A KAM Theorem for Higher Dimensional Reversible Nonlinear Schr\"{o}dinger Equations

TL;DR

This paper establishes a KAM theorem for infinite-dimensional reversible systems and demonstrates the existence and stability of quasi-periodic solutions for certain nonlinear Schrödinger equations on multi-dimensional tori.

Contribution

It introduces a new KAM theorem applicable to infinite-dimensional reversible systems, enabling analysis of nonlinear Schrödinger equations beyond Hamiltonian cases.

Findings

Existence of quasi-periodic solutions for nonlinear Schrödinger systems.

Linear stability of these solutions.

Extension of KAM theory to reversible, non-Hamiltonian systems.

Abstract

In the paper, we prove an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional reversible systems. Using this KAM theorem, we obtain the existence and linear stability of quasi-periodic solutions for a class of reversible (non-Hamiltonian) coupled nonlinear Schr\"{o}dinger systems on $d-$torus $\mathbb{T}^d$.

Equations

$$\{\begin{array}{l}\mathrm{i}{u}_t-\Delta u+M_{\xi }u+{\partial }_{\bar{u}}{G}_1(|u{|}^2,|v{|}^2)=0,\\ \mathrm{i}{v}_t-\Delta v+M_{\tilde{\xi }}v+{\partial }_{\bar{v}}{G}_2(|u{|}^2,|v{|}^2)=0,x\in {\mathbb{T}}^d:={\mathbb{R}}^d/2\pi {\mathbb{Z}}^d,\end{array}$$

$${\lambda }_{i^{(j)}}={\omega }_j=|i^{(j)}{|}^2+{\xi }_j,1\le j\le n,$$

$${\lambda }_i=|i{|}^2,i\notin {\mathcal{I}}_1,$$

$${\tilde{\lambda }}_{{\tilde{i}}^{(j)}}={\tilde{\omega }}_j=|{\tilde{i}}^{(j)}{|}^2+{\tilde{\xi }}_j,1\le j\le m,$$

$${\tilde{\lambda }}_i=|i{|}^2,i\notin {\mathcal{I}}_2,$$

$$\{\begin{array}{l}u(t,x)=\sum _{i\in {\mathbb{Z}}^d}{u}_i({\acute{\omega }}_{1}t,\cdots ,{\acute{\omega }}_{n}t){\varphi }_i(x),\\ v(t,x)=\sum _{i\in {\mathbb{Z}}^d}{v}_i({\acute{\stackrel{~}{\omega }}}_{1}t,\cdots ,{\acute{\stackrel{~}{\omega }}}_{m}t){\varphi }_i(x),\end{array}$$

$$\Vert z{\Vert }_{\rho }=\sum _{j\in {\mathbb{Z}}^d\setminus \mathcal{I}}{\mathrm{e}}^{|j|\rho }|z_j|<\infty ,$$

$${\mathcal{P}}_{\rho }:={\mathbb{T}}^n\times {\mathbb{T}}^m\times {\mathbb{R}}^n\times {\mathbb{R}}^m\times {\ell }_1^{\rho }\times {\ell }_2^{\rho }\times {\ell }_1^{\rho }\times {\ell }_2^{\rho }\ni y:=(\theta ,\phi ,I,J,z,w,\bar{z},\bar{w}).$$

$$\begin{array}{rl}{D}_{\rho }(r,s)=& \{y:|\text{Im}\theta |<r,|\text{Im}\phi |<r,|I|<s,|J|<s,\\ & \Vert z{\Vert }_{\rho }<s,\Vert w{\Vert }_{\rho }<s,\Vert \bar{z}{\Vert }_{\rho }<s,\Vert \bar{w}{\Vert }_{\rho }<s\}\end{array}$$

$$\begin{array}{rl}f(y;\zeta )=& \sum _{\genfrac{}{}{0ex}{}{k\in {\mathbb{Z}}^n,l\in {\mathbb{N}}^n,\alpha ,\beta \in {\mathbb{N}}^{{\mathbb{Z}}_1^d},}{\tilde{k}\in {\mathbb{Z}}^m,\tilde{l}\in {\mathbb{N}}^m,\tilde{\alpha },\tilde{\beta }\in {\mathbb{N}}^{{\mathbb{Z}}_2^d}}}{f}_{kl\alpha \beta ,\tilde{k}\tilde{l}\tilde{\alpha }\tilde{\beta }}(\zeta ){\mathrm{e}}^{\mathrm{i}(⟨k,\theta ⟩+⟨\tilde{k},\phi ⟩)}{I}^l{J}^{\tilde{l}}{z}^{\alpha }{\bar{z}}^{\beta }{w}^{\tilde{\alpha }}{\bar{w}}^{\tilde{\beta }},\end{array}$$

$$\begin{array}{rl}\Vert f{\Vert }_{D_{\rho }(r,s)\times \mathcal{O}}=& \underset{\genfrac{}{}{0ex}{}{|z{|}_{\rho }<s,|\bar{z}{|}_{\rho }<s}{|w{|}_{\rho }<s,|\bar{w}{|}_{\rho }<s}}{\sup }\sum _{\genfrac{}{}{0ex}{}{k,l,\alpha ,\beta ,}{\tilde{k},\tilde{l},\tilde{\alpha },\tilde{\beta }}}|f_{kl\alpha \beta ,\tilde{k}\tilde{l}\tilde{\alpha }\tilde{\beta }}{|}_{\mathcal{O}}{\mathrm{e}}^{(|k|+|\tilde{k}|)r}{s}^{(|l|+|\tilde{l}|)}|z^{\alpha }||{\bar{z}}^{\beta }||w^{\tilde{\alpha }}||{\bar{w}}^{\tilde{\beta }}|.\end{array}$$

$$z_j^{\varrho }=\{\begin{array}{ll}{z}_j,& \varrho =+,\\ {\bar{z}}_j,& \varrho =-,\end{array}$$

$$\begin{array}{rl}X(y)=& X^{(\theta )}(y)\frac{\partial }{\partial \theta }+X^{(\phi )}(y)\frac{\partial }{\partial \phi }+X^{(I)}(y)\frac{\partial }{\partial I}+X^{(J)}(y)\frac{\partial }{\partial J}\\ & +X^{(z)}(y)\frac{\partial }{\partial z}+X^{(w)}(y)\frac{\partial }{\partial w}+X^{(\bar{z})}(y)\frac{\partial }{\partial \bar{z}}+X^{(\bar{w})}(y)\frac{\partial }{\partial \bar{w}}\\ =& \sum _{\text{v}\in \mathcal{V}}{X}^{(\text{v})}(y)\frac{\partial }{\partial \text{v}},\end{array}$$

$$\begin{array}{rl}\Vert X{\Vert }_{s;D_{\rho }(r,s)\times \mathcal{O}}=& \Vert X^{(\theta )}{\Vert }_{D_{\rho }(r,s)\times \mathcal{O}}+\Vert X^{(\phi )}{\Vert }_{D_{\rho }(r,s)\times \mathcal{O}}\\ & +\frac{1}{s}\Vert X^{(I)}{\Vert }_{D_{\rho }(r,s)\times \mathcal{O}}+\frac{1}{s}\Vert X^{(J)}{\Vert }_{D_{\rho }(r,s)\times \mathcal{O}}\\ & +\frac{1}{s}\sum _{\sigma =\pm }(\sum _{i\in {\mathbb{Z}}_1^d}{\mathrm{e}}^{|i|\rho }\Vert X^{(z_i^{\sigma })}{\Vert }_{D_{\rho }(r,s)\times \mathcal{O}}+\sum _{i\in {\mathbb{Z}}_2^d}{\mathrm{e}}^{|i|\rho }\Vert X^{(w_i^{\sigma })}{\Vert }_{D_{\rho }(r,s)\times \mathcal{O}}).\end{array}$$

$$X^0(y;\zeta )=N(y;\zeta )+\mathcal{A}(y;\zeta )$$

$$\begin{array}{rl}N=& \omega (\zeta )\frac{\partial }{\partial \theta }+\tilde{\omega }(\zeta )\frac{\partial }{\partial \phi }+\mathrm{i}\Omega (\zeta )z\frac{\partial }{\partial z}+\mathrm{i}\tilde{\Omega }(\zeta )w\frac{\partial }{\partial w}-\mathrm{i}\Omega (\zeta )\bar{z}\frac{\partial }{\partial \bar{z}}-\mathrm{i}\tilde{\Omega }(\zeta )\bar{w}\frac{\partial }{\partial \bar{w}}\\ =& \sum _{b=1}^n{\omega }_b(\zeta )\frac{\partial }{\partial {\theta }_b}+\sum _{b=1}^m{\tilde{\omega }}_b(\zeta )\frac{\partial }{\partial {\phi }_b}\\ & +\sum _{j\in {\mathbb{Z}}_1^d}(\mathrm{i}{\Omega }_j(\zeta )z_j\frac{\partial }{\partial z_j}-\mathrm{i}{\Omega }_j(\zeta ){\bar{z}}_j\frac{\partial }{\partial {\bar{z}}_j})+\sum _{j\in {\mathbb{Z}}_2^d}(\mathrm{i}{\tilde{\Omega }}_j(\zeta )w_j\frac{\partial }{\partial w_j}-\mathrm{i}{\tilde{\Omega }}_j(\zeta ){\bar{w}}_j\frac{\partial }{\partial {\bar{w}}_j}),\end{array}$$

$$\begin{array}{rl}\mathcal{A}=& \mathrm{i}A(\zeta )w\frac{\partial }{\partial z}+\mathrm{i}\tilde{A}(\zeta )z\frac{\partial }{\partial w}-\mathrm{i}A(\zeta )\bar{w}\frac{\partial }{\partial \bar{z}}-\mathrm{i}\tilde{A}(\zeta )\bar{z}\frac{\partial }{\partial \bar{w}}\\ =& \sum _{j\in {\mathbb{Z}}_1^d\cap {\mathbb{Z}}_2^d}(\mathrm{i}{A}_j(\zeta )w_j\frac{\partial }{\partial z_j}+\mathrm{i}{\tilde{A}}_j(\zeta )z_j\frac{\partial }{\partial w_j}-\mathrm{i}{A}_j(\zeta ){\bar{w}}_j\frac{\partial }{\partial {\bar{z}}_j}-\mathrm{i}{\tilde{A}}_j(\zeta ){\bar{z}}_j\frac{\partial }{\partial {\bar{w}}_j}),\end{array}$$

$$\begin{cases}\dot{\theta}=\omega,\,\,\dot{\varphi}=\tilde{\omega},\\ \dot{I}=0,\,\,\dot{J}=0,\\ \dot{z}^{\sigma}_{j}=\sigma\mathrm{i}\Omega_{j}z^{\sigma}_{j},\quad&\sigma=\pm,\,j\in\mathbb{Z}^{d}_{1}\setminus\mathbb{Z}^{d}_{2},\\ \dot{w}^{\sigma}_{j}=\sigma\mathrm{i}\tilde{\Omega}_{j}w^{\sigma}_{j},\quad&j\in\mathbb{Z}^{d}_{2}\setminus\mathbb{Z}^{d}_{1},\\ \left(\begin{array}[]{c}\dot{z}^{\sigma}_{j}\\ \dot{w}^{\sigma}_{j}\\ \end{array}\right)=\sigma\mathrm{i}\left(\begin{array}[]{cc}\Omega_{j}&A_{j}\\ \tilde{A}_{j}&\tilde{\Omega}_{j}\\ \end{array}\right)\left(\begin{array}[]{c}z^{\sigma}_{j}\\ w^{\sigma}_{j}\\ \end{array}\right),\quad&j\in\mathbb{Z}^{d}_{1}\cap\mathbb{Z}^{d}_{2}.\\ \end{cases}$$

$$X=X^0+P=N+\mathcal{A}+P(y;\zeta ).$$

$${\Omega }_j=|j{|}^2+{\Omega }_j^0,j\in {\mathbb{Z}}_1^d,{\tilde{\Omega }}_j=|j{|}^2+{\tilde{\Omega }}_j^0,j\in {\mathbb{Z}}_2^d,$$

$$M_{j}\>=\left(\begin{array}[]{cc}\Omega_{j}&A_{j}\\ \tilde{A}_{j}&\tilde{\Omega}_{j}\\ \end{array}\right),\,j\in\mathbb{Z}^{d}_{1}\cap\mathbb{Z}^{d}_{2}.$$

$$|⟨k,\omega ⟩+⟨\tilde{k},\tilde{\omega }⟩|\ge \frac{\gamma }{(|k|+|\tilde{k}|{)}^{\tau }},(k,\tilde{k})\ne 0,$$

$$|⟨k,\omega ⟩+⟨\tilde{k},\tilde{\omega }⟩+{\Omega }_i|\ge \frac{\gamma }{(|k|+|\tilde{k}|{)}^{\tau }},i\in {\mathbb{Z}}_1^d\setminus {\mathbb{Z}}_2^d$$

$$|⟨k,\omega ⟩+⟨\tilde{k},\tilde{\omega }⟩+{\tilde{\Omega }}_i|\ge \frac{\gamma }{(|k|+|\tilde{k}|{)}^{\tau }},i\in {\mathbb{Z}}_2^d\setminus {\mathbb{Z}}_1^d,$$

$$|⟨k,\omega ⟩+⟨\tilde{k},\tilde{\omega }⟩+{\Omega }_i\pm {\Omega }_j|\ge \frac{\gamma }{(|k|+|\tilde{k}|{)}^{\tau }},(k,\tilde{k})\ne 0,i,j\in {\mathbb{Z}}_1^d\setminus {\mathbb{Z}}_2^d,$$

$$|⟨k,\omega ⟩+⟨\tilde{k},\tilde{\omega }⟩+{\Omega }_i\pm {\tilde{\Omega }}_j|\ge \frac{\gamma }{(|k|+|\tilde{k}|{)}^{\tau }},i\in {\mathbb{Z}}_1^d\setminus {\mathbb{Z}}_2^d,j\in {\mathbb{Z}}_2^d\setminus {\mathbb{Z}}_1^d,$$

$$|⟨k,\omega ⟩+⟨\tilde{k},\tilde{\omega }⟩+{\tilde{\Omega }}_i\pm {\tilde{\Omega }}_j|\ge \frac{\gamma }{(|k|+|\tilde{k}|{)}^{\tau }},(k,\tilde{k})\ne 0,i,j\in {\mathbb{Z}}_2^d\setminus {\mathbb{Z}}_1^d,$$

$$|\det ((⟨k,\omega ⟩+⟨\tilde{k},\tilde{\omega }⟩)I_2\pm M_i)|\ge \frac{\gamma }{(|k|+|\tilde{k}|{)}^{\tau }},i\in {\mathbb{Z}}_1^d\cap {\mathbb{Z}}_2^d,$$

$$|\det ((⟨k,\omega ⟩+⟨\tilde{k},\tilde{\omega }⟩+{\Omega }_i)I_2\pm M_j)|\ge \frac{\gamma }{(|k|+|\tilde{k}|{)}^{\tau }},$$

$$|\det ((⟨k,\omega ⟩+⟨\tilde{k},\tilde{\omega }⟩+{\tilde{\Omega }}_i)I_2\pm M_j)|\ge \frac{\gamma }{(|k|+|\tilde{k}|{)}^{\tau }},$$

Full text

Keywords. KAM theorem, reversible vector field, NLS, quasi-periodic solution.

2010 Mathematics Subject Classification: 37K55, 35B15.

1 Introduction and Main Result

1.1 Introduction

Among various techniques for studying the existence of quasi-periodic solutions of nonlinear partial differential equations (PDEs), KAM theory is one of the most powerful tools. Kuksin [12] and Wayne [18] first developed Newtonian scheme to investigate quasi-periodic solutions of Hamiltonian PDEs in one dimensional space. The general idea is that Hamiltonian function is thought of as a normal form plus a real analytic perturbation, then constructing an infinite symplectic transformation sequences to make the perturbation smaller and smaller and construct a converged local normal form. The normal form is helpful to understand the dynamics around the quasi–periodic solutions, for example, one sees the linear stability and zero Lyapunov exponents.

The feasibility of KAM method in one dimensional Hamiltonian PDEs, however, depends crucially on the second Melnikov condition. Due to multiple eigenvalues of linear operator, such condition is not naturally available in higher dimensional case, and KAM method is in general not easy to apply. In 1998, Bourgain [3] first made a breakthrough. He used multi-scale analysis method to avoid the cumbersome second Melnikov condition and thus obtained small-amplitude quasi-periodic solutions of two dimensional nonlinear Schrödinger equations (NLS). Later, he improved his method and studied quasi-periodic solutions of NLS and nonlinear wave equations in any dimensional space. Following the idea and method in [3], abundant works [4, 2, 17] have been done.

There are strong hopes to develop KAM theory for higher dimensional PDEs, because multi-scale analysis can not help us to understand the dynamics around quasi–periodic solutions. Geng and You [9] first built a KAM theorem for higher dimensional beam equation and nonlocal smooth NLS. They used momentum conservation condition which means the nonlinearity is independent of spatial variable $x$ to avoid the difficulty of multiple eigenvalues. In 2010, Eliasson and Kuksin [6] studied a class of higher dimensional NLS with convolution type potential and nonlinearity containing spatial variable $x$. They used the block diagonal normal form structure to deal with multiple eigenvalues of linear operator. Besides, they introduced Lipschitz domain property of perturbation to handle infinitely many resonances at each KAM step. By developing Töplitz-Lipschitz property of perturbation and constructing appropriate tangential sites on $\mathbb{Z}^{2}$, Geng, Xu and You [8] got the quasi-periodic solutions of two-dimensional completely resonant NLS . Later on, Geng and You [11] simplified the proof of [6] via momentum conservation condition. C.Procesi and M.Procesi [15] extended the result in [8] to the $d$-dimensional case by a very ingenious choice of tangential sites. See also [5, 16, 19, 10] for further studies.

Recently, KAM theory for Hamiltonian PDEs has been generalized to reversible ones in one dimensional space [20, 1]. To the best of our knowledge, there is not any KAM result for higher dimensional reversible PDEs yet. In fact, reversible PDEs are a class of physically important PDEs as well as Hamiltonian ones. For example, the following coupled NLS system arising from nonlinear optics (see [13]):

[TABLE]

where $M_{\xi}$ and $M_{\widetilde{\xi}}$ are real Fourier multiplier, $G_{i}=o(|u|^{3}+|v|^{3}),\,i=1,2$ are real analytic functions near $(u,v)=(0,0).$ When $G_{1}=G_{2},$ equation (1.1) is not only reversible (with respect to the involution S_{0}(u(x),v(x))$$=(\bar{u}(-x),\bar{v}(-x)) ) but also Hamiltonian, and quasi-periodic solutions for this case were recently obtained via Hamiltonian KAM theory in [21]. When $G_{1}\neq G_{2},$ equation (1.1) is no longer Hamiltonian but still reversible. This motives us to develop reversible KAM theory for equation (1.1).

As in the Hamiltonian case, the major difficulty in constructing KAM scheme for equation (1.1) is also to deal with infinitely many resonances. In this paper, by introducing the class of Töplitz-Lipschitz vector fields (inspired by [6, 8, 1]) and momentum conservation condition, the difficulty can be overcome. Töplitz-Lipschitz vector field plays the most essential role and it reduces infinitely many resonances to only finitely many ones. momentum conservation condition can simplify the proof. We mention that Töplitz-Lipschitz vector field introduced here is the generalization of Töplitz-Lipschitz functions in [8].

Following [6], we could study more general equation (1.1) with nonlinearities $G_{i}$ containing the spatial variable $x$ explicitly, but the proof would be more complicated since we have to deal with block diagonal normal form. Our present paper is working on NLS with the external parameters, and the more interesting completely resonant case (i.e. no Fourier multiplier in equation (1.1)) will be in our forthcoming paper [7]. As in the Hamiltonian case (see [8, 15]), the construction of Birkhoff normal form will be a new challenge.

1.2 Main result

Let $\mathcal{I}_{1}=\{i^{(1)},i^{(2)},\cdots,i^{(n)}\}\subset\mathbb{Z}^{d}$ and $\mathcal{I}_{2}=\{\tilde{i}^{(1)},\tilde{i}^{(2)},\cdots,\tilde{i}^{(m)}\}\subset\mathbb{Z}^{d}$ be two sets of distinguished sites of Fourier modes. For some technical convenience, we suppose $0\in\mathcal{I}_{1}\cap\mathcal{I}_{2}.$ Denote by $\lambda_{i},\,i\in\mathbb{Z}^{d}$ ($resp.$ $\widetilde{\lambda}_{i}$) the eigenvalues of $-\Delta+M_{\xi}$ ($resp.$ $-\Delta+M_{\widetilde{\xi}}$ ) under periodic boundary conditions:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and the corresponding eigenfunctions $\phi_{i}(x)=\frac{1}{\sqrt{(2\pi)^{d}}}\mathrm{e}^{\mathrm{i}\langle i,x\rangle}.$ Assume the parameters $(\xi,\tilde{\xi})\in\mathcal{O}:=[0,1]^{n}\times[0,1]^{m}\subset\mathbb{R}^{n+m}.$

Then we have the following main result.

Theorem 1.1**.**

For any $0<\gamma\ll 1,$ there exists a Cantor subset $\mathcal{O}_{\gamma}\subset\mathcal{O}$ with $\operatorname{meas}(\mathcal{O}\setminus\mathcal{O}_{\gamma})=O(\gamma^{\frac{1}{4}})$, such that for any $(\xi,\tilde{\xi})\in\mathcal{O}_{\gamma},$ equation (1.1) with reversible perturbation $G_{1}\neq G_{2}$ possesses a small amplitude quasi-periodic solution of the form

[TABLE]

*where $u_{i}:\mathbb{T}^{n}\rightarrow\mathbb{R}$ *(*resp. $v_{i}:\mathbb{T}^{m}\rightarrow\mathbb{R}$ ) and $\acute{\omega}_{1},\cdots,\acute{\omega}_{n}$ *(*resp. $\acute{\tilde{\omega}}_{1},\cdots,\acute{\tilde{\omega}}_{m}$ ) are close to the unperturbed frequencies $\omega_{1},\cdots,\omega_{n}$ *(resp. $\tilde{\omega}_{1},\cdots,\tilde{\omega}_{m}$ ). Moreover, the quasi-periodic solutions are real analytic and linearly stable.

The rest of the paper is organized as follows. In Section 2, we give the definitions of weighted norms for functions and vector fields. An abstract KAM theorem (Theorem 3.1) for infinite dimensional reversible systems is presented in Section 3. In Section 4, we use the KAM theorem to prove Theorem 1.1. The proof of Theorem 3.1 is given in Section 5. Some properties of reversible system and technical lemmas are listed in the Appendix.

2 Preliminary

For the sake of completeness, we first introduce some definitions and notations.

Let $\mathcal{I}\subset\mathbb{Z}^{d}$ be a finite subset and $\rho>0$, we introduce the Banach space $\ell^{\rho}_{\mathcal{I}}$ of all complex sequences $z=(z_{j})_{j\in\mathbb{Z}^{d}\setminus\mathcal{I}}$ with

[TABLE]

where $|j|=\sqrt{|j_{1}|^{2}+\cdots+|j_{d}|^{2}}$.

Given two finite subsets of $\mathbb{Z}^{d}:$ $\mathcal{I}_{1}=\{i^{(1)},i^{(2)},\cdots,i^{(n)}\}$ and $\mathcal{I}_{2}=\{\tilde{i}^{(1)},\tilde{i}^{(2)},\cdots,\tilde{i}^{(m)}\}$. Denote $\mathbb{Z}^{d}_{l}:=\mathbb{Z}^{d}\setminus\mathcal{I}_{l}$ and $\ell^{\rho}_{l}:=\ell^{\rho}_{\mathcal{I}_{l}},\,(l=1,2).$ Consider the phase space

[TABLE]

We introduce a complex neighborhood

[TABLE]

of $\mathcal{T}^{n+m}_{0}:=\mathbb{T}^{n}\times\mathbb{T}^{m}\times\{I=0\}\times\{J=0\}\times\{z=0\}\times\{w=0\}\times\{\bar{z}=0\}\times\{\bar{w}=0\},$ where $|\cdot|$ is the sup-norm for vectors.

Suppose $\mathcal{O}\subset\mathbb{R}^{n+m}$ is a compact parameter subset. A function $f:D_{\rho}(r,s)\times\mathcal{O}\rightarrow\mathbb{C}$ is real analytic in $y$ and $C^{4}_{W}$ (i.e., $C^{4}-$smooth in the sense of Whitney) in $\zeta\in\mathcal{O}$ and has Taylor-Fourier series expansion

[TABLE]

where $\langle k,\theta\rangle=\sum\limits^{n}_{i=1}k_{i}\theta_{i},$ $I^{l}=\prod^{n}\limits_{i=1}I^{l_{i}}_{i}$ and $z^{\alpha}\bar{z}^{\beta}=\prod\limits_{j\in\mathbb{Z}^{d}_{1}}z^{\alpha_{j}}_{j}\bar{z}^{\beta_{j}}_{j}.$ $\alpha,\beta$ have only finitely many nonzero components, and similarly for the other indexes. We define the weighted norm of $f$ as follows

[TABLE]

where $|f_{kl\alpha\beta,\tilde{k}\tilde{l}\tilde{\alpha}\tilde{\beta}}|_{\mathcal{O}}=\sup\limits_{\zeta\in\mathcal{O}}\sum\limits_{0\leq b\leq 4}|\partial^{b}_{\zeta}f_{kl\alpha\beta,\tilde{k}\tilde{l}\tilde{\alpha}\tilde{\beta}}|.$

Let

[TABLE]

and similarly for $z^{\varrho}=(z^{\varrho}_{j})_{j\in\mathbb{Z}^{d}_{1}},$ $w^{\varrho}_{j}$ and $w^{\varrho}$.

Consider a vector field $X(y),y\in D_{\rho}(r,s):$

[TABLE]

where $\mathscr{V}=\{\theta_{a},\varphi_{b},\,z_{i},w_{j},\bar{z}_{i},\bar{w}_{j}:a=1,\cdots,n;b=1,\cdots,m;i\in\mathbb{Z}^{d}_{1};j\in\mathbb{Z}^{d}_{2}\}.$ Suppose $X$ is real analytic in $y$ and depends $C^{4}_{W}$ smoothly on parameters $\zeta\in\mathcal{O},$ we define the weighted norm of $X$ as follows444The norm of vector valued function $G:D_{\rho}(r,s)\times\mathcal{O}\rightarrow\mathbb{C}^{n}$, $n<\infty,$ is defined as $\|G\|_{D_{\rho}(r,s)\times\mathcal{O}}=\sum\limits^{n}_{b=1}\|G_{b}\|_{D_{\rho}(r,s)\times\mathcal{O}}$.

[TABLE]

The Lie bracket of two vector fields $X$ and $Y$ is defined as $[X,Y]=YX-XY.$

3 A KAM Theorem for Infinite Dimensional Reversible Systems

In this section, we give an abstract KAM theorem for infinite dimensional reversible systems. The definition and properties of reversible system are listed in the Appendix.

Given an involution $S:(\theta,\varphi,I,J,z,w,\bar{z},\bar{w})\mapsto(-\theta,-\varphi,I,J,\bar{z},\bar{w},z,w).$ We consider a family of $S-$reversible vector fields

[TABLE]

[TABLE]

[TABLE]

where $\omega_{b},\tilde{\omega}_{b},\Omega_{j},\,\tilde{\Omega}_{j},\,A_{j},\,\tilde{A}_{j}\in\mathbb{R}$ and $A_{j}=0,(j\in\mathbb{Z}^{d}_{1}\setminus\mathbb{Z}^{d}_{2}),$ $\tilde{A}_{j}=0,(j\in\mathbb{Z}^{d}_{2}\setminus\mathbb{Z}^{d}_{1}).$

For each $\zeta\in\mathcal{O}$, the motion equation governed by the vector field $X^{0}$ is

[TABLE]

Obviously, $\{(\theta+\omega t,\varphi+\tilde{\omega}t,0,0,0,0,0,0,):t\in\mathbb{R}\}$ forms an invariant torus of the above system.

Consider now the perturbed $S-$reversible vector field

[TABLE]

We will prove that, for typical (in the sense of Lebesgue measure) $\zeta\in\mathcal{O}$, the vector fields (3.3) still admit invariant tori for sufficiently small $P.$ For this purpose, we need the following six assumptions:

**(A1) Non-degeneracy: **

The map $\zeta\mapsto(\omega(\zeta),\tilde{\omega}(\zeta))$ is a $C^{4}_{W}$ diffeomorphism between $\mathcal{O}$ and its image.

(A2) Asymptotics of normal frequencies:

[TABLE]

where $\Omega^{0}_{j},\,\tilde{\Omega}^{0}_{j}\in C^{4}_{W}(\mathcal{O})$ with $C^{4}_{W}-$norm bounded by some small positive constant $L.$

**(A3) Non-resonance conditions: **

Denote

[TABLE]

Suppose $A_{j},\tilde{A}_{j}\in C^{4}_{W}(\mathcal{O})$ and there exist $\gamma,\tau>0,$ such that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

$(i,j)\ \hbox{or}\ (j,i)\in(\mathbb{Z}^{d}_{1}\cap\mathbb{Z}^{d}_{2})\times(\mathbb{Z}^{d}_{1}\setminus\mathbb{Z}^{d}_{2}),$

[TABLE]

$(i,j)\ \hbox{or}\ (j,i)\in(\mathbb{Z}^{d}_{1}\cap\mathbb{Z}^{d}_{2})\times(\mathbb{Z}^{d}_{2}\setminus\mathbb{Z}^{d}_{1}),$

[TABLE]

where $I_{b}$ is $b\times b$ identity matrix. $\det(\cdot)$, $\otimes$ and $(\cdot)^{T}$ denotes the determinant, the tensor product and the transpose of matrices, respectively.

**(A4) Regularity: **

$\mathcal{A}+P$ is real analytic in $y$ and $C^{4}_{W}-$smooth in $\zeta$. Moreover, $\|\mathcal{A}\|_{s;D(r,s)\times\mathcal{O}}<1,$ $\varepsilon_{0}:=\|P\|_{s;D(r,s)\times\mathcal{O}}<\infty.$

**(A5) Momentum conservation: **

The perturbation $P$ satisfies $[P,\mathbb{M}_{l}]=0,\,\,(l=1,\cdots,d),$ where

[TABLE]

**(A6) Töplitz-Lipschitz property: **

Let

[TABLE]

For fixed $i,j\in\mathbb{Z}^{d}$ $c\in\mathbb{Z}^{d}\setminus\{0\},$ the following limits exist and satisfy:

[TABLE]

[TABLE]

[TABLE]

Furthermore, there exists $K>0$ such that when $|t|>K,$ the following estimates hold.

[TABLE]

$x=\theta_{b},\varphi_{b},I_{b},J_{b};u=z,w.$

[TABLE]

$u=z,w.$

[TABLE]

$(u,v)=(z,w),(w,z).$

Remark 3.1**.**

In (A6), the conditions (3.16)-(3.17) and (3.19)-(3.20) are the most important for measure estimates. The role played by the conditions (3.15) and (3.18) is to preserve Töplitz-Lipschitz property after the KAM iteration (see Lemmas 5.4 and 5.5).

Now we state our KAM theorem.

Theorem 3.1**.**

Suppose the $S-$reversible vector field $X=N+\mathcal{A}+P$ in (3.3) satisfies $(A1)-(A6).$ $\gamma>0$ is small enough. Then there exists a positive $\varepsilon$ depending only on $n,m,L,K,\tau,r,s$ and $\rho$ such that if $\|P\|_{s;D(r,s)\times\mathcal{O}}\leq\varepsilon,$ the following holds: There exist (1) a Cantor subset $\mathcal{O}_{\gamma}\subset\mathcal{O}$ with Lebesgue measure $\operatorname{meas}(\mathcal{O}\setminus\mathcal{O}_{\gamma})=O(\gamma^{1/4});$ (2) a $C^{4}_{W}-$smooth family of real analytic torus embeddings

[TABLE]

which is $\frac{\varepsilon}{\gamma^{20}}-$close to the trivial embedding $\Psi_{0}:\mathbb{T}^{n+m}\times\mathcal{O}\rightarrow\mathcal{T}^{n+m}_{0};$ (3) a $C^{4}_{W}-$smooth map $\phi:\mathcal{O}_{\gamma}\rightarrow\mathbb{R}^{n+m}$ which is $\varepsilon-$close to the unperturbed frequency $(\omega,\tilde{\omega})$ such that for every $\zeta\in\mathcal{O}_{\gamma}$ and $(\theta,\varphi)\in\mathbb{T}^{n+m}$ the curve $t\mapsto\Psi((\theta,\varphi)+\phi(\zeta)t;\zeta)$ is a quasi-periodic solution of the equation governed by the vector field $X=N+\mathcal{A}+P.$

4 Application to the Coupled NLS

4.1 Lattice form of equation (1.1)

Let $\mathcal{I}_{1}=\{i^{(1)},i^{(2)},\cdots,i^{(n)}\}\subset\mathbb{Z}^{d}$ and $\mathcal{I}_{2}=\{\tilde{i}^{(1)},\tilde{i}^{(2)},\cdots,\tilde{i}^{(m)}\}\subset\mathbb{Z}^{d}$ and $0\in\mathcal{I}_{1}\cap\mathcal{I}_{2}.$ Under periodic boundary conditions, we denote the eigenvalues of $-\Delta+M_{\xi}$ and $-\Delta+M_{\widetilde{\xi}}$ by $\lambda_{i},\,i\in\mathbb{Z}^{d}$ and $\widetilde{\lambda}_{i},\,i\in\mathbb{Z}^{d}$, respectively, satisfying

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and the corresponding eigenfunctions $\phi_{i}(x)=(2\pi)^{-\frac{d}{2}}\mathrm{e}^{\mathrm{i}\langle i,x\rangle}.$

Without loss of generality, we consider the equation (1.1) when $G_{1}=|u|^{4}|v|^{2},G_{2}=|u|^{2}|v|^{2}$ since the higher order terms of nonlinearities will not make any difference.

Let $u(t,x)=\sum\limits_{h\in\mathbb{Z}^{d}}q_{h}(t)\phi_{h}(x),$ $v(t,x)=\sum\limits_{h\in\mathbb{Z}^{d}}p_{h}(t)\phi_{h}(x),$ then we obtain the equivalent lattice reversible equations

[TABLE]

which is reversible with respect to $S(q,p,\bar{q},\bar{p})=(\bar{q},\bar{p},q,p),$ where

[TABLE]

and

[TABLE]

with

[TABLE]

and

[TABLE]

By direct computation, one can verify that the perturbations $Q^{(q)}=(Q^{(q_{h})})_{h\in\mathbb{Z}^{d}_{1}}$ and $\tilde{Q}^{(p)}=(\tilde{Q}^{(p_{h})})_{h\in\mathbb{Z}^{d}_{2}}$ have the following regularity property.

Lemma 4.1**.**

For any fixed $\rho>0$, $Q^{(q)}$ (resp. $\tilde{Q}^{(p)}$) is real analytic as a map in a neighborhood of the origin with

[TABLE]

Let $P^{0}=\sum\limits_{\varrho=\pm}(Q^{(q^{\varrho})}\frac{\partial}{\partial q^{\varrho}}+\tilde{Q}^{(p^{\varrho})}\frac{\partial}{\partial p^{\varrho}}),$ then we have the following lemma.

Lemma 4.2**.**

(1)* $[P^{0},\mathbb{M}^{0}_{l}]=0,\,l=1,\cdots,d,$ where*

[TABLE]

(2)* $P^{0}$ satisfies Töplitz-Lipschitz property.*

Proof.

(1) If we write

[TABLE]

then by (4.4) and (4.5), we have $Q^{(q^{\varrho}_{h})}_{\alpha\beta\tilde{\alpha}\tilde{\beta}}=0$ and $\tilde{Q}^{(p^{\varrho}_{h})}_{\alpha\beta\tilde{\alpha}\tilde{\beta}}=0$ when $\pi_{l}(\alpha\beta,\tilde{\alpha}\tilde{\beta};v)\neq 0,v=q^{\varrho}_{h},p^{\varrho}_{h}$. where

[TABLE]

Note that by elementary computation, we have $[q^{\alpha}\bar{q}^{\beta}p^{\tilde{\alpha}}\bar{p}^{\tilde{\beta}},\mathbb{M}^{0}_{l}]=\mathrm{i}\pi_{l}(\alpha\beta,\tilde{\alpha}\tilde{\beta};v)q^{\alpha}\bar{q}^{\beta}p^{\tilde{\alpha}}\bar{p}^{\tilde{\beta}},$ which implies $[P^{0},\mathbb{M}^{0}_{l}]=0.$

(2) We only consider $\lim\limits_{t\rightarrow\infty}\frac{\partial Q^{(q_{i+tc})}}{\partial q_{j+tc}}.$ It follows from (4.2) and (4.4) that

[TABLE]

then $\frac{\partial Q^{(q_{i+tc})}}{\partial q_{j+tc}}=\frac{\partial Q^{(q_{i})}}{\partial q_{j}}=\lim\limits_{t\rightarrow\infty}\frac{\partial Q^{(q_{i+tc})}}{\partial q_{j+tc}}.$

∎

4.2 Verification of assumptions (A1)-(A6)

We introduce action-angle variables $(\theta,\varphi,I,J)$ and normal coordinates $(z,w,\bar{z},\bar{w})$ by the following transformation $\Psi$ on some $D(r,s),$ ($r,s>0$):

[TABLE]

where the $I^{0}_{j},\,\,J^{0}_{j}$ are fixed numbers satisfying $0<s<I^{0}_{j},\,J^{0}_{j}<2s.$

We obtain a new vector field

[TABLE]

where

[TABLE]

and

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

$X$ is reversible with respect to the involution

[TABLE]

Now we give the verification of assumptions (A1)-(A6) for (4.9).

*Verifying *(A1): Set $\zeta=(\xi,\tilde{\xi})$, it is obvious as the Jacobian matrix $\frac{\partial(\omega,\tilde{\omega})}{\partial\zeta}=\frac{\partial(\omega,\tilde{\omega})}{\partial(\xi,\tilde{\xi})}=I_{n+m}.$

*Verifying *(A2): It is also obvious.

*Verifying *(A3): One can refer to Section 3.2 in [8] since the proof is similar.

*Verifying *(A4): Suppose vector field (4.9) is defined on the domain $D(r,s)$ with $0<r<1,s=\varepsilon^{2}.$ It follows from (4.11)-(4.16) that

[TABLE]

[TABLE]

then

[TABLE]

*Verifying *(A5): Through the transformation $\Psi$ in (4.8), the vector fields $\mathbb{M}^{0}_{l}$ in (4.6) are transformed into $\mathbb{M}_{l}=\Psi^{*}\mathbb{M}^{0}_{l},$ then

[TABLE]

*Verifying *(A6): We only consider $\frac{\partial P^{(\theta_{b})}}{\partial z_{j+tc}}$ and $\frac{\partial P^{(z_{i+tc})}}{\partial z_{j+tc}}$ and the others can be verified similarly.

[TABLE]

then

[TABLE]

It follows from (4.13) and (2) in Lemma 4.2 that $\frac{\partial P^{(z_{i})}}{\partial z_{j}}=O(s^{\frac{5}{2}})\mathrm{e}^{-\rho|i-j|}$ and $\frac{\partial P^{(z_{i+tc})}}{\partial z_{j+tc}}=\frac{\partial P^{(z_{i})}}{\partial z_{j}}=\lim\limits_{t\rightarrow\infty}\frac{\partial P^{(z_{i+tc})}}{\partial z_{j+tc}}.$ then

[TABLE]

5 Proof of Theorem 3.1

At the $\nu$th step of KAM iteration, we consider an $S-$reversible vector field on $D_{\rho_{\nu}}(r_{\nu},s_{\nu})\times\mathcal{O}_{\nu}:$

[TABLE]

satisfying (A1)-(A6), where $N_{\nu}$ and $\mathcal{A}_{\nu}$ have the same form as $N$ and $\mathcal{A}$ in (3.1) and (3.2).

We shall construct an $S-$invariant transformation

[TABLE]

such that $\Phi^{*}_{\nu}X_{\nu}:=(D\Phi_{\nu})^{-1}\cdot X_{\nu}\circ\Phi_{\nu}=N_{\nu+1}+\mathcal{A}_{\nu+1}+P_{\nu+1}$ with new normal form $N_{\nu+1},$ $\mathcal{A}_{\nu+1}$ and a much smaller perturbation term $P_{\nu+1}$ and still satisfies (A1)-(A6).

In the sequel, for simplicity, we drop the subscript $\nu$ and write the symbol ‘$+$’ for ‘$\nu+1$’. Then we have vector field

[TABLE]

with

[TABLE]

[TABLE]

Let $0<r_{+}<r$ and

[TABLE]

where $c$ is some suitable (possible different) constant independent of the iterative steps. Then our goal is to find a set $\mathcal{O}_{+}\subset\mathcal{O}$ and an $S-$invariant transformation $\Phi:D_{\rho}(r_{+},s_{+})\times\mathcal{O}_{+}\rightarrow D_{\rho}(r,s)\times\mathcal{O}$ such that it transforms above $X$ in (5.1) into

[TABLE]

with

[TABLE]

and

[TABLE]

5.1 Solving the homological equations

In the sequel, for $K>0,$ we define the truncation operator $\mathcal{T}_{K}$ as follows: for $f$ on $D(r)=\{(\theta,\varphi)\in\mathbb{C}^{n}\times\mathbb{C}^{m}:|\hbox{Im}\theta|<r,|\hbox{Im}\varphi|<r\},$

[TABLE]

The average of $f$ with respect to $(\theta,\varphi)$ is defined as

[TABLE]

We write the reversible vector field $P$ as Taylor-Fourier series

[TABLE]

Let $R=\sum\limits_{\textsf{v}\in\mathscr{V}}R^{(\textsf{v})}(y;\zeta)\frac{\partial}{\partial\textsf{v}}$ be the truncation of $P$, i.e., for $\textsf{v}\in\{\theta_{b},\varphi_{b}\}$, $R^{(\textsf{v})}=\mathcal{T}_{K}P^{(\textsf{v})}_{000,000}(\theta,\varphi)$ and for $\textsf{v}\in\{I_{b},J_{b},z_{j},\bar{z}_{j},w_{j},\bar{w}_{j}\}$,

[TABLE]

We rewrite $R^{(\textsf{v})}$ as follows: for $\textsf{v}\in\{\theta_{b},\varphi_{b}\}$, $R^{(\textsf{v})}=R^{\textsf{v}}(\theta,\varphi)$ and for $\textsf{v}\in\{I_{b},J_{b},z_{j},\bar{z}_{j},w_{j},\bar{w}_{j}\}$,

[TABLE]

We define the normal form of $R$ as

[TABLE]

In the sequel, denote by $\phi^{t}_{X}$ the flow generated by vector field $X$ and $\phi^{1}_{X}=\phi^{t}_{X}|_{t=1}.$

Suppose vector field $F$ has the same form as $R,$ $\Phi^{*}X=(\phi^{1}_{F})^{*}X,$

[TABLE]

We solve the homological equation

[TABLE]

Denote $\partial_{(\omega,\tilde{\omega})}f(\theta,\varphi):=\partial_{\omega}f(\theta,\varphi)+\partial_{\tilde{\omega}}f(\theta,\varphi).$ By the definition of Lie bracket, the homological equation (5.2) is equivalent to the following scalar forms $\eqref{sheq1}-\eqref{sheq4}:$

[TABLE]

where

[TABLE]

$u\in\{\theta_{a},\varphi_{a}\};$ $v\in\{I_{a},J_{a},I_{a}I_{b},I_{a}J_{b},J_{a}I_{b},J_{a}J_{b}\}.$

[TABLE]

where

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

here $\delta_{\mu\nu}=1,\hbox{if}\ \mu=\nu,\hbox{and}\ 0,\hbox{otherwise}.$

Suppose for $\zeta\in\mathcal{O}$, $|k|+|\tilde{k}|\leq K,$

[TABLE]

Lemma 5.1**.**

Suppose that uniformly on $\mathcal{O},$ non-resonance conditions in (5.7) hold, then there exist a positive $c=c(n,m,\tau)$ such that the equation (5.2) has a unique solution $F$ with $[F]=0,$ which is regular on $D(r,s)\times\mathcal{O}.$ Moreover,

1. (1)

$\|F\|_{s;D_{\rho}(r,s)\times\mathcal{O}}\leq c\gamma^{-5}K^{5\tau+19}\varepsilon;$** 2. (2)

$F\circ S=DS\cdot F;$** 3. (3)

$[F,\mathbb{M}_{l}]=0,\,\,l=1,\cdots,d;$** 4. (4)

$F$* satisfies (A6) with $\varepsilon^{\frac{2}{3}}$ in place of $\varepsilon$ on $D(r-\delta,s/2)$, where $0<\delta<r/2.$*

Proof.

(1) As we have mentioned above, the equation (5.2) is equivalent to $\eqref{sheq1}-\eqref{sheq4}.$ Below we only consider the most difficult equation (5.6) with $\varrho=\sigma,i\neq j$ since the other ones can be solved similarly.

By Fourier expansion, we have

[TABLE]

where

[TABLE]

Using non-resonance conditions in (5.7), we obtain for $|k|+|\tilde{k}|\leq K,$

[TABLE]

[TABLE]

then according to the definition of vector fields, we have

[TABLE]

(2) $F\circ S=DS\cdot F$ can be implied by the uniqueness of solutions of homological equation.

(3) We verify $[F,\mathbb{M}_{l}]=0.$ Consider for $l=1,\cdots,d,$

[TABLE]

where

[TABLE]

As in the proof of Lemma (4.2), one can verify that a vector field $X$ satisfying $[X,\mathbb{M}_{l}]=0$ is equivalent to $X^{(\textsf{v})}_{kl\alpha\beta,\tilde{k}\tilde{l}\tilde{\alpha}\tilde{\beta}}=0,$ if $\pi_{l}(k\alpha\beta;\tilde{k}\tilde{\alpha}\tilde{\beta};\textsf{v})\neq 0.$ Thus in order to prove $[F,\mathbb{M}_{l}]=0,$ it suffices to verify that $F^{(\textsf{v})}_{kl\alpha\beta,\tilde{k}\tilde{l}\tilde{\alpha}\tilde{\beta}}=0,$ if $\pi_{l}(k\alpha\beta;\tilde{k}\tilde{\alpha}\tilde{\beta};\textsf{v})\neq 0.$ This can be implied by $[P,\mathbb{M}]=0$ since $F$ is determined by $R.$

(4) The proof can follow that of Lemma 4.3 in [8] since there is no essential difference.

∎

5.2 Estimates on the coordinate transformation

Lemma 5.2**.**

If $\varepsilon\ll\delta\gamma^{5}K^{-5\tau-19},$ then for every $-1\leq t\leq 1,$

[TABLE]

and

[TABLE]

[TABLE]

Proof.

Using Cauchy’s inequality, we obtain

[TABLE]

then if $\varepsilon\ll\delta\gamma^{5}K^{-5\tau-19},$ for every $-1\leq t\leq 1,$

[TABLE]

is well-defined. Thus by Gronwall’s inequality and the estimate for $DF$, we have

[TABLE]

and

[TABLE]

∎

5.3 The new normal form

Through the time-1 map $\Phi=\phi^{1}_{F}$ defined above, vector field $X$ is transformed into $X_{+}=\Phi^{*}X=N_{+}+\mathcal{A}_{+}+P_{+}$ with new normal form $N_{+},\mathcal{A}_{+}$ and new perturbation $P_{+}.$

In this subsection, we consider the new normal form

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

It follows form (5.9) that $\|\hat{N}\|_{s;D_{\rho}(r,s)\times\mathcal{O}}\leq\|R\|_{s;D_{\rho}(r,s)\times\mathcal{O}},$ then $|[R^{z^{\sigma}_{i}z^{\sigma}_{i}}]|_{\mathcal{O}}\leq\|R\|_{s;D_{\rho}(r,s)\times\mathcal{O}}\leq\varepsilon,$ thus

[TABLE]

Similarly,

[TABLE]

5.4 The new perturbation

The new perturbation

[TABLE]

with $R(t)=(1-t)[R]+tR.$

Let $\eta=\varepsilon^{\frac{1}{3}}.$ We now give the estimate of $\|P_{+}\|_{\eta s;D(r-2\delta,\eta s/4)\times\mathcal{O}}.$

[TABLE]

Consider the estimate for $\|(\phi^{1}_{F})^{*}(P-R)\|_{\eta s;D(r-2\delta,\eta s/4)\times\mathcal{O}}.$ Rewrite $P-R$ as

[TABLE]

where

[TABLE]

[TABLE]

Then

[TABLE]

[TABLE]

This implies that

[TABLE]

Therefore,

[TABLE]

The following lemma ensures that the new perturbation $P_{+}$ satisfies reversibility and momentum conservation condition.

Lemma 5.3**.**

1. (1)

$P_{+}$* is *$S-$reversible; 2. (2)

$[P_{+},\mathbb{M}_{l}]=0,\,l=1,\cdots,d;$**

Proof.

1. (1)

We conclude from (2) in Lemma 5.1 that $\Phi\circ S=S\circ\Phi,$ which implies $X_{+}\circ S=-DS\cdot X_{+}.$ It is obvious that $N_{+}\circ S=-DS\cdot N_{+}$, thus $P_{+}\circ S=-DS\cdot P_{+}.$ 2. (2)

We know that $[N,\mathbb{M}_{l}]=0$ and $[P,\mathbb{M}_{l}]=0.$ From Lemma 5.1, we get $[F,\mathbb{M}_{l}]=0$. This together with

[TABLE]

implies that $[P_{+},\mathbb{M}_{l}]=0$.

∎

Lemma 5.4**.**

Suppose $P$ satisfies (A6), $F$ satisfies (A6) with $\varepsilon^{\frac{2}{3}}$ in place of $\varepsilon.$ Moreover, for $\sigma=\pm,$ $|j|>K,$

[TABLE]

and for $|i\mp j|>K,$

[TABLE]

then $[P,F]$ also satisfies (A6) with $\varepsilon_{+}$ in place of $\varepsilon.$

Proof.

By the definition of Lie bracket, the $z_{i}$-component of $[P,F]$ is

[TABLE]

where $\mathscr{V}=\{\theta_{a},\varphi_{b},\,z_{i},w_{j},\bar{z}_{i},\bar{w}_{j}:a=1,\cdots,n;b=1,\cdots,m;i\in\mathbb{Z}^{d}_{1};j\in\mathbb{Z}^{d}_{2}\}.$

To verify $[P,F]$ satisfies (A6), we only consider $\frac{\partial}{\partial z_{j}}[P,F]^{(z_{i})}$ and the derivatives with respect to the other components are similarly analyzed.

In the following, it suffices to consider $\sum\limits_{h}\frac{\partial^{2}P^{(z_{i})}}{\partial z_{h}\partial z_{j}}F^{(z_{h})}$ and $\sum\limits_{h}\frac{\partial P^{(z_{i})}}{\partial z_{h}}\frac{\partial F^{(z_{h})}}{\partial z_{j}}$ in $\frac{\partial}{\partial z_{j}}[P,F]^{(z_{i})}$ since the other terms can be similarly studied.

Let $p^{zz}_{ij}=\lim\limits_{t\rightarrow\infty}\frac{\partial P^{(z_{i+tc})}}{\partial z_{j+tc}},\,\,f^{zz}_{ij}=\lim\limits_{t\rightarrow\infty}\frac{\partial F^{(z_{i+tc})}}{\partial z_{j+tc}}.$ Then

[TABLE]

[TABLE]

Note that $h$ is bounded by $cK^{d}$ in the above inequality since $|i-h|\leq K$ and $|j-h|\leq K.$

∎

By (5.10) and Lemma 5.4 above, we obtain:

Lemma 5.5**.**

$P_{+}$* satisfies (A6) with $K_{+},\varepsilon_{+},\rho_{+}$ in place of $K,\varepsilon,\rho$.*

5.5 Iteration and Convergence

For given $r>0,s>0,L>0$ and $0<\gamma,\varepsilon<1.$ $c$ is a positive constant depending only on $n,m,\tau.$ For $\nu\geq 0,$ we define the iterative sequences as follows:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

5.5.1 Iteration lemma

According to the preceding analysis, we obtain the following iterative lemma.

Lemma 5.6** (Iterative Lemma).**

Let $\varepsilon$ be small enough and $\nu\geq 0.$ Suppose that

1. (1)

The normal form

[TABLE]

$(\zeta\in\mathcal{O}_{\nu})$* satisfies (5.7) with $\omega_{\nu},\tilde{\omega}_{\nu},\Omega_{\nu},\tilde{\Omega}_{\nu},A_{\nu},\tilde{A}_{\nu}$ and $K_{\nu};$* 2. (2)

$\omega_{\nu},\tilde{\omega}_{\nu},\Omega_{\nu,j},\tilde{\Omega}_{\nu,j}$* are $C^{4}_{W}$ smooth in $\zeta$ and satisfy*

[TABLE] 3. (3)

$N_{\nu}+\mathcal{A}_{\nu}+P_{\nu}$* satisfies $(A5)$ and $(A6)$ with $K_{\nu},\varepsilon_{\nu},\rho_{\nu}$ and*

[TABLE]

Then there exists a real analytic, $S-$invariant transformation

[TABLE]

satisfying

[TABLE]

[TABLE]

and a closed subset

[TABLE]

where $\mathcal{R}^{\nu+1}_{k\tilde{k}}(\gamma)$ is defined in (5.17), such that $X_{\nu+1}=(\Phi_{\nu})^{*}X_{\nu}=N_{\nu+1}+\mathcal{A}_{\nu+1}+P_{\nu+1}$ satisfies the same assumptions as $X_{\nu}$ with ‘$\nu+1$’ in place of ‘$\nu$’.

5.5.2 Convergence

We now finish the proof of Theorem 3.1. Let

[TABLE]

be initial $S-$reversible vector field and satisfies the assumptions of Theorem 3.1. Recall that $\varepsilon_{0}=\varepsilon,$ $r_{0}=r,$ $s_{0}=s,$ $\rho_{0}=\rho,$ $L_{0}=L.$ Suppose $\mathcal{O}$ is a compact set of positive Lebesgue measure and all the conditions in the iterative lemma with $\nu=0$ hold. Then we inductively obtain the following sequences

[TABLE]

[TABLE]

[TABLE]

Let $\widetilde{\mathcal{O}}=\cap^{\infty}_{\nu=0}\mathcal{O}_{\nu}.$ Using (5.13), (5.14) and following from [14], we obtain that $N_{\nu}+A_{\nu},\Psi^{\nu},D\Psi^{\nu}$ converge uniformly on $D_{\frac{\rho}{2}}(\frac{r}{2},0)\times\widetilde{\mathcal{O}}$ with

[TABLE]

By the choice of $\varepsilon_{\nu}$ and $K_{\nu},$ we have $\varepsilon_{\nu+1}=O(\varepsilon^{\frac{7}{6}}_{\nu}),$ thus $\varepsilon_{\nu}\rightarrow 0,\nu\rightarrow\infty.$ And we also have $\sum^{\infty}_{\nu=0}\varepsilon_{\nu}\leq 2\varepsilon.$ Consider the flow $\phi^{t}_{X}$ of $X.$ It follows from $X_{\nu+1}=(\Psi^{\nu})^{*}X$ that

[TABLE]

Thanks to the uniform converge of $X_{\nu},\Psi^{\nu}\ \hbox{and}\ D\Psi^{\nu},$ we can take the limits on both sides of (5.16). Therefore, on $D_{\frac{\rho}{2}}(\frac{r}{2},0)\times\widetilde{\mathcal{O}}$, we have

[TABLE]

and

[TABLE]

It follows that for each $\zeta\in\widetilde{\mathcal{O}},$ $\Psi^{\infty}(\mathbb{T}^{n+m}\times\{\zeta\})$ is and embedded torus which is invariant for the original perturbed reversible system at $\zeta\in\widetilde{\mathcal{O}}.$

5.6 Measure Estimate

Let $\mathcal{O}_{-1}=\mathcal{O},$ $K_{-1}=0.$ At the $\nu$th step of KAM iteration, the following resonant set $\mathcal{R}^{\nu}\subset\mathcal{O}_{\nu-1}$ need to be excluded.

[TABLE]

with

[TABLE]

where

[TABLE]

[TABLE]

$i\in\mathbb{Z}^{d}_{1}.$

[TABLE]

$i\in\mathbb{Z}^{d}_{2}.$

[TABLE]

$i,j\in\mathbb{Z}^{d}_{1}\setminus\mathbb{Z}^{d}_{2}.$

[TABLE]

$i\in\mathbb{Z}^{d}_{1}\setminus\mathbb{Z}^{d}_{2},j\in\mathbb{Z}^{d}_{2}\setminus\mathbb{Z}^{d}_{1}.$

[TABLE]

$i,j\in\mathbb{Z}^{d}_{2}\setminus\mathbb{Z}^{d}_{1}.$

[TABLE]

$i\in\mathbb{Z}^{d}_{1}\cap\mathbb{Z}^{d}_{2}.$

[TABLE]

$(i,j)\ \hbox{or}\ (i,j)\in(\mathbb{Z}^{d}_{1}\cap\mathbb{Z}^{d}_{2})\times(\mathbb{Z}^{d}_{1}\setminus\mathbb{Z}^{d}_{2}).$

[TABLE]

$(i,j)\ \hbox{or}\ (i,j)\in(\mathbb{Z}^{d}_{1}\cap\mathbb{Z}^{d}_{2})\times(\mathbb{Z}^{d}_{2}\setminus\mathbb{Z}^{d}_{1}).$

[TABLE]

Note that $\mathcal{R}^{11,-\nu}_{k\tilde{k},ij},$ $\mathcal{R}^{22,-\nu}_{k\tilde{k},ij}$ and $\mathcal{R}^{34,-\nu}_{k\tilde{k},ij}$ are the most complicated three case, and the former two have been studied in [11], thus it suffices to consider the last case .

Denote $D_{\nu}=(\langle k,\omega_{\nu}\rangle+\langle\tilde{k},\tilde{\omega}_{\nu}\rangle)I_{4}+M_{\nu,i}\otimes I_{2}-I_{2}\otimes M^{T}_{\nu,j}.$

Lemma 5.7**.**

For any given $i,j\in\mathbb{Z}^{d}_{1}\cap\mathbb{Z}^{d}_{2}$ with $|i-j|\leq K_{\nu},$ either $|\det(D_{\nu})|\geq 1$ or there are $i_{0},j_{0},c_{1},c_{2},\cdots,c_{d-1}\in\mathbb{Z}^{d}$ with $|i_{0}|,|j_{0}|,|c_{1}|,|c_{2}|,\cdots,|c_{d-1}|\leq 3K^{2}_{\nu}$ and $t_{1},t_{2},\cdots,t_{d-1}\in\mathbb{Z}$ such that $i=i_{0}+t_{1}c_{1}+t_{2}c_{2}+\cdots+t_{d-1}c_{d-1},j=j_{0}+t_{1}c_{1}+t_{2}c_{2}+\cdots+t_{d-1}c_{d-1}.$

Proof.

See [11]. ∎

In the following, for the convenience of notation, let $\textbf{t}:=(t_{1},t_{2},\cdots,t_{d-1}),$ $\textbf{c}:=(c_{1},c_{2},\cdots,c_{d-1})$ and $\textbf{t}\cdot\textbf{c}:=t_{1}c_{1}+t_{2}c_{2}+\cdots+t_{d-1}c_{d-1}.$

By Lemma 5.7, we have

Lemma 5.8**.**

[TABLE]

where $|i_{0}|,|j_{0}|,|c_{1}|,|c_{2}|,\cdots,|c_{d-1}|\leq 3K^{2}_{\nu}$.

Lemma 5.9**.**

Let $\tau>\frac{4(d-1)(d+1)!}{(d-1)!(d+1)-1}.$ Then for fixed $k,\tilde{k},i_{0},j_{0},c_{1},c_{2},\cdots,c_{d-1}$,

[TABLE]

Proof.

Without loss of generality, we assume $|t_{1}|\leq|t_{2}|\leq\cdots\leq|t_{d-1}|.$ Let $\Omega_{\nu,j}=|j|^{2}+\Omega^{0}_{\nu,j},\,\,\tilde{\Omega}_{\nu,j}=|j|^{2}+\tilde{\Omega}^{0}_{\nu,j}$ and $D_{\nu}(\textbf{t})=(\langle k,\omega_{\nu}\rangle+\langle\tilde{k},\tilde{\omega}_{\nu}\rangle)I_{4}+M_{\nu,i_{0}+\textbf{t}\cdot\textbf{c}}\otimes I_{2}-I_{2}\otimes M^{T}_{\nu,j_{0}+\textbf{t}\cdot\textbf{c}}.$

Using Töplitz-Lipschitz property of $\mathcal{A}_{\nu}+P_{\nu},$ we have for $l=i_{0},j_{0},1\leq j\leq d-1,$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

then we have

[TABLE]

Consider the resonant set

[TABLE]

For fixed $k,\tilde{k},i_{0},j_{0},\textbf{c},$ its Lebesgue measure

[TABLE]

and for $\zeta\in\mathcal{O}_{\nu-1}\setminus\mathcal{R}^{34,-\nu}_{k\tilde{k},i_{0}j_{0}\textbf{c}\infty^{d-1}},$

[TABLE]

Below we consider the following $d$ cases.

Case (1): $|t_{1}|>K_{\nu}^{\frac{\tau}{d!}+4}$. For $\zeta\in\mathcal{O}_{\nu-1}\setminus\mathcal{R}^{34,-\nu}_{k\tilde{k},i_{0}j_{0}\textbf{c}\infty^{d-1}},$ we have

[TABLE]

In general, for $2\leq l\leq d-1,$ consider

Case (l): $|t_{1}|\leq K_{\nu}^{\frac{\tau}{d!}+4},|t_{2}|\leq K_{\nu}^{\frac{2\tau}{d!}+4},\cdots,|t_{l-1}|\leq K_{\nu}^{\frac{(l-1)!\tau}{d!}+4},|t_{l}|>K_{\nu}^{\frac{l!\tau}{d!}+4}$. We define the resonant set

[TABLE]

Then for fixed $k,\tilde{k},i_{0},j_{0},\textbf{c},t_{1},t_{2},\cdots,t_{l-1},$ its Lebesgue measure

[TABLE]

and

[TABLE]

Thus for $|t_{1}|\leq K_{\nu}^{\frac{\tau}{d!}+4},|t_{2}|\leq K_{\nu}^{\frac{2\tau}{d!}+4},\cdots,|t_{l-1}|\leq K_{\nu}^{\frac{(l-1)!\tau}{d!}+4},|t_{l}|>K_{\nu}^{\frac{l!\tau}{d!}+4}$, $\zeta\in\mathcal{O}_{\nu-1}\setminus\mathcal{R}^{34,-\nu}_{k\tilde{k},i_{0}j_{0}\textbf{c}t_{1}\cdots t_{l-1}\infty^{d-l}},$

[TABLE]

At last, for

Case (d): $|t_{1}|\leq K_{\nu}^{\frac{\tau}{d!}+4},|t_{2}|\leq K_{\nu}^{\frac{2\tau}{d!}+4},\cdots,|t_{d-1}|\leq K_{\nu}^{\frac{(d-1)!\tau}{d!}+4}$. We define the resonant set

[TABLE]

For fixed $k,\tilde{k},i_{0},j_{0},\textbf{c},t_{1},t_{2},\cdots,t_{d-1},$ its Lebesgue measure

[TABLE]

and

[TABLE]

Therefore, if $\tau>\frac{4(d-1)(d+1)!}{(d-1)!(d+1)-1},$ we obtain

[TABLE]

∎

According to the above analysis , we obtain the following lemma.

Lemma 5.10**.**

Let $\tau>d!(2d(d+1)+n+m+1)+\frac{4(d-1)(d+1)!}{(d-1)!(d+1)-1}.$ Then the total measure of resonant set should be excluded during the KAM iteration is

[TABLE]

6 Acknowledgments

The research was supported by National Natural Science Foundation of China (Grant No. 11271180).

7 Appendix

Suppose vector field $X(\theta,I,z,\bar{z})$ is defined on $D_{\rho}(r,s)=\{y=(\theta,I,z,\bar{z}):|\textmd{Im}\theta|<r,|I|<s,\|z\|_{\rho}<s,\|\bar{z}\|_{\rho}<s\}.$

Definition 7.1** (Reversible vector field ).**

Suppose $S$ is an involution map: $S^{2}=id.$ Vector field $X$ is called reversible with respect to $S$ (or $S-$reversible), if

[TABLE]

i.e.,

[TABLE]

where $DS$ is the tangent map of $S.$

Definition 7.2**.**

Suppose $S$ is an involution map: $S^{2}=id.$ Vector field $X$ is called invariant with respect to $S$ (or $S-$invariant), if

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Definition 7.3**.**

A transformation $\Phi$ is called invariant with respect to above involution $S$ (or $S-$invariant), if $\Phi\circ S=S\circ\Phi.$

Lemma 7.1**.**

1. (1)

*If $X$ and $Y$ are both *$S-$*reversible (or *$S-$*invariant), then $[X,Y]$ is *$S-$invariant. 2. (2)

*If $X$ is *$S-$*reversible, $Y$ is *$S-$*invariant and the transformation $\Phi$ is *$S-$*invariant, then $[X,Y]$ and $\Phi^{*}{X}$ are both *$S-$*reversible. In particular, the flow $\phi_{Y}^{t}$ of $Y$ are *$S-$*invariant, thus $(\phi_{Y}^{t})^{*}{X}$ is *$S-$reversible.

Lemma 7.2** (Cauchy’s inequality, [11]).**

Let $0<\delta<r.$ For an analytic function $f(\theta,I,z,\bar{z})$ on $D_{\rho}(r,s),$

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and

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