]

The new normal form $N_{+}$ is given in (2.13). Note that $[R_{0}]$ (in view of (2.8)) is a constant which does not affect the Hamiltonian vector field. Moreover, in view of (2.9), we denote by

[TABLE]

where the terms $\sum_{a\in\mathbb{N}^{\mathbb{Z}}}B_{a00}^{(n)}\mathcal{M}_{a00}$ is the so-called frequency shift. The estimate of $\left|\sum_{a\in\mathbb{N}^{\mathbb{Z}}}B_{a00}^{(n)}\mathcal{M}_{a00}\right|$ will be given in the next section (see (3.28) for the details).

Finally, we give the estimate of the Hamiltonian vector field.

Lemma 2.6**.**

Given a Hamiltonian

[TABLE]

then for any $\mu>r>(\frac{1}{2-2^{\theta}}+3)\rho,||q||_{r,\infty}<1$ and $||I(0)||_{r,\infty}<1$, one has

[TABLE]

where $C(r,\rho,\mu,\theta)$ is a positive constant depending on $r,\rho,\mu$ and $\theta$ only, and

[TABLE]

Proof.

The details of the proof will be given in the Appendix. ∎

3. Iteration and Convergence

Now we give the precise set-up of iteration parameters. Let $s\geq 1$ be the $s$-th KAM step.

• $\rho_{0}=\rho,$ $r\geq\frac{100\rho}{2-2^{\theta}}$, $\mu_{0}\geq 2r$,
• $\delta_{s}=\frac{\rho}{s^{2}}$,
• $\rho_{s+1}=\rho_{s}+3\delta_{s}$,
• $\mu_{s+1}=\mu_{s}-6\delta_{s}$
• $\epsilon_{s}=\epsilon_{0}^{(\frac{3}{2})^{s}}$, which dominates the size of the perturbation,
• $\lambda_{s}=e^{-C(\theta)(\ln{\frac{1}{\epsilon_{s+1}}})^{\frac{4}{\theta+4}}}$,
• $\eta_{s+1}=\frac{1}{20}\lambda_{s}\eta_{s}$,
• $d_{0}=0,\,d_{s+1}=d_{s}+\frac{1}{\pi^{2}(s+1)^{2}}$,
• $D_{s}=\{(q_{n})_{n\in\mathbb{Z}}:\frac{1}{2}+d_{s}\leq|q_{n}|e^{r|n|^{\theta}}\leq 1-d_{s}\}$.

Denote the complex cube of size $\lambda>0$:

[TABLE]

Lemma 3.1**.**

Suppose $H_{s}=N_{s}+R_{s}$ is real analytic on $D_{s}\times\mathcal{C}_{\eta_{s}}(V^{*}_{s})$, where

[TABLE]

is a normal form with coefficients satisfying

[TABLE]

and $R_{s}=R_{0,s}+R_{1,s}+R_{2,s}$ satisfying

[TABLE]

Then for all $V\in\mathcal{C}_{\eta_{s}}(V_{s}^{*})$ satisfying $\widetilde{V}_{s}(V)\in\mathcal{C}_{\lambda_{s}}(\omega)$, there exist real analytic symplectic coordinate transformations $\Phi_{s+1}:D_{s+1}\rightarrow D_{s}$ satisfying

[TABLE]

such that for $H_{s+1}=H_{s}\circ\Phi_{s+1}=N_{s+1}+R_{s+1}$, the same assumptions as above are satisfied with ‘$s+1$’ in place of ‘$s$’, where $\mathcal{C}_{\eta_{s+1}}(V_{s+1}^{*})\subset\widetilde{V}_{s}^{-1}(\mathcal{C}_{\lambda_{s}}(\omega))$ and

[TABLE]

[TABLE]

Proof.

In the step $s\rightarrow s+1$, there is saving of a factor

[TABLE]

By (4.1), one has

[TABLE]

Recalling after this step, we need

[TABLE]

Consequently, in $R_{i,s}\ (i=0,1)$, it suffices to eliminate the nonresonant monomials $\mathcal{M}_{akk^{\prime}}$ for which

[TABLE]

that is

[TABLE]

On the other hand, in the small divisors analysis (see Lemma 4.2), one has

[TABLE]

Hence we need only impose condition on $(\widetilde{V}_{n})_{|n|\leq\mathcal{N}_{s}}$, where

[TABLE]

Correspondingly, the Diophantine condition becomes

[TABLE]

We finished the truncation step.

Next we will show (3.15) preserves under small perturbation of $(\widetilde{V}_{n})_{|n|\leq\mathcal{N}_{s}}$ and this is equivalent to get lower bound on the right hand side of (3.15). Let

[TABLE]

then we have

[TABLE]

where the last inequality is based on $\epsilon_{0}$ is small enough.

Assuming $V\in\mathcal{C}_{\lambda_{s}}(\omega)$, from the lower bound (3.17), the relation (3.15) remains true if we substitute $V$ for $\omega$. Moreover, there is analyticity on $\mathcal{C}_{\lambda_{s}}(\omega)$. The transformations $\Phi_{s+1}$ is obtained as the time-1 map $X_{F_{s}}^{t}|_{t=1}$ of the Hamiltonian vector field $X_{F_{s}}$ with $F_{s}=F_{0,s}+F_{1,s}$. Taking $\rho=\rho_{s}$, $\delta=\delta_{s}$ in Lemma 2.3, we get

[TABLE]

where $i=0,1$. By Lemma 2.2, we get

[TABLE]

Combining (3.4), (3.5), (3.18) and (3.19), we get

[TABLE]

By Lemma 2.6, we get

[TABLE]

where noting that $0<\epsilon_{0}\ll 1$ small enough and depending on $\rho,\theta$ only.

Since $\epsilon_{s}^{0.55}\ll\frac{1}{\pi^{2}(s+1)^{2}}=d_{s+1}-d_{s}$, we have $\Phi_{s+1}:D_{s+1}\rightarrow D_{s}$ with

[TABLE]

which is the estimate (3.7). Moreover, from (3.22) we get

[TABLE]

and thus the estimate (3.8) follows.

Moreover, under the assumptions (3.4)-(3.6) at stage $s$, we get from (2.28), (2.29) and (2.30) that

[TABLE]

and

[TABLE]

which are just the assumptions (3.4)-(3.6) at stage $s+1$.

If $V\in\mathcal{C}_{\frac{\eta_{s}}{2}}(V_{s}^{*})\subset\mathcal{C}_{{\eta_{s}}}(V_{s}^{*})$ and using Cauchy’s estimate, for any $m$ one has

[TABLE]

Let $V\in\mathcal{C}_{\frac{1}{10}\lambda_{s}\eta_{s}}(V_{s}^{*})$, then

[TABLE]

that is

[TABLE]

Note that

[TABLE]

Assuming further

[TABLE]

and for any $s$,

[TABLE]

we obtain

[TABLE]

By (3.28), we have

[TABLE]

which verifies (3.9). Further applying Cauchy’s estimate on $\mathcal{C}_{\lambda_{s}\eta_{s}}(V_{s}^{*})$, one gets

[TABLE]

Since

[TABLE]

it follows that

[TABLE]

and hence by iterating (3.31) implies

[TABLE]

On $\mathcal{C}_{\frac{1}{10}\lambda_{s}\eta_{s}}(V_{s}^{*})$ and for any $m$, we deduce from (3.30), (3.32) and the assumption (3.3) that

[TABLE]

and consequently

[TABLE]

which verifies (3.3) for $s+1$.

Finally, we will freeze $\omega$ by invoking an inverse function theorem. Consider the following functional equation

[TABLE]

from (3.33) and the standard inverse function theorem implies (3.34) having a solution $V_{s+1}^{*}$, which verifies (3.2) for $s+1$. Rewriting (3.34) as

[TABLE]

and using (3.29) (3.33) implies

[TABLE]

which verifies (3.10) and completes the proof of the iterative lemma. ∎

We are now in a position to prove the convergence. To apply iterative lemma with $s=0$, set

[TABLE]

and consequently (3.2)–(3.6) with $s=0$ are satisfied. Hence, the iterative lemma applies, and we obtain a decreasing sequence of domains $D_{s}\times\mathcal{C}_{\eta_{s}}(V_{s}^{*})$ and a sequence of transformations

[TABLE]

such that $H\circ\Phi^{s}=N_{s}+P_{s}$ for $s\geq 1$. Moreover, the estimates (3.7)–(3.10) hold. Thus we can show $V_{s}^{*}$ converge to a limit $V_{*}$ with the estimate

[TABLE]

and $\Phi^{s}$ converge uniformly on $D_{*}\times\{V_{*}\}$, where $D_{*}=\{(q_{n})_{n\in\mathbb{Z}}:\frac{2}{3}\leq|q_{n}|e^{r|n|^{\theta}}\leq\frac{5}{6}\}$, to $\Phi:D_{*}\times\{V_{*}\}\rightarrow D_{0}$ with the estimates

[TABLE]

Hence

[TABLE]

where

[TABLE]

and

[TABLE]

By (2.33), the Hamiltonian vector field $X_{R_{2,*}}$ is a bounded map from $\mathfrak{H}_{r,\infty}$ into $\mathfrak{H}_{r,\infty}$. Taking

[TABLE]

we get an invariant torus $\mathcal{T}$ with frequency $(n^{2}+\omega_{n})_{n\in\mathbb{Z}}$ for ${X}_{H_{*}}$. Finally, by $X_{H}\circ\Phi=D\Phi\cdot{X}_{H_{*}}$, $\Phi(\mathcal{T})$ is the desired invariant torus for the NLS (1.1). Moreover, we deduce the torus $\Phi(\mathcal{T})$ is linearly stable from the fact that (3.37) is a normal form of order 2 around the invariant torus.

4. Appendix

4.1. Technical Lemmas

Lemma 4.1**.**

Denote $(n^{*}_{i})_{i\geq 1}$ the decreasing rearrangement of

[TABLE]

Then for any $\theta\in(0,1)$, one has

[TABLE]

Proof.

Without loss of generality, denote $(n_{i})_{i\geq 1},\ |n_{1}|\geq|n_{2}|\geq\cdots$, the system \{\mbox{n is repeated}\ 2a_{n}+k_{n}+k_{n}^{\prime}\ \mbox{times}\} and we have $n_{i}^{*}=|n_{i}|\ \mbox{for}\ \forall\ i\geq 1$. There exists $(\mu_{i})_{i\geq 1}$ with $\mu_{i}\in\{-1,1\}$ such that

[TABLE]

and hence

[TABLE]

Consequently

[TABLE]

Thus the inequality (4.1) will follow from the inequality

[TABLE]

To prove the inequality (4.2), one just needs the following fact: consider the function

[TABLE]

and one has

[TABLE]

which is based on

[TABLE]

Hence, for any $a\geq b>0$, we have

[TABLE]

where the last inequality is based on (4.3). That is

[TABLE]

By iteration and in view of (4.4), one obtains

[TABLE]

where the last inequality is based on

[TABLE]

for all $a,b$ and $0<\theta<1$. ∎

Lemma 4.2**.**

Let $\theta\in(0,1)$ and $k_{n},k^{\prime}_{n}\in\mathbb{N},|{\widetilde{V}}_{n}|\leq 2\ \mbox{for}\ \forall\ n\in\mathbb{Z}$. Assume further

[TABLE]

Then one has

[TABLE]

where $(n_{i})_{i\geq 1},|n_{1}|\geq|n_{2}|\geq|n_{3}|\geq\cdots$, denote the system {$n$: $n$ is repeated $k_{n}+k^{\prime}_{n}$ times}.

Proof.

From the definition of $(n_{i})_{i\geq 1}$, there exist $(\mu_{i})_{i\geq 1}$ with $\mu_{i}\in\{-1,1\}$ such that

[TABLE]

and

[TABLE]

In view of (4.5), (4.8) and $|\widetilde{V}_{n}|\leq 2$, one has

[TABLE]

which implies

[TABLE]

On the other hand, by (4.7), we obtain

[TABLE]

To prove the inequality (4.6), we will distinguish two cases:

Case. 1. $\frac{\mu_{2}}{\mu_{1}}=-1$.

Case. 1.1. $n_{1}=n_{2}$.

Then it is easy to show that

[TABLE]

Case. 1.2. $n_{1}\neq n_{2}$.

Then one has

[TABLE]

Hence

[TABLE]

For $j=1,2,$ one has

[TABLE]

where the last inequality is based on the fact that the function $|x|^{\theta/2}$ is a concave function for $0<\theta<1$. Therefore,

[TABLE]

Now one has

[TABLE]

Case. 2. $\frac{\mu_{2}}{\mu_{1}}=1$.

In view of (4.9), one has

[TABLE]

which implies

[TABLE]

Therefore,

[TABLE]

Following the proof of (4.13), we have

[TABLE]

∎

4.2. Proof of Lemma 2.2

Proof.

Firstly, we will prove the inequality (2.19). Write $\mathcal{M}_{akk^{\prime}}$ in the form of

[TABLE]

where

[TABLE]

and $l_{n}l_{n}^{\prime}=0$ for all $n$.

Express the term

[TABLE]

by the monomials of the form

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Now we will estimate the bounds for the coefficients respectively.

Consider the term $\mathcal{M}_{akk^{\prime}}=\prod_{n}I_{n}(0)^{a_{n}}q_{n}^{k_{n}}\bar{q}_{n}^{k_{n}^{\prime}}$ with fixed $a,k,k^{\prime}$ satisfying $k_{n}k_{n}^{\prime}=0$ for all $n$. It is easy to see that $\mathcal{M}_{akk^{\prime}}$ comes from some parts of the terms $\mathcal{M}_{\alpha\kappa\kappa^{\prime}}$ with no assumption for $\kappa$ and $\kappa^{\prime}$. For any given $n$ one has

[TABLE]

Hence,

[TABLE]

and

[TABLE]

Therefore, if $0\leq\alpha_{n}\leq a_{n}$ is chosen, so $\beta_{n},k_{n},k_{n}^{\prime}$ are determined. On the other hand,

[TABLE]

Hence,

[TABLE]

Similarly,

[TABLE]

In view of (2.16) and (4.17), we have

[TABLE]

Now we will show that

[TABLE]

Case 1. $n_{1}^{*}=n_{2}^{*}=n_{3}^{*}.$ Then one has

[TABLE]

Case 2. $n_{1}^{*}>n_{2}^{*}=n_{3}^{*}.$ In this case, $a_{n}=1$ for $n=n_{1}$. Then we have

[TABLE]

Case 3. $n_{1}^{\ast}\geq n_{2}^{*}>n_{3}^{*}.$ In this case, $a_{n}=1$ or $2$ for $n\in\{n_{1},n_{2}\}$. Hence

[TABLE]

We finished the proof of (4.19).

Similarly, one has

[TABLE]

and hence

[TABLE]

On the other hand, the coefficient of $\mathcal{M}_{abll^{\prime}}$ increases by at most a factor

[TABLE]

then

[TABLE]

where the last inequality is based on Lemma 7.5 in [8] with $p=2$. ∎

4.3. Proof of Lemma 2.3

Proof.

We distinguish two cases:

Case. 1.

[TABLE]

Since $|\widetilde{{V}}_{n}|\leq 2$, we have

[TABLE]

where the last inequality is based on $\mbox{supp}\ k\bigcap\mbox{supp}\ k^{\prime}=\emptyset$. There is no small divisor and (2.21) holds trivially.

Case. 2.

[TABLE]

In this case, we always assume

[TABLE]

otherwise there is no small divisor.

Firstly, one has

[TABLE]

where the last inequality is based on Lemma 4.1.

Since $\sum_{n\in\mathbb{Z}}(k_{n}-k^{\prime}_{n})n^{2}\in\mathbb{Z},$ the Diophantine property of $(\widetilde{V}_{n})$ implies

[TABLE]

Hence,

[TABLE]

where $C(\theta)$ is a positive constant depending on $\theta$ only.

Therefore, in view of (2.16) and (4.23), we finish the proof of (2.21) for $i=0$.

It is easy to verify the following two facts that

[TABLE]

with $f(x)=(-\delta x^{\theta/2}+8\ln x)$, and when $|n|>N=\left(\frac{16}{\theta\delta}\right)^{4/\theta},k_{n}\neq k^{\prime}_{n}$, one has

[TABLE]

Similarly, one can prove (2.21) for $i=1$. ∎

4.4. Proof of Lemma 2.4

Proof.

Let

[TABLE]

and

[TABLE]

It follows easily that

[TABLE]

where

[TABLE]

Then the coefficient of

[TABLE]

is given by

[TABLE]

where

[TABLE]

and

[TABLE]

In view of (2.3) and Lemma 4.1, one has

[TABLE]

and

[TABLE]

Substitution of (4.4) and (4.4) in (4.26) gives

[TABLE]

Noting that

[TABLE]

and

[TABLE]

Then one has

[TABLE]

where

[TABLE]

To show (2.22) holds, it suffices to prove

[TABLE]

where

[TABLE]

To this end, we first note some simple facts:

$\textbf{1}.$ If $j\notin\ \mbox{supp}\ (k+k^{\prime})\bigcap\ \mbox{supp}\ (K+K^{\prime})$, then

[TABLE]

Hence we always assume $j\in\ \mbox{supp}\ (k+k^{\prime})\bigcap\ \mbox{supp}\ (K+K^{\prime})$. Therefore one has

[TABLE]

$\textbf{2}.$ The following inequality always holds

[TABLE]

and then one has

[TABLE]

$\textbf{3}.$ It is easy to see

[TABLE]

and

[TABLE]

Based on (4.33) and (4.34), we obtain

[TABLE]

In view of (4.29) and (4.34), we have

[TABLE]

$\textbf{4}.$ It is easy to see

[TABLE]

Hence,

[TABLE]

Moreover, one has

[TABLE]

which implies

[TABLE]

Now we will prove the inequality (4.31) holds:

$\textbf{Case. 1.}\ \nu_{1}^{*}\leq N_{1}^{*}$.

$\textbf{Case. 1.1.}\ |j|\leq n_{3}^{*}.$

Then one has

[TABLE]

if

[TABLE]

Hence one obtains

[TABLE]

Remark 4.3*.*

Note that if $j,a,k,k^{\prime}$ are specified, and then $A,K,K^{\prime}$ are uniquely determined.

In view of (4.37) and (4.40), we have

[TABLE]

where the last inequality is based on $0<\delta_{1},\delta_{2}\ll 1$ and $C(\theta),C_{1}(\theta),C_{2}(\theta)$ are positive constants depending on $\theta$ only.

$\textbf{Case. 1.2.}\ j\in\{n_{1},n_{2}\},\ |n_{1}|=n_{1}^{*},\ |n_{2}|=n_{2}^{*}.$

If $2a_{j}+k_{j}+k^{\prime}_{j}>2$, then $|j|\leq n_{3}^{*}$, we are in Case. 1.1.. Hence in what follows, we always assume

[TABLE]

which implies

[TABLE]

and

[TABLE]

From (4.41) and in view of $j\in\{n_{1},n_{2}\}$, it follows that

[TABLE]

Since

[TABLE]

one has

[TABLE]

where $\delta_{1}\wedge\delta_{2}=\min\{\delta_{1},\delta_{2}\}$.

Remark 4.4*.*

Obviously, $\{n_{1},n_{2}\}\bigcap\mathrm{supp}\ \mathcal{M}_{\alpha\kappa\kappa^{\prime}}\neq\emptyset$, and if $n_{1}$ (resp. $n_{2}$), $\{n_{i}\}_{i\geq 3}$ and $m(k,k^{\prime})=l$ is specified, then $n_{2}$ (resp. $n_{1}$) is determined uniquely. Thus $n_{1},n_{2}$ range in a set of cardinality no more than

[TABLE]

Also, if $\{n_{i}\}_{i\geq 1}$ is given, then $\{2a_{n}+k_{n}+k^{\prime}_{n}\}_{n\in\mathbb{Z}}$ is specified, and hence $(a,k,k^{\prime})$ is specified up to a factor of

[TABLE]

where

[TABLE]

Following the inequality (4.4), we thus obtain

[TABLE]

where the last equality is based on Lemma 7.4 in [8] and $C_{3}(\theta),C_{4}(\theta),C_{5}(\theta)$ are positive constants depending on $\theta$ only.

$\textbf{Case. 2.}\ \nu_{1}^{*}>N_{1}^{*}.$

In view of (4.32), one has $n_{1}^{*}=\nu_{1}^{*}$. Hence, $n_{2}$ is determined by $n_{1}$, $\{n_{i}\}_{i\geq 3}$ and the momentum $m(k,k^{\prime})$. Similar to Case 1.2, we have

[TABLE]

where $C_{6}(\theta)$ is some positive constant depending on $\theta$ only.

Therefore, we finish the proof of (2.22). ∎

4.5. Proof of Lemma 2.6

Proof.

In view of (2.32) and for each $j\in\mathbb{Z}$, one has

[TABLE]

Now we would like to estimate

[TABLE]

Based on (2.3), one has

[TABLE]

In view of $||q||_{r,\infty}<1$ and $||I(0)||_{r,\infty}<1$, one has

[TABLE]

and

[TABLE]

Substituting (4.47) and (4.48) into (4.46), one has

[TABLE]

Now we will estimate the last inequality in the following two cases:

Case 1. $|j|\leq n_{3}^{*}$.

Then one has

[TABLE]

where the last inequality is based on Lemma 7.4 in [8] and $C_{1}(\theta)$ is a positive constant depending on $\theta$ only.

Case 2. $|j|>n_{3}^{*}$, which implies $k_{j}\leq 2$.

Then one has

[TABLE]

If $\{n_{i}\}_{i\geq 1}$ is given, then $\{2a_{n}+k_{n}+k^{\prime}_{n}\}_{n\in\mathbb{Z}}$ is specified, and hence $(a,k,k^{\prime})$ is specified up to a factor of

[TABLE]

where

[TABLE]

Since $|j|>n_{3}^{*}$, then $j\in\{n_{1},n_{2}\}$. Hence, if $(n_{i})_{i\geq 3}$ and $j,m^{\ast}(k,k^{\prime})$ are given, then $n_{1}$ and $n_{2}$ are uniquely determined. Then, one has

[TABLE]

where the last equality is based on Lemma 7.2 and Lemma 7.4 in [8], and $C_{1}(\theta),C_{2}(\theta),C_{3}(\theta)$ are positive constants depending on $\theta$ only.

Hence, we finished the proof of (2.33). ∎