A KAM theorem for 2-dimensional nonlinear Schrödinger equations with forcing terms and (2p+1)-nonlinearities

Abstract

In this paper, we prove an infinite dimensional KAM theorem and apply it to study 2-dimensional nonlinear Schrödinger equations with different large forcing terms and (2p + 1)-nonlinearities

　　　　　 \mathrm{i}{u}_t-\mathrm{\Delta }u+{\phi }_1\left({\overline{\omega }}_{1}t\right)u+{\phi }_2\left({\overline{\omega }}_{2}t\right)|u{|}^{2p}u=0,t\in \mathbb{R},x\in {\mathbb{T}}^2

under periodic boundary conditions. As a result, the existence of a Whitney smooth family of small-amplitude reducible quasi-periodic solutions is obtained.

Keywords

Schrödinger equation / reducible KAM tori / small divisor / quasi-periodic solution

1 Introduction

In the past few decades, significant progress has been made in the study of the KAM theory for nonlinear Hamiltonian partial differential equations. Researchers have been focusing on the existence of quasi-periodic solutions of a linear or integrable equation under Hamiltonian perturbations. Kuksin [17] and Wayne [24] were among the first to make breakthroughs in this regard, studying periodic or quasi-periodic solutions of one-dimensional semi-linear wave equations and Schrödinger equations with Dirichlet boundary conditions. Bourgain [2] provided the existence of periodic solutions of one-dimensional semi-linear wave equations and Schrödinger equations with periodic boundary conditions. Bourgain [3-8] improved the Newton iteration technique introduced by Craig and Wayne [10] by developing it into the CWB method. These papers primarily used Lyapunov-Schmidt bifurcation analysis and Nash-Moser implicit function iteration process. Compared with Dirichlet boundary conditions, the normal frequency under periodic boundary conditions is no longer single, and the CWB method overcomes the difficulty caused by multiple normal frequencies by solving a cohomological equation with variable coefficients (related to angular variables), bypassing the second Melnikov condition (i.e., this condition is not necessary). However, this method only provides an irreducible canonical form near the torus, so it cannot present the stability information for quasi-periodic solutions.

Later, Chierchia and You [9] provided a linearly stable reducible KAM torus for the one-dimensional semi-linear wave equation under periodic boundary conditions using another approach based on classical KAM theory (see [1, 11, 12, 16, 18, 19]). In simple terms, these papers all require establishing an infinite iterative process, constructing a symplectic transformation at each iteration step to transform the previous Hamiltonian function into a “better” canonical form with a smaller perturbation. This involves solving a constant-coefficient cohomological equation (system) at each step while ensuring that the process does not encounter the “small divisor” phenomenon (which is inherent to KAM theory). Therefore, it is necessary to assume the second Melnikov non-resonance condition, which inevitably requires a measure digging operation on the parameter set. Ultimately, convergence of the iterations must be ensured, and the measure of the dug parameter set must be small. The canonical form analysis can provide different frequencies and amplitudes for the desired quasi-periodic solutions, thus ensuring that the remaining parameter set is an almost full measure set and also providing linear stability forthe quasi-periodic solutions of the equation. This is an advantage over the CWB method (see [26] for KAM theory on finite and infinite-dimensional spaces).

For the case where the spatial dimension equals 1, a very ideal framework has been established. However, for cases where the spatial dimension is greater than or equal to 2, the research results are far from ideal, primarily because as n\to \mathrm{\infty }, the multiplicity of normal frequencies also tends to infinity, leading to significant difficulties. Among them, Bourgain [3] pioneered the work by using Lyapunov-Schmidt and other techniques and pre-resolved identities to control the inverse of an infinite-dimensional matrix with small eigenvalues, while also demonstrating that the inverse matrix has exponential decay away from the diagonal. This proved the existence of small-amplitude quasi-periodic solutions for the 2-dimensional nonlinear Schrödinger equation. Eliasson and Kuksin [12] presented another landmark achievement, where the authors used a modified KAM method to construct small-amplitude linearly stable quasi-periodic solutions for more interesting high-dimensional nonlinear Schrödinger equations [12]. The most significant contribution was the establishment of the Töplitz-Lipschitz property, proving that the neighborhood of infinity \mathrm{\infty } can be covered by a finite number of Lipschitz domains. This pioneering work provided a feasible method for measure estimation in high-dimensional scenarios (the generalized quasi-Töplitz function is referenced in [22]). Here, it is also mentioned that Geng et al. [13] extended Bourgain [3] results on two-dimensional invariant tori, proving the existence of invariant tori of any finite dimension and obtaining a better canonical form [13]. Essentially, by utilizing the Töplitz-Lipschitz property of Eliasson and Kuksin, they employed a more understandable repeated limit form for critical measure estimation. The introduction of the concept of “admissible set” in [13] aimed to simplify the form of the canonical form, ensuring that the system of cohomological equations is at most fourth-order, thus facilitating the solution of the cohomological equation system. The equations in [3, 12] contained Fourier multipliers or convolution potentials as external parameters. These parameterized equations are more convenient for avoiding the occurrence of resonance phenomena through parameter selection. On the other hand, [13] presented a completely resonant model without external parameters. Geng et al. [13] adopted Bourgain’s idea and defined the admissible set of cubic Schrödinger equations by selecting an appropriate set of tangent points and only exciting the Fourier indices of these tangent points. They also provided specific construction methods. Additionally, it was mentioned that the results of Procesi [20, 21] were extended to cubic Schrödinger equations with translational invariance of any finite dimension, and Wang [23] proved the existence of small-amplitude quasi-periodic solutions for energy-supercritical d-dimensional nonlinear Schrödinger equations. To some extent, the general properties of tangent points in [20, 21] and the assumption conditions on tangent points in [23] are parallel.

This paper will generalize the form of the admissible set to deal with the more general case of nonlinear terms in the Schrödinger equation. It should be noted that the form of the admissible set generalized in this paper is to some extent consistent with [20, 21].

Reference [14] has already proven the existence of quasi-periodic solutions for the 2D Schrödinger equation with a large forcing term and nonlinear terms up to cubic order:

\mathrm{i}{u}_t-\mathrm{\Delta }u+\phi \left(\overline{\omega }t\right)u+\phi \left(\overline{\omega }t\right)|u{|}^{2}u=0,t\in \mathbb{R},x\in {\mathbb{T}}^2.

The author eliminated the large forcing term (leaving only the mean term) through a step of symplectic transformation before arranging the canonical form, following the concept of admissible set for cubic Schrödinger equations as in [13], while retaining the condition that the characteristic values corresponding to the second type of resonance have imaginary roots, resulting in partially hyperbolic tori. Regarding the extension of the admissible set, Reference [15] has demonstrated the existence of small-amplitude quasiperiodic solutions for the 2D Schrödinger equation with nonlinear terms up to quintic order under periodic boundary conditions:

\mathrm{i}{u}_t-\mathrm{\Delta }u+|u{|}^{4}u=0,t\in \mathbb{R},x\in {\mathbb{T}}^2.

Here, the author extends the admissible set applicable to cubic nonlinear terms to quintic nonlinear terms. In the process of extension, it is found that the canonical form becomes more complex, making it difficult to accurately verify the existence of the admissible set. It can be imagined that if the order of the nonlinear terms is directly extended to a general 2p + 1, it would be almost impossible to verify the relevant properties, which is also the starting point of this paper.

The admissible set in [13] and the generality condition in [20, 21] can only be applied to the case where the spatial dimension is 2. When the spatial dimension is greater than 2, it is difficult to accurately organize the canonical form. Therefore, to develop the KAM theory, this paper addresses the periodic boundary conditions:

u\left(t,x_1+2\pi ,x_2\right)=u\left(t,x_1,x_2+2\pi \right)=u\left(t,x_1,x_2\right).

For the 2D Schrödinger equation with different large forcing terms and nonlinear terms of 2p + 1 order:

(1.1)\mathrm{i}{u}_t-\mathrm{\Delta }u+{\phi }_1\left({\overline{\omega }}_{1}t\right)u+{\phi }_2\left({\overline{\omega }}_{2}t\right)|u{|}^{2p}u=0,t\in \mathbb{R},x\in {\mathbb{T}}^2,

where {\phi }_1\left({\overline{\omega }}_{1}t\right) and {\phi }_2\left({\overline{\omega }}_{2}t\right) are real-analytic in {\overline{\theta }}_1={\overline{\omega }}_{1}t and {\overline{\theta }}_2={\overline{\omega }}_{2}t, respectively, and the forcing frequency \overline{\omega }=({\overline{\omega }}_1,{\overline{\omega }}_2) satisfies the condition

{\Vert ⟨\overline{k},\overline{\omega }⟩\Vert }_{\mathbb{Z}}\triangleq \left|⟨\overline{k},\overline{\omega }⟩+l\right|\ge \frac{\gamma }{|\overline{k}{|}^{\tau }},\overline{k}\ne 0,l\in \mathbb{Z}

for fixed Diophantine vectors.

We emphasize the following points. First, compared to the scenario in [14] with two identical large forcing terms, in this paper, {\phi }_1\left({\overline{\omega }}_{1}t\right) and {\phi }_2\left({\overline{\omega }}_{2}t\right) are both non-negligible and distinct. This will cause some changes in the functions used to construct transformations in this paper compared to the cases presented in previous literature. Second, we have removed the condition in [14] regarding the presence of imaginary roots for eigenvalue corresponding to the second type of resonance. This will lead to a more complex small divisor problem. The small divisor condition, which was originally avoided, needs to be reconsidered and dealt with through a measure digging operation in this paper. Last, it is important to emphasize that the nonlinear term in this paper is of the general form 2p + 1. This is because many models from physics in reality have nonlinear terms of this general form. This inevitably increases the complexity of the first step of organizing the canonical form and makes it more difficult to verify relevant properties. This difficulty arises from the inherent complexity of the Schrödinger equation itself. Organizing the canonical form is the most important part of this paper. Therefore, the main work of this paper is to address the above-mentioned difficulties encountered in proving the required KAM theorem. For other framework-based proofs, due to space limitations, please refer to our previous papers [14, 15].

Express (1.1) as an infinite-dimensional Hamiltonian system:

(1.2)u_t=\mathrm{i}\frac{\mathrm{\partial }H}{\mathrm{\partial }\overline{u}}.

Let

u\left(x\right)=\sum _{n\in {\mathbb{Z}}^2} q_n{\varphi }_n\left(x\right).

Then the Hamiltonian system (1.2) is equivalent to the lattice Hamiltonian equation:

{\dot{q}}_n=\mathrm{i}\left({\lambda }_n{q}_n+{\phi }_1({\overline{\omega }}_{1}t)q_n+\frac{\mathrm{\partial }G}{\mathrm{\partial }{\overline{q}}_n}\right),

where

G\equiv \frac{1}{(p+1)(2\pi {)}^{2p}}{\phi }_2\left({\overline{\omega }}_{2}t\right)\sum _{k_1-k_2+k_3-k_4+\cdots +k_{2p+1}-k_{2p+2}=0} q_{k_1}{\overline{q}}_{k_2}{q}_{k_3}{\overline{q}}_{k_4}\cdots q_{k_{2p+1}}{\overline{q}}_{k_{2p+2}}.

The corresponding Hamiltonian function is

H=\sum _{n\in {\mathbb{Z}}^2} {\lambda }_n|q_n{|}^2+{\phi }_1({\overline{\omega }}_{1}t\left)\right|q_n{|}^2+G=N+G,

where \left\{{\lambda }_n\right\}=|n{|}^2=|n_1{|}^2+|n_2{|}^2, n=(n_1,n_2)\in {\mathbb{Z}}^2 are the eigenvalues of the operator A = −\mathrm{\Delta } under periodic boundary conditions, and the corresponding eigenvectors {\varphi }_n\left(x\right)=\frac{1}{2\pi }{\mathrm{e}}^{\mathrm{i}⟨n,x⟩} form a group of basis.

To facilitate the definition of the admissible set, we introduce two linear mappings and one set as follows:

\pi :{\mathbb{Z}}^b\to \mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\left(S\right),\pi \left(e_j\right)=i_j,

\eta :{\mathbb{Z}}^b\to \mathbb{Z},\eta \left(e_j\right)=1,

where Span(S) is the spanned by the basis \{i_1,\dots ,i_b\}\subset {\mathbb{Z}}^2, and e_1,\dots ,e_{\mathrm{b}} are the canonical basis of the lattice {\mathbb{Z}}^b.

Definition 1.1　Consider the set

X_p:=\left\{l:=\sum _{k=1}^{2p} \pm e_{j_k}=\sum _{j=1}^b l_j{e}_j,l\ne 0,-2e_j,\mathrm{\forall }j,\eta \left(l\right)\in \{0,-2\}\right\},

denoting the elements in X_p satisfying \eta \left(l\right)=0 as X_p^0 and those satisfying \eta \left(l\right)=-2 as X_p^{-2}.

Before presenting the main theorem of this paper, we first provide the properties of the admissible set.

Definition 1.2　A finite set of points S=\{i_1,\dots ,i_b\}\subset {\mathbb{Z}}^2(b>2) is called an admissible set if the following conditions are satisfied.

(1) For any selection of 2p + 1 vectors from S, if there exists a vector w\in {\mathbb{Z}}^2 satisfying:

\{\begin{array}{rl}& i_1-i_2+i_3-i_4+\cdots +i_{2p+1}-w=0,\\ & |i_1{|}^2-|i_2{|}^2+|i_3{|}^2-|i_4{|}^2+\cdots +|i_{2p+1}{|}^2-|w{|}^2=0,\end{array}

then w\in S.

(2) For all n_j\in \mathbb{Z} such that \sum _{j=1}^b n_j=0 and 2\le \sum _{j=1}^b \left|n_j\right|\le 2p+2, it holds that:

\sum _{j=1}^b n_j{i}_j\ne 0.

(3) For any n\in {\mathbb{Z}}^2\setminus S, there exists at most one point \{l,m\}\in {\mathbb{Z}}^{b+2}, where \begin{array}{c}l=\sum _{j=1}^b l_j{e}_j\in X_p\end{array}, m\in {\mathbb{Z}}^2\setminus S, and one of the following conditions is satisfied:

\{\begin{array}{rl}& n-m+\sum _{j=1}^b l_j{i}_j=0,\\ & |n{|}^2-|m{|}^2+\sum _{j=1}^b l_j|i_j{|}^2=0,\end{array}l=\sum _{j=1}^b l_j{e}_j\in X_p^0,

or

\{\begin{array}{rl}& n+m+\sum _{j=1}^b l_j{i}_j=0,\\ & |n{|}^2+|m{|}^2+\sum _{j=1}^b l_j|i_j{|}^2=0,\end{array}l=\sum _{j=1}^b l_j{e}_j\in X_p^{-2}.

The above n and m referred to as the first type of resonance (denoted as n, m \in {\mathcal{L}}_1) and the second type of resonance (denoted as n, m \in {\mathcal{L}}_2), respectively. It is easy to see that n and m uniquely determine each other. Furthermore, for all l=\sum _{j=1}^b l_j{e}_j\in X_p^{-2}, it follows that

2\sum _{j=1}^b l_j|i_j{|}^2+{\left|\sum _{j=1}^b l_j{i}_j\right|}^2\ne 0.

(4) For all n\in {\mathbb{Z}}^2\setminus S, it cannot be both the first type of resonance and the second type of resonance. This is equivalent to the non-existence of l=\sum _{j=1}^b l_j{e}_j\in X_p^0, l{}^{\prime }=\sum _{j=1}^b l_j{}^{\prime }{e}_j\in X_p^{-2}, and m, m{}^{\prime }\in {\mathbb{Z}}^2\setminus S satisfying

\{\begin{array}{rl}& n-m+\sum _{j=1}^b l_j{i}_j=0,\\ & |n{|}^2-|m{|}^2+\sum _{j=1}^b l_j|i_j{|}^2=0,\\ & n+m{}^{\prime }+\sum _{j=1}^b l_j{}^{\prime }{i}_j=0,\\ & |n{|}^2+|m{}^{\prime }{|}^2+\sum _{j=1}^b l_j{}^{\prime }|i_j{|}^2=0.\end{array}

Note 1.1　In the first step of canonical form rearrangement, property (1) ensures that if the nonlinear term of degree 2p + 2 involves contributions from 2p + 1 terms from the tangent point admissible set, then the other index of resonance must be in S. Combined with property (2), it is known that if all 2p + 2 degree terms come from S, then the same index must appear in pairs conjugately, indicating a resonance of the form q_{i_1}{\overline{q}}_{i_1}{q}_{i_2}{\overline{q}}_{i_2}\cdots q_{i_{p+1}}{\overline{q}}_{i_{p+1}}. Properties (3) and (4) describe that the occurrence of resonance can only happen in one specific way at most, while also ensuring that a resonance of the form q_{i_1}{\overline{q}}_{i_2}{q}_{i_3}{\overline{q}}_{i_4}\cdots q_{i_{2p-3}}{\overline{q}}_{i_{2p-2}}{\overline{q}}_{i_{2p-1}}{\overline{q}}_{i_{2p}}{z}_n{z}_n does not occur. By assuming the admissible set, this paper ensures that the first step of canonical form rearrangement is sufficiently concise. For the existence of the admissible set, please refer to [21, Proposition 14].

The main result of this paper is as follows.

Theorem 1.1　 Let S=\{i_1,\dots ,i_b\}\subset {\mathbb{Z}}^2(b>2) be an admissible set. Let {\phi }_2^0={\int }_{{\mathbb{T}}^m}{\phi }_2\left({\overline{\theta }}_2\right)\mathrm{d}{\overline{\theta }}_2\ne 0. Then there exists a Cantor set C of positive measure such that for any \xi =({\xi }_1,\dots ,{\xi }_b)\in \mathcal{C}, the nonlinear Schrödinger equation (1.1) admits a family of small-amplitude analytic quasiperiodic solutions of the following form:

\begin{array}{rl}u(\overline{\omega }t,{\tilde{\omega }}_{1}t,{\tilde{\omega }}_{2}t,\dots ,{\tilde{\omega }}_{b}t,x)& ={\sum }_{j=1}^b\sqrt{{\xi }_j}{\mathrm{e}}^{\mathrm{i}{\tilde{\omega }}_{j}t}{\varphi }_{i_j}+O\left(\right|\xi {|}^{\frac{3}{2}}),{\tilde{\omega }}_j\\ & ={\epsilon }^{-3p}(|i_j{|}^2+{\phi }_1^0)+O\left(\right|\xi {|}^p).\end{array}

The above theorem is a direct corollary of Theorem 4.1. To present this infinite-dimensional KAM theorem, this paper will provide the necessary function spaces and corresponding norms in Section 2, perform the first step of canonical form rearrangement for the given equation in Section 3, present an infinite-dimensional KAM theorem suitable for the given equation in Section 4, and demonstrate the proof framework for the infinite-dimensional KAM theorem in the following two sections.

2 Function space

First, let’s denote the set {i_1,\dots ,i_{\mathrm{b}}} as b given vectors in {\mathbb{Z}}^2. Let {\mathbb{Z}}_1^2={\mathbb{Z}}^2\setminus \{i_1,\dots ,i_b\}, z=(\dots ,z_n,\dots {)}_{n\in {\mathbb{Z}}_1^2}, and the complex conjugate \overline{z}=(\dots ,{\overline{z}}_n,\dots {)}_{n\in {\mathbb{Z}}_1^2}.

Define the weighted norm as

{\Vert z\Vert }_{\rho }=\sum _{n\in {\mathbb{Z}}_1^2} \left|z_n\right|{\mathrm{e}}^{\left|n\right|\rho },

where \left|n\right|=\sqrt{n_1^2+n_2^2}, n=(n₁, n₂), and \rho >0.

Define a neighborhood of {\mathbb{T}}^{b+m}\times \{I=0\}\times \{z=0\}\times \{\overline{z}=0\} as

Here, |·| denotes the supremum norm of complex vectors, and \mathcal{O} represents a positive measure parameter set in {\mathbb{R}}^b. Let \alpha \equiv (\dots ,{\alpha }_n,\dots {)}_{n\in {\mathbb{Z}}_1^2}, \beta \equiv (\dots ,{\beta }_n,\dots {)}_{n\in {\mathbb{Z}}_1^2}, where α_n, {\beta }_n\in \mathbb{N}, and α and β are vectors containing only finitely many non-negative integer elements. The product z^{\alpha }{\overline{z}}^{\beta } denotes \prod _n z_n^{{\alpha }_n}{\overline{z}}_n^{{\beta }_n}. For any given function

F(\theta ,I,z,\overline{z})=\sum _{\alpha ,\beta ,k\in {\mathbb{Z}}^{b+m},l\in {\mathbb{N}}^b} F_{kl\alpha \beta }\left(\xi \right)I^l{\mathrm{e}}^{\mathrm{i}⟨k,\theta ⟩}{z}^{\alpha }{\overline{z}}^{\beta }

in the Whitney sense, it is C_W^{4p}-smooth with respect to the parameter ξ (later derivatives with respect to ξ are taken in the Whitney sense). Define

{\Vert F\Vert }_{\mathcal{O}}=\sum _{\alpha ,\beta ,k,l} \left(\underset{\xi \in \mathcal{O}}{sup} \sum _{0⩽d⩽4p} \left|{\mathrm{\partial }}_{\xi }^d{F}_{kl\alpha \beta }\right|\right)\left|I^l\right|{\mathrm{e}}^{\left|k\right|\left|\mathrm{I}\mathrm{m}\theta \right|}\left|z^{\alpha }\right|\left|{\overline{z}}^{\beta }\right|,

then the weighted norm of F is given by

(2.1){\Vert F\Vert }_{D_{\rho }(r,s),\mathcal{O}}\equiv \underset{D_{\rho }(r,s)}{sup} {\Vert F\Vert }_{\mathcal{O}}.

For a function F, the norm of the vector field generated by F, X_F=(F_I,-F_{\theta },\{\mathrm{i}{F}_{z_n}{\}}_{n\in {\mathbb{Z}}_1^2},\{-\mathrm{i}{F}_{{\overline{z}}_n}{\}}_{n\in {\mathbb{Z}}_1^2}) is defined as:

\begin{array}{rl}{\Vert X_F\Vert }_{D_{\rho }(r,s),\mathcal{O}}\equiv & {\Vert F_I\Vert }_{D_{\rho }(r,s),\mathcal{O}}+\frac{1}{s^2}{\Vert F_{\theta }\Vert }_{D_{\rho }(r,s),\mathcal{O}}\\ & +\underset{D_{\rho }(r,s)}{sup}\frac{1}{s}\left[\sum _{n\in {\mathbb{Z}}_1^2} {\Vert F_{z_n}\Vert }_{\mathcal{O}}{\mathrm{e}}^{\left|n\right|\rho }+\sum _{n\in {\mathbb{Z}}_1^2} {\Vert F_{{\overline{z}}_n}\Vert }_{\mathcal{O}}{\mathrm{e}}^{\left|n\right|\rho }\right].\end{array}

3 Birkhoff canonical form

In this section, four successive transformations are carried out. The first step involves a symplectic transformation aimed at removing the large forcing terms, leaving only the mean terms (similar to the process in [14]). The second step involves another symplectic transformation to eliminate the non-resonant parts of the nonlinear terms. The extended admissible set here minimizes the non-integrable terms in the canonical form, making the cohomological equation easier to solve. The third step involves a scaling transformation through the action-angle variable transformation, subtly introducing parameters to address the frequency issue of the invariant tori independently from the KAM steps. Although careful selection of tangent directions has been made, due to the complexity of the nonlinear term orders in the equation, the coefficients in the current canonical form are variable. Therefore, a coordinate transformation is still needed to make the coefficients constant (similar to the transformation in [15], but more complex). Thus, the Birkhoff canonical form before the KAM iterative steps is obtained.

Below are the specific transformations involved.

(1) First, introduce the action-angle variables corresponding to the external frequencies:

\begin{array}{rl}(\overline{\theta },J)& =\left(\right({\overline{\theta }}_1,{\overline{\theta }}_2),(J_1,J_2\left)\right)\\ & =\left(\right({\overline{\theta }}_{1_1},\dots ,{\overline{\theta }}_{1_{m1}},{\overline{\theta }}_{2_1},\dots ,{\overline{\theta }}_{2_{m2}}),(J_{1_1},\dots ,J_{1_{m1}},J_{2_1},\dots ,J_{2_{m2}}\left)\right)\in {\mathbb{R}}^m\times {\mathbb{R}}^m,\end{array}

where m=m_1+m_2,{\overline{\theta }}_1={\overline{\omega }}_{1}t,{\overline{\theta }}_2={\overline{\omega }}_{2}t. Then, we have:

\dot{\overline{\theta }}=\frac{\mathrm{\partial }H}{\mathrm{\partial }J}=\overline{\omega },\dot{J}=-\frac{\mathrm{\partial }H}{\mathrm{\partial }\overline{\theta }},{\dot{q}}_n=\mathrm{i}\frac{\mathrm{\partial }H}{\mathrm{\partial }{\overline{q}}_n},{\dot{\overline{q}}}_n=-\mathrm{i}\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}_n},n\in {\mathbb{Z}}^2.

The Hamiltonian function is given by

H=⟨{\overline{\omega }}_1,J_1⟩+⟨{\overline{\omega }}_2,J_2⟩+\sum _{n\in {\mathbb{Z}}^2} ({\lambda }_n+{\phi }_1({\overline{\theta }}_1\left)\right)q_n{\overline{q}}_n+P({\overline{\theta }}_2,q,\overline{q})={\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+P,

where \overline{\omega }=({\overline{\omega }}_1,{\overline{\omega }}_2),J=(J_1,J_2),{\mathrm{\Lambda }}_1\equiv ⟨\overline{\omega },J⟩+\sum _{n\in {\mathbb{Z}}^2} {\lambda }_n{q}_n{\overline{q}}_n,{\mathrm{\Lambda }}_2\equiv \sum _{n\in {\mathbb{Z}}^2} {}_1\left({\overline{\theta }}_1\right)q_n{\overline{q}}_n.

Next, construct the function F₁:

F_1({\overline{\theta }}_1,q,\overline{q})=\sum _{{\overline{k}}_1\in {\mathbb{Z}}^{m_1},{\overline{k}}_1\ne 0,n\in {\mathbb{Z}}^2} \frac{\mathrm{i}{\phi }_1^{{\overline{k}}_1}}{⟨{\overline{k}}_1,{\overline{\omega }}_1⟩}{\overline{q}}_n{\mathrm{e}}^{\mathrm{i}⟨{\overline{k}}_1,{\overline{\theta }}_1⟩},

where the first symplectic transformation gives rise to the Hamiltonian flow mapping X_{F_1}^1 with time equal to 1.

Note 3.1　Similar to the treatment in [14], it can be shown that F₁ is a bounded function, i.e.,

{\Vert F_1\Vert }_{D_{\rho }(r_{+},s_{+}),\mathcal{O}}\le c{\gamma }^{-2}{\left(\frac{2\tau +m+1}{r}\right)}^{2\tau +m+1}{\mathrm{e}}^{-(2\tau +m+1)}.

Hence, the flow mapping X_{F_1}^1 corresponding to the function F₁ is well-defined.

(2) For the given admissible set in this paper, the second symplectic transformation is provided, which also yields a bounded function and a Hamiltonian flow mapping X_{F_2}^1 with time equal to 1. The construction of F₂ is as follows:

\begin{array}{rl}{F}_2=& \sum _{\begin{array}{c}\alpha ,\beta \in ({\mathbb{Z}}^2{)}^N:|\alpha |=|\beta |=p+1\\ \begin{array}{c}\sum _{k\in {\mathbb{Z}}^{m+b}} ({\alpha }_k-{\beta }_k)k=0,\sum _{k\in {\mathbb{Z}}^{m+b}} ({\alpha }_k-{\beta }_k)|k{|}^2\ne 0\\ ♯S\cap {\left\{\frac{\pi \left({\alpha }_k\right)}{\eta \left({\alpha }_k\right)},\frac{\pi \left({\beta }_k\right)}{\eta \left({\beta }_k\right)}\right\}}_{k\in {\mathbb{Z}}^{m+b}}\ge 2p,{\overline{k}}_2\in {\mathbb{Z}}^{m}2\end{array}\end{array}} \\ & \frac{\mathrm{i}{\varphi }_2^{{\overline{k}}_2}}{(p+1)\left(2\pi {)}^{2p}\right(\sum _{k\in {\mathbb{Z}}^{m+b}} ({\alpha }_k-{\beta }_k)\left|k{|}^2\right)}{q}^{\alpha }{\overline{q}}^{\beta }{\mathrm{e}}^{\mathrm{i}(k,\theta )}.\end{array}

Note 3.2　In comparison to the paper [14], since this paper deals with two distinct forcing terms, the numerators of the coefficients in F₁ and F₂ are distinguished based on the different actions of {\phi }_1 and {\phi }_2. It is also emphasized that due to the general 2p+1 order of nonlinearity in this paper, the expression for F₂ is complex. In comparison to [14-15], this paper demonstrates the conciseness achieved by introducing some notational expressions from [21].

(3) Next, introduce action-angle variables in the tangent space:

q_i=\sqrt{I_i+{\xi }_i}{\mathrm{e}}^{\mathrm{i}{\theta }_i},{\overline{q}}_i=\sqrt{I_i+{\xi }_i}{\mathrm{e}}^{-\mathrm{i}{\theta }_i},i\in S,q_n=w_n,{\overline{q}}_n={\overline{w}}_n,n\in {\mathbb{Z}}_1^2

and perform a time-scale transformation:

\xi \to {\epsilon }^3\xi ,J\to {\epsilon }^{5}J,I\to {\epsilon }^{5}I,\overline{\theta }\to \overline{\theta },\tilde{\theta }\to \tilde{\theta },w\to {\epsilon }^{\frac{5}{2}}w,\overline{w}\to {\epsilon }^{\frac{5}{2}}\overline{w}.

(4) Finally, perform a coordinate transformation to convert the canonical form to constant coefficients. The specific form of the transformation \mathrm{\Psi } is given below:

\{\begin{array}{rl}& {\theta }_{+}=\theta ,\\ & I_{+}=I-\sum _{n\in {\mathcal{L}}_1}(w_n{\overline{w}}_n{l}^{+}+w_m{\overline{w}}_m{l}^{-})-\sum _{n\in {\mathcal{L}}_2}(w_n,w_n,l^{+}-w_m,w_m,l^{-}),\\ & \left(\begin{array}{c}{z}_n\\ z_m\end{array}\right)=S\left(\begin{array}{cc}{\mathrm{e}}^{\mathrm{i}}(l^{+},\theta )& 0\\ 0& {\mathrm{e}}^{i(-\theta )}\end{array}\right)\left(\begin{array}{c}{w}_n\\ w_m\end{array}\right),\\ & \left(\begin{array}{c}{\overline{z}}_n\\ {\overline{z}}_m\end{array}\right)=\overline{S}\left(\begin{array}{cc}{\mathrm{e}}^{-\mathrm{i}(l^{+},\theta )}& 0\\ 0& {\mathrm{e}}^{-\mathrm{i}(l^{-},\theta )}\end{array}\right)\left(\begin{array}{c}{\overline{w}}_n\\ {\overline{w}}_m\end{array}\right),n\in {\mathcal{L}}_1,\\ & \left(\begin{array}{c}{z}_n\\ z_m\end{array}\right)=\left(\begin{array}{cc}{\mathrm{e}}^{\mathrm{i}(l^{+},\theta )}& 0\\ 0& {\mathrm{e}}^{-\mathrm{i}(l^{-},\theta )}\end{array}\right)\left(\begin{array}{c}{w}_n\\ w_m\end{array}\right),\\ & \left(\begin{array}{c}{\overline{z}}_n\\ {\overline{z}}_m\end{array}\right)=\left(\begin{array}{cc}{\mathrm{e}}^{-\mathrm{i}(l^{+},\theta )}& 0\\ 0& {\mathrm{e}}^{\mathrm{i}(l^{-},\theta )}\end{array}\right)\left(\begin{array}{c}{\overline{w}}_n\\ {\overline{w}}_m\end{array}\right),n\in {\mathcal{L}}_2,\\ & z_n=w_n,{\overline{z}}_n={\overline{w}}_n,n\in {\mathbb{Z}}_1^2\setminus ({\mathcal{L}}_1\cup {\mathcal{L}}_2).\end{array}

Note 3.3　The extension of the admissible set ensures smooth progress in the first step of canonical form arrangement, allowing for the specific form of the twisted symplectic transformation Ψ. The existence of the matrix S in the transformation can be found in reference [25].

Thus, through four crucial and complex transformations, the first step of arranging the Birkhoff canonical form is completed, preparing for the KAM iteration steps. For the convenience of writing the Hamiltonian function, define \iota ={\epsilon }^{-3p}\left(\right|i_1{|}^2+{\phi }_1^0,|i_2{|}^2+{\phi }_1^0,\dots ,|i_b{|}^2+{\phi }_1^0)|, \xi =({\xi }_{i_1},{\xi }_{i_2},\dots ,{\xi }_{i_b}),\left|l\right|=l^{+}+l^{-},

A_r({\xi }_1,\dots ,{\xi }_b)=\sum _{\sum _j{k}_j=r}{\left(\begin{array}{c}r\\ k_1,\dots ,k_b\end{array}\right)}^2\prod _j{\xi }_j^{k_j},

G=⟨l,{\mathrm{\nabla }}_{\xi }{A}_{p+1}\left(\xi \right){⟩}^2+4\sum _{\begin{array}{c}l=l^{+}-l^{-}\in X_p^0\\ \alpha \in {\mathbb{N}}^b,|l^{+}+\alpha {|}_1=p\end{array}}(p+1{)}^4{\left[\left(\begin{array}{c}p\\ l+\alpha \end{array}\right)\left(\begin{array}{c}p\\ l^{-}+\alpha \end{array}\right)\right]}^2{\xi }^{l+2\alpha }.

Thus, after four steps of complex calculations, the Hamiltonian function becomes

(3.1)H=H_0+P=N+\mathcal{B}+\overline{\mathcal{B}}+P.

Here,

\begin{array}{rl}N=& ⟨{\epsilon }^{-3p}\overline{\omega },J⟩+⟨\tilde{\omega }\left(\xi \right),I⟩+\sum _{n\in {\mathbb{Z}}_1^2\setminus {\mathcal{L}}_2} {\mathrm{\Omega }}_n\left(\xi \right)z_n{\overline{z}}_n\\ & +\sum _{n\in {\mathcal{L}}_2} \left[({\mathrm{\Omega }}_n+⟨l^{+},\tilde{\omega }⟩)z_n{\overline{z}}_n+({\mathrm{\Omega }}_m-⟨l^{-},\tilde{\omega }⟩)z_m{\overline{z}}_m\right],\end{array}

\{\begin{array}{rl}& {\tilde{\omega }}_i\left(\xi \right)={\epsilon }^{-3p}\left(\right|i{|}^2+{\phi }_1^0)+\frac{1}{(p+1)(2\pi {)}^{2p}}{\phi }_2^0\mathrm{\partial }\xi ,A_{p+1}(\xi ),\\ & {\mathrm{\Omega }}_n={\epsilon }^{-3p}\left(\right|n{|}^2+{\phi }_1^0)+\frac{p+1}{(2\pi {)}^{2p}}{\phi }_2^0{A}_p(\xi ),n\in {\mathbb{Z}}_1^2\setminus {\mathcal{L}}_1,\\ & {\mathrm{\Omega }}_n={\epsilon }^{-3p}\left(\right|n{|}^2+{\phi }_1^0)+(l^{+},\iota )+\frac{(p+1){\phi }_2^0}{(2\pi {)}^{2p}}{A}_p(\xi )+\frac{{\phi }_2^0}{2(p+1)(2\pi {)}^{2p}}(|l|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}(\xi )⟩\\ & +\frac{\left|{\phi }_2^0\right|}{2(p+1)(2\pi {)}^{2p}}\sqrt{G},n\in {\mathcal{L}}_1,\\ & {\mathrm{\Omega }}_m={\epsilon }^{-3p}\left(\right|m{|}^2+{\phi }_1^0)+(l^{-},\iota )+\frac{(p+1){\phi }_2^0}{(2\pi {)}^{2p}}{A}_p(\xi )+\frac{{\phi }_2^0}{2(p+1)(2\pi {)}^{2p}}(\left|l\right|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}\left(\xi \right))\\ & -\frac{\left|{\phi }_2^0\right|}{2(p+1)(2\pi {)}^{2p}}\sqrt{G},n\in {\mathcal{L}}_1,\end{array}

and

\mathcal{B}=\sum _{\begin{array}{c}l=l^{+}-l^{-}\in X_p^{-2}\\ \alpha \in {\mathbb{N}}^b,|l^{+}+\alpha {|}_1=p-1\end{array}}\frac{p{\phi }_2^0}{(2\pi {)}^{2p}}\left(\begin{array}{c}p-1\\ l^{+}+\alpha \end{array}\right)\left(\begin{array}{c}p+1\\ l^{-}+\alpha \end{array}\right)\sum _{n\in {\mathcal{L}}_2}{\xi }^{\frac{l}{2}+\alpha }{z}_n{z}_m,

\overline{\mathcal{B}}=\sum _{\begin{array}{c}l=l^{+}-l^{-}\in X_p^{-2}\\ \alpha \in {\mathbb{N}}^b,|l^{+}+\alpha {|}_1=p-1\end{array}}\frac{p{\phi }_2^0}{(2\pi {)}^{2p}}\left(\begin{array}{c}p-1\\ l^{+}+\alpha \end{array}\right)\left(\begin{array}{c}p+1\\ l^{-}+\alpha \end{array}\right)\sum _{n\in {\mathcal{L}}_2} {\xi }^{\frac{l}{2}+\alpha }{\overline{z}}_n{\overline{z}}_m,

where the parameter \xi \in \mathcal{O}, and the symplectic structure of the phase space is \mathrm{d}I\wedge \mathrm{d}\theta +\mathrm{i}\sum _{n\in {\mathbb{Z}}_1^2} \mathrm{d}{z}_n\wedge \mathrm{d}{\overline{z}}_n.

4 Infinite-dimensional KAM theorem

For any \xi \in \mathcal{O}, and \begin{array}{c}{H}_0,(\theta ,0,0,0)\mapsto (\theta +\omega t,0,0,0)\end{array} is a particular solution of the Hamiltonian equations corresponding to the invariant torus of the phase space. Considering the Hamiltonian function with the perturbation P, the goal of this paper is to prove that, in terms of Lebesgue measure, for the vast majority of parameter values \xi \in \mathcal{O}, if {\Vert X_P\Vert }_{D_{\rho }(r,s),\mathcal{O}} is sufficiently small, then the Hamiltonian function H=N+\mathcal{B}+\overline{\mathcal{B}}+P still possesses invariant tori.

Below are the descriptions of the properties that the tangent frequency ω(ξ), the normal frequency {\mathrm{\Omega }}_n\left(\xi \right), and the perturbation P need to satisfy.

(A1) Non-degeneracy condition. Suppose for all \mathrm{\forall }\xi \in \mathcal{O},

\{\begin{array}{rl}& \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\frac{\mathrm{\partial }{\tilde{\omega }}_1}{\mathrm{\partial }\xi },\dots ,\frac{\mathrm{\partial }{\tilde{\omega }}_b}{\mathrm{\partial }\xi }\right)=\kappa ,\\ & \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\frac{{\mathrm{\partial }}^{\left|\beta \right|}\tilde{\omega }}{\mathrm{\partial }{\xi }^{\beta }}|\mathrm{\forall }\beta ,1\le \left|\beta \right|\le b-\kappa +1\right)=b,\end{array}

where κ is an integer given by 1⩽\kappa ⩽b, \frac{\mathrm{\partial }{\tilde{\omega }}_1}{\mathrm{\partial }\xi },\dots ,\frac{\mathrm{\partial }{\tilde{\omega }}_b}{\mathrm{\partial }\xi } are all first-order partial derivatives of the vectors with respect to the parameters \xi. For a fixed β, \begin{array}{c}\frac{{\mathrm{\partial }}^{\left|\beta \right|}\tilde{\omega }}{\mathrm{\partial }{\xi }^{\beta }}=(\frac{{\mathrm{\partial }}^{\left|\beta \right|}{\tilde{\omega }}_1}{\mathrm{\partial }{\xi }^{\beta }},\dots ,\frac{{\mathrm{\partial }}^{\left|\beta \right|}{\tilde{\omega }}_b}{\mathrm{\partial }{\xi }^{\beta }})\end{array}.

(A2) Asymptotic growth of the normal frequency

{\mathrm{\Omega }}_n={\epsilon }^{-a}\left(\right|n{|}^2+{\varphi }_1^0)+{\tilde{\mathrm{\Omega }}}_n,a\ge 0,n\in {\mathbb{Z}}_1^2\setminus {\mathcal{L}}_1,

where {\phi }_1^0={\int }_{{\mathbb{T}}^m} {\phi }_1\left({\overline{\theta }}_1\right)\mathrm{d}{\overline{\theta }}_1, {\tilde{\mathrm{\Omega }}}_n is a C_W^{4p}\left(\mathcal{O}\right)\text{-}smooth function with respect to \xi, and the C_W^{4p}\left(\mathcal{O}\right)\text{-}norm is bounded above by L(L > 0).

(A3) Melnikov non-resonance condition. Let

{\mathcal{M}}_n=\left(\begin{array}{cc}{\mathrm{\Omega }}_n+⟨l^{+},\tilde{\omega }⟩& -C(l,\alpha ){\xi }^{\frac{l}{2}+\alpha }\\ C(l,\alpha ){\xi }^{\frac{l}{2}+\alpha }& -({\mathrm{\Omega }}_m-⟨l^{-},\tilde{\omega }⟩)\end{array}\right),n\in {\mathcal{L}}_2,

where \begin{array}{c}C(l,\alpha )=\sum _{\begin{array}{c}l=l^{+}-l^{-}\in X_p^{-2}\\ \alpha \in {\mathbb{N}}^b,|l^{+}+\alpha {|}_1=p-1\end{array}} \frac{p{\varphi }_2^0}{(2\pi {)}^{2p}}\left(\genfrac{}{}{0pt}{}{p-1}{l^{+}+\alpha }\right)\left(\genfrac{}{}{0pt}{}{p+1}{l^{-}+\alpha }\right)\end{array}. Suppose \tilde{\omega }\left(\xi \right), {\mathcal{M}}_n\left(\xi \right)\in C_W^{4p}\left(\mathcal{O}\right), there exist γ, τ > 0 satisfying (I₂ refers to the 2×2 identity matrix, and I₄ refers to the 4×4 identity matrix).

\begin{array}{rl}& k,\omega ⟩|\ge \frac{\gamma }{|k{|}^{\tau }},k=(\overline{k},\tilde{k}),\overline{k}\in {\mathbb{Z}}^m,\tilde{k}\in {\mathbb{Z}}^b,|k|=|\overline{k}|+|\tilde{k}|>0,\omega =({\epsilon }^{-3p}\overline{\omega },\tilde{\omega }),\\ & |⟨k,\omega ⟩\pm {\mathrm{\Omega }}_n|\ge \frac{\gamma }{|k{|}^{\tau }},n\in {\mathbb{Z}}_1^2\setminus {\mathcal{L}}_2,\\ & |⟨k,\omega ⟩\pm {\mathrm{\Omega }}_n\pm {\mathrm{\Omega }}_m|\ge \frac{\gamma }{|k{|}^{\tau }},n,m\in {\mathbb{Z}}_1^2\setminus {\mathcal{L}}_2,\\ & |⟨k,\omega ⟩\pm {\mathrm{\Omega }}_i\pm {\mu }_j|\ge \frac{\gamma }{|k{|}^{\tau }},k\ne 0,i\in {\mathbb{Z}}_1^2\setminus {\mathcal{L}}_2,j\in \left\{1,2\right\},\\ & ⟨k,\omega ⟩\pm {\mu }_j|\ge \frac{\gamma }{|k{|}^{\tau }},k\ne 0,j\in \{1,2\},\end{array}

where μ₁ and μ₂ are the two eigenvalues of the second-order matrix {\mathcal{M}}_{n{}^{\prime }},

\left|\mathrm{d}\mathrm{e}\mathrm{t}\right(⟨k,\omega ⟩I_4\pm {\mathcal{M}}_n\otimes I_2\pm I_2\otimes {\mathcal{M}}_{n{}^{\prime }}\left)\right|\ge \frac{\gamma }{|k{|}^{\tau }},k\ne 0,n,n{}^{\prime }\in {\mathcal{L}}_2.

Note 4.1　This paper removes the condition in [14] where the eigenvalues of {\mathcal{M}}_n(n\in {\mathcal{L}}_2) have nonzero imaginary parts, hence the non-resonance condition in this paper will be more complex.

(A4) Regularity of \mathcal{B}+\overline{\mathcal{B}}+P. \mathcal{B}+\overline{\mathcal{B}}+P is real-analytic with respect to I, θ, z, \overline{z}, and C_W^{4p}-smooth with respect to ξ, and satisfies

(A5) Properties of zero momentum and partial gauge invariance of the perturbation. \text{B}+\overline{\mathcal{B}}+P has the following properties of zero momentum and partial gauge invariance:

\mathcal{D}=\left\{\mathcal{B}+\overline{\mathcal{B}}+P:\mathcal{B}+\overline{\mathcal{B}}+P=\sum _{k,l,\alpha ,\beta } (\mathcal{B}+\overline{\mathcal{B}}+P{)}_{kl\alpha \beta }\left(\xi \right)I^l{\mathrm{e}}^{\mathrm{i}(\mathrm{k},\theta )}{z}^{\alpha }{\overline{z}}^{\beta }\right\},

where \begin{array}{c}k=(\overline{k},\tilde{k})\in {\mathbb{Z}}^{b+m}\end{array}, \overline{k}\in {\mathbb{Z}}^m, \tilde{k}\in {\mathbb{Z}}^b, l\in {\mathbb{Z}}^b, α, β have the following relationship:

\sum _{j=1}^b {\tilde{k}}_j{i}_j+\sum _{n\in {\mathbb{Z}}_1^d} ({\alpha }_n-{\beta }_n)n=0,

which is referred to as the property of zero momentum

\sum _{j=1}^b {\tilde{k}}_j+\sum _{n\in {\mathbb{Z}}_1^d} ({\alpha }_n-{\beta }_n)=0,

which is referred to as the property of partial gauge invariance.

Note 4.2　Since this paper also involves external frequencies, it is necessary to generalize the properties of zero momentum and gauge invariance, which is still aimed at avoiding terms that are difficult to handle.

(A6) Töplitz-Lipschitz property: For any fixed n, m\in {\mathbb{Z}}^2, c\in {\mathbb{Z}}^2\{0}, the limits

\underset{t\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}} \frac{{\mathrm{\partial }}^2(\mathcal{B}+P)}{\mathrm{\partial }{z}_{n+tc}\mathrm{\partial }{z}_{m-tc}},\underset{t\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}} \frac{{\mathrm{\partial }}^2(\sum _{n\in {\mathbb{Z}}_1^2} {\tilde{\mathrm{\Omega }}}_n{z}_n{\overline{z}}_n+P)}{\mathrm{\partial }{z}_{n+tc}\mathrm{\partial }{\overline{z}}_{m+tc}},\underset{t\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}} \frac{{\mathrm{\partial }}^2(\overline{\mathcal{B}}+P)}{\mathrm{\partial }{\overline{z}}_{n+tc}\mathrm{\partial }{\overline{z}}_{m-tc}}

all exist. In addition, there exists K > 0 such that for |t| > K, \mathcal{B}+\overline{\mathcal{B}}+P satisfies

\begin{array}{rl}& {\Vert \frac{{\mathrm{\partial }}^2(\mathcal{B}+P)}{\mathrm{\partial }{z}_{n+tc}\mathrm{\partial }{z}_{m-tc}}-\underset{t\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{{\mathrm{\partial }}^2(\mathcal{B}+P)}{\mathrm{\partial }{z}_{n+tc}\mathrm{\partial }{z}_{m-tc}}\Vert }_{D_{\rho }(r,s),\mathcal{O}}\le \frac{\epsilon }{\left|t\right|}{\mathrm{e}}^{-|n+m|\rho },\\ & {\Vert \frac{{\mathrm{\partial }}^2(\sum _{n\in {\mathbb{Z}}_1^2} {\tilde{\mathrm{\Omega }}}_n{z}_n{\overline{z}}_n+P)}{\mathrm{\partial }{z}_{n+tc}\mathrm{\partial }{\overline{z}}_{m+tc}}-\underset{t\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{{\mathrm{\partial }}^2(\sum _{n\in {\mathbb{Z}}_1^2} {\tilde{\mathrm{\Omega }}}_n{z}_n{\overline{z}}_n+P)}{\mathrm{\partial }{z}_{m+tc}\mathrm{\partial }{\overline{z}}_{m+tc}}\Vert }_{D_{\rho }(r,s),\mathcal{O}}\le \frac{\epsilon }{\left|t\right|}{\mathrm{e}}^{-|n-m|\rho },\\ & {\Vert \frac{{\mathrm{\partial }}^2(\overline{\mathcal{B}}+P)}{\mathrm{\partial }{\overline{z}}_{n+tc}\mathrm{\partial }{\overline{z}}_{m-tc}}-\underset{t\to \mathrm{\infty }}{lim} \frac{{\mathrm{\partial }}^2(\overline{\mathcal{B}}+P)}{\mathrm{\partial }{\overline{z}}_{n+tc}\mathrm{\partial }{\overline{z}}_{m-tc}}\Vert }_{D_o(r,s),\mathcal{O}}\le \frac{\epsilon }{\left|t\right|}{\mathrm{e}}^{-|n+m|\rho }.\end{array}

Note 4.3　Property (A6) is used for measure estimation. To ensure the existence of symplectic transformations required for each iteration step, the existence of non-resonance conditions is needed. Therefore, a parameter digging operation is performed to ensure that the final parameter set remains a set of positive measure, and it is required that an infinite number of non-resonance conditions cannot be added. Property (A6) ensures that only a finite number of non-resonance conditions need to be added after a sufficiently large number, which is obtained through successive limits.

Finally, we state the infinite-dimensional KAM theorem.

Theorem 4.1　 Suppose the Hamiltonian function H₀ + P in (3.1) satisfies assumptions (A1) to (A6). Let \gamma >0 be sufficiently small. Then, there exists a positive number \epsilon =\epsilon (b,K,\tau ,\gamma ,r,s,\rho ) such that when , the following conclusion holds.

There exists a Cantor set {\mathcal{O}}_{\gamma }\subset \mathcal{O}, and \mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}(\mathcal{O}\setminus {\mathcal{O}}_{\gamma })=O\left({\gamma }^{\frac{1}{4p}}\right), and two θ analysis, and for parameter \xi =\left({\xi }_1,\dots ,{\xi }_b\right)C_W^{4p}-smooth mapping:

\mathrm{\Psi }:{\mathbb{T}}^{b+m}\times {\mathcal{O}}_{\gamma }\to D_{\rho }(r,s),\hat{\omega }:{\mathcal{O}}_{\gamma }\to {\mathbb{R}}^{b+m},

where \mathrm{\Psi } is \frac{\epsilon }{{\gamma }^{4}p}-close to the trivial embedding {\mathrm{\Psi }}_0:{\mathbb{T}}^{b+m}\times \mathcal{O}\to {\mathbb{T}}^{b+m}\times \{0,0,0\}; and \hat{\omega } is ε-close to the unperturbed frequency \omega =({\epsilon }^{-3p}\overline{\omega },\tilde{\omega }). For any \xi \in {\mathcal{O}}_{\gamma } and \theta \in {\mathbb{T}}^{b+m}, the curve t\mapsto \mathrm{\Psi }(\theta +\hat{\omega }(\xi )t,\xi ) is a quasi-periodic solution to the Hamiltonian equations H=H_0+P.

By applying Theorem 4.1, the main result Theorem 1.1 of this paper has been proven. For the completeness of the paper, only the main framework theorem and propositions for proving the infinite-dimensional KAM theorem are provided later in the paper. For the relevant proofs, please refer to the proofs in [14, 15]. In these proofs, our main focus is to verify that the equations in this paper indeed satisfy the most challenging properties (A1) and (A3) of Theorem 4.1. In addition, after solving the cohomological equations, ensuring the boundedness of the functions F constructed at each step guarantees that the symplectic transformations required for each iteration step are reasonable.

5 Verification of the properties for initial iteration

Verification of (A1): The elements of the Jacobian matrix of \tilde{\omega } are polynomial functions of degree p − 1 in ξ, with integer coefficients. The coefficients of \frac{\mathrm{\partial }{\tilde{\omega }}_i}{\mathrm{\partial }{\xi }_j} and \frac{\mathrm{\partial }{\tilde{\omega }}_i}{\mathrm{\partial }{\xi }_i} are

\frac{p{\phi }_2^0}{(2\pi {)}^{2p}}\left(\begin{array}{c}p-1\\ k_1,\dots ,k_i-1,\dots ,k_j-1,\dots ,k_b\end{array}\right)\left(\begin{array}{c}p+1\\ k_1,\dots ,k_b\end{array}\right),

\frac{p{\phi }_2^0}{(2\pi {)}^{2p}}\left(\begin{array}{c}p-1\\ k_1,\dots ,k_i-2,\dots ,k_j,\dots ,k_b\end{array}\right)\left(\begin{array}{c}p+1\\ k_1,\dots ,k_b\end{array}\right).

There exists a prime number r that can divide p+1, namely p+1=r^ts, where it is required that r does not divide s and \left(\begin{array}{c}p+1\\ k_1,\dots ,k_b\end{array}\right). First, divide the Jacobian matrix of \tilde{\omega } by p, then take the modulo r. The matrix becomes a direct sum of some matrices with nonzero diagonal elements. Therefore, it is easy to verify that det\left(\frac{\mathrm{\partial }\tilde{\omega }}{\mathrm{\partial }\xi }\right)\ne 0. Hence, (A1) is verified.

Verification of (A3): This part of the proof is similar to [14], but due to the generality of the nonlinear terms in this paper, there are differences in the details of the treatment. Next, the proof for the most complex case will be presented.

Case 1　 n,n{}^{\prime }\in {\mathcal{L}}_1.

Consider ⟨k,\omega ⟩\pm {\mathrm{\Omega }}_n\pm {\mathrm{\Omega }}_{n{}^{\prime }}, there are resonance conditions:

\{\begin{array}{rl}& n-m+\sum _{j=1}^b l_j{i}_j=0,\\ & |n{|}^2-|m{|}^2+\sum _{j=1}^b l_j|i_j{|}^2=0,\end{array}l=l^{+}-l^{-}=\sum _{j=1}^b l_j{e}_j\in X_p^0.

Then the eigenvalues are:

\begin{array}{rl}& ⟨\overline{k},{\epsilon }^{-3p}\overline{\omega }⟩+⟨\tilde{k},\iota ⟩\pm \left({\epsilon }^{-3p}\left(\right|n{|}^2+{\phi }_1^0)+⟨l^{+},\iota ⟩+\frac{p+1}{(2\pi {)}^{2p}}{\phi }_2^0{A}_p(\xi )\right)\\ & +\frac{{\phi }_2^0}{2(p+1)(2\pi {)}^{2p}}⟨2k\pm \left|l\right|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}\left(\xi \right)⟩\\ & \pm \left({\epsilon }^{-3p}\left(\right|n{}^{\prime }{|}^2+{\phi }_1^0)+⟨l^{\prime }+,\iota ⟩+\frac{p+1}{(2\pi {)}^{2p}}{\phi }_2^0{A}_p(\xi )+\frac{{\phi }_2^0}{2(p+1)(2\pi {)}^{2p}}⟨|l{}^{\prime }|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}(\xi )⟩\right)\\ & \pm \frac{\left|{\phi }_2^0\right|}{2(p+1)(2\pi {)}^{2p}}(\sqrt{G}\pm \sqrt{G{}^{\prime }}).\end{array}

If l^{+}\ne l^{\prime +}, due to the existence of square root terms, all eigenvalues must not be constantly equal to 0. Otherwise, if l^{+}=l^{\prime +}, simultaneously l^{-}=l^{\prime -}. If the eigenvalues are in the following form:

\begin{array}{rl}& ⟨\overline{k},{\epsilon }^{-3p}\overline{\omega }⟩+⟨\tilde{k},\iota ⟩+\left({\epsilon }^{-3p}\left(\right|n{|}^2+{\phi }_1^0)+⟨l^{+},\iota ⟩+\frac{p+1}{(2\pi {)}^{2p}}{\phi }_2^0{A}_p(\xi )\right)\\ & +\frac{{\phi }_2^0}{2(p+1)(2\pi {)}^{2p}}⟨2k+\left|l\right|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}\left(\xi \right)⟩\\ & -\left({\epsilon }^{-3p}\left(\right|n{}^{\prime }{|}^2+{\phi }_1^0)+⟨l^{+},\iota ⟩+\frac{p+1}{(2\pi {)}^{2p}}{\phi }_2^0{A}_p(\xi )+\frac{{\phi }_2^0}{2(p+1)(2\pi {)}^{2p}}⟨|l|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}(\xi )⟩\right)\\ & +\frac{\left|{\phi }_2^0\right|}{2(p+1)(2\pi {)}^{2p}}(\sqrt{G}-\sqrt{G})\\ =& ⟨\overline{k},{\epsilon }^{-3p}\overline{\omega }⟩+⟨\tilde{k},\iota ⟩+{\epsilon }^{-3p}\left(\right|n{|}^2-\left|n{}^{\prime }{|}^2\right)+\frac{{\phi }_2^0}{p(p+1)(2\pi {)}^{2p}}⟨H\left(A_{p+1}\right(\xi \left)\right)k,\xi ⟩,\end{array}

where H\left(A_{p+1}\right(\xi \left)\right) is the Hessian Matrix of A_{p+1}\left(\xi \right), we modulo this Hessian matrix by a prime number r, and the matrix becomes a direct sum of some matrices with non-zero diagonal elements. Therefore, the Hessian matrix is non-degenerate, and it is easy to verify that for k ≠ 0, H\left(A_{p+1}\right(\xi \left)\right)k\ne 0. If the eigenvalues are in the following form:

\begin{array}{rl}& ⟨\overline{k},{\epsilon }^{-3p}\overline{\omega }⟩+⟨\tilde{k},\iota ⟩+\left({\epsilon }^{-3p}\left(\right|n{|}^2+{\phi }_1^0)+⟨l^{+},\iota ⟩+\frac{p+1}{(2\pi {)}^{2p}}{\phi }_2^0{A}_p(\xi )\right)\\ & +\frac{{\phi }_2^0}{2(p+1)(2\pi {)}^{2p}}⟨2k+\left|l\right|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}\left(\xi \right)⟩\\ & +\left({\epsilon }^{-3p}\left(\right|n{}^{\prime }{|}^2+{\phi }_1^0)+⟨l^{+},\iota ⟩+\frac{p+1}{(2\pi {)}^{2p}}{\phi }_2^0{A}_p(\xi )+\frac{{\phi }_2^0}{2(p+1)(2\pi {)}^{2p}}⟨|l|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}(\xi )⟩\right)\\ & +\frac{\left|{\phi }_2^0\right|}{2(p+1)(2\pi {)}^{2p}}(\sqrt{G}-\sqrt{G})\\ =& ⟨\overline{k},{\epsilon }^{-3p}\overline{\omega }⟩+⟨\tilde{k},\iota ⟩+{\epsilon }^{-3p}\left(\right|n{|}^2+|n{}^{\prime }{|}^2+2{\phi }_1^0)+2⟨l^{+},\iota ⟩+\frac{{\phi }_2^0}{p(p+1)(2\pi {)}^{2p}}\\ & \times ⟨H\left(A_{p+1}\right(\xi \left)\right)(k+|l\left|\right)+2(p+1{)}^2{\mathrm{\nabla }}_{\xi }{A}_p(\xi ),\xi ⟩,\end{array}

when \begin{array}{c}H\left(A_{p+1}\right(\xi \left)\right)(k+|l\left|\right)+2(p+1{)}^2{\mathrm{\nabla }}_{\xi }{A}_p(\xi )=0\end{array}. We can find the system of equations

\begin{array}{rl}& \left(\begin{array}{c}p+1\\ p+1,\dots ,0\end{array}\right)(k+|l|{)}_1+\left(\begin{array}{c}p+1\\ p,1,\dots \end{array}\right)(k+\left|l\right|{)}_2+\cdots +\left(\begin{array}{c}p+1\\ p,\dots ,1\end{array}\right)(k+|l|{)}_b\\ =& -2(p+1)\left(\begin{array}{c}p\\ p,\dots ,0\end{array}\right),\\ & \left(\begin{array}{c}p+1\\ 1,p,\dots \end{array}\right)(k+|l|{)}_1+\left(\begin{array}{c}p+1\\ 0,p+1,\dots \end{array}\right)(k+\left|l\right|{)}_2+\cdots +\left(\begin{array}{c}p+1\\ 0,p,\dots ,1\end{array}\right)(k+|l|{)}_b\\ =& -2(p+1)\left(\begin{array}{c}p\\ 0,p,\dots ,0\end{array}\right),\\ \cdots & \\ & \left(\begin{array}{c}p+1\\ 1,\dots ,p\end{array}\right)(k+|l|{)}_1+\left(\begin{array}{c}p+1\\ 0,1,\dots ,p\end{array}\right)(k+\left|l\right|{)}_2+\cdots +\left(\begin{array}{c}p+1\\ 0,\dots ,p+1\end{array}\right)(k+|l|{)}_b\\ =& -2(p+1)\left(\begin{array}{c}p\\ 0,\dots ,p\end{array}\right).\end{array}

Therefore, all elements of k+|l| are equal and satisfy [1+(p+1)(b-1\left)\right](k+|l|{)}_1=-2(p+1). Obviously, this system of equations has no integer solutions. Therefore, all eigenvalues must not be constantly equal to 0.

Case 2　 n\in {\mathcal{L}}_1,n{}^{\prime }\in {\mathcal{L}}_2·

Let

\begin{array}{rl}M=& {\left(⟨l,{\mathrm{\nabla }}_{\xi }{A}_{p+1}\left(\xi \right)⟩+2(p+1)A_p\left(\xi \right)\right)}^2\\ & +4\sum _{\begin{array}{c}\begin{array}{c}l=l^{+}-l^{-}\in X_p^{-2}\end{array}\\ \alpha \in {\mathbb{N}}^b,|l^{+}+\alpha {|}_1|=p-1\end{array}}{p}^2(p+1{)}^2[\left(\begin{array}{c}p-1\\ l^{+}+\alpha \end{array}\right)\left(\begin{array}{c}p+1\\ l^{-}+\alpha \end{array}\right){]}^2{\xi }^{l+2\alpha }.\end{array}

In this case, the eigenvalues are:

\begin{array}{rl}& ⟨\overline{k},{\epsilon }^{-3p}\overline{\omega }⟩+⟨\tilde{k},\iota ⟩\pm \left({\epsilon }^{-3p}\left(\right|n{|}^2+{\phi }_1^0)+⟨l^{+},\iota ⟩+\frac{p+1}{(2\pi {)}^{2p}}{\phi }_2^0{A}_p(\xi )\right)\\ & +\frac{{\phi }_2^0}{2(p+1)(2\pi {)}^{2p}}⟨2k\pm \left|l\right|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}\left(\xi \right)⟩\\ & \pm \left({\epsilon }^{-3p}\left(\right|n{}^{\prime }{|}^2+{\phi }_1^0)+⟨l{}^{\prime }+,\iota ⟩+\frac{{\phi }_2^0}{2(p+1)(2\pi {)}^{2p}}⟨|l{}^{\prime }|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}(\xi )⟩\right)\\ & \pm \frac{\left|{\phi }_2^0\right|}{2(p+1)(2\pi {)}^{2p}}(\sqrt{G}\pm \sqrt{M{}^{\prime }}).\end{array}

Note 5.1　If \sqrt{M{}^{\prime }} is purely imaginary, because there are no small divisors in this resonance case, there is no need to perform a parameter set measure digging operation. It is emphasized in this paper that the condition ensuring purely imaginary values has been removed here, so it is necessary to reapply the parameter set measure digging operation, which also reflects the complexity of the non-resonance conditions in this paper.

If square root terms exist, all eigenvalues must not be constantly equal to 0. Otherwise, the square root terms vanish. If the eigenvalues are:

\begin{array}{rl}⟨\overline{k},{\epsilon }^{-3p}\overline{\omega }⟩& +⟨\tilde{k},\iota ⟩+\left({\epsilon }^{-3p}\right(|n{|}^2+{\phi }_1^0\pm |n{}^{\prime }{|}^2\pm {\phi }_1^0)+⟨l^{+}\pm l{}^{\prime }+,\iota ⟩)\\ & +\frac{{\phi }_2^0}{2p(p+1)(2\pi {)}^{2p}}⟨H\left(A_{p+1}\right(\xi \left)\right)(2k+|l|\pm |l{}^{\prime }\left|\right)+2(p+1{)}^2{\mathrm{\nabla }}_{\xi }{A}_p(\xi ),\xi ⟩,\end{array}

when H\left(A_{p+1}\right(\xi \left)\right)(2k+|l|\pm |l{}^{\prime }\left|\right)+2(p+1{)}^2{\mathrm{\nabla }}_{\xi }{A}_p(\xi )=0, there is also a fact that all elements of 2k+\left|l\right|\pm \left|l{}^{\prime }\right| are equal, and satisfy

[1+(p+1\left)\right(b-1\left)\right](2k+|l|\pm |l{}^{\prime }|{)}_1=-2(p+1).

Obviously, this system of equations has no integer solutions. Therefore, all eigenvalues must not be constantly equal to 0.

Case 3　 n,n{}^{\prime }\in {\mathcal{L}}_2.

In this case, the eigenvalues of ⟨k,\omega ⟩I\pm {\mathcal{M}}_n\otimes I_2\pm I_2\otimes {\mathcal{M}}_{n{}^{\prime }} are

\begin{array}{rl}⟨\overline{k},{\epsilon }^{-3p}\overline{\omega }⟩& +⟨\tilde{k},\iota ⟩\pm \left({\epsilon }^{-3p}\left(\right|n{|}^2+{\phi }_1^0)+(l^{+},\iota )+\frac{{\phi }_2^0}{2(p+1)(2\pi {)}^{2p}}⟨2k\pm |l|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}(\xi )⟩\right)\\ & \pm \left({\epsilon }^{-3p}\left(\right|n{}^{\prime }{|}^2+{\phi }_1^0)+(l{}^{\prime }+,\iota )+\frac{{\phi }_2^0}{2(p+1)(2\pi {)}^{2p}}⟨|l{}^{\prime }|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}(\xi )⟩\right)\\ & \pm \frac{{\phi }_2^0}{2(p+1)(2\pi {)}^{2p}}(\sqrt{M}\pm \sqrt{M{}^{\prime }}).\end{array}

If l^{+}\ne l^{\prime +}, due to the presence of square root terms, all eigenvalues must not be constantly equal to 0. Otherwise, if l^{+}\ne l^{\prime +}, then l^{-}\ne l^{\prime -}. If the eigenvalues are in the following form:

\begin{array}{rl}⟨\overline{k},{\epsilon }^{-3p}\overline{\omega }⟩& +⟨\tilde{k},\iota ⟩+\left({\epsilon }^{-3p}\left(\right|n{|}^2+{\varphi }_1^0)+⟨l^{+},\iota ⟩+\frac{{\varphi }_2^0}{2(p+1)(2\pi {)}^{2p}}⟨2k+|l|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}(\xi )⟩\right)\\ & -\left({\epsilon }^{-3p}\left(\right|n{}^{\prime }{|}^2+{\varphi }_1^0)+⟨l^{+},\iota ⟩+\frac{{\varphi }_2^0}{2(p+1)(2\pi {)}^{2p}}⟨|l|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}(\xi )⟩\right)\\ & +\frac{{\varphi }_2^0}{2(p+1)(2\pi {)}^{2p}}(\sqrt{M}-\sqrt{M})\\ =& ⟨\overline{k},{\epsilon }^{-3p}\overline{\omega }⟩+⟨\tilde{k},\iota ⟩+{\epsilon }^{-3p}\left(\right|n{|}^2-\left|n{}^{\prime }{|}^2\right)+\frac{{\varphi }_2^0}{p(p+1)(2\pi {)}^{2p}}⟨H\left(A_{p+1}\right(\xi \left)\right)k,\xi ⟩,\end{array}

where H\left(A_{p+1}\right(\xi \left)\right) is the Hessian matrix of A_{p+1}\left(\xi \right), we take it modulo a prime number r. This matrix becomes a direct sum of some matrices with only nonzero diagonal elements. Therefore, the Hessian matrix is non-degenerate, and it can be easily verified that for k ≠ 0, H\left(A_{p+1}\right(\xi \left)\right)k\ne 0. If the eigenvalues are in the following form

\begin{array}{rl}⟨\overline{k},{\epsilon }^{-3p}\overline{\omega }⟩& +⟨\tilde{k},\iota ⟩+\left({\epsilon }^{-3p}\left(\right|n{|}^2+{\phi }_1^0)+⟨l^{+},\iota ⟩+\frac{{\phi }_2^0}{2(p+1)(2\pi {)}^{2p}}⟨2k+|l|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}(\xi )⟩\right)\\ & +\left({\epsilon }^{-3p}\left(\right|n{}^{\prime }{|}^2+{\phi }_1^0)+⟨l^{+},\iota ⟩+\frac{{\phi }_2^0}{2(p+1)(2\pi {)}^{2p}}⟨|l|,{\mathrm{\nabla }}_{\xi }{A}_{p+1}(\xi )⟩\right)\\ & +\frac{1}{2(p+1)(2\pi {)}^{2p}}(\sqrt{M}-\sqrt{M})\\ =& ⟨\overline{k},{\epsilon }^{-3p}\overline{\omega }⟩+⟨\tilde{k},\iota ⟩+{\epsilon }^{-3p}\left(\right|n{|}^2+|n{}^{\prime }{|}^2+2{\phi }_1^0)\\ & +2⟨l^{+},\iota ⟩+\frac{{\phi }_2^0}{p(p+1)(2\pi {)}^{2p}}⟨H\left(A_{p+1}\right(\xi \left)\right)(k+|l\left|\right),\xi ⟩,\end{array}

when H\left(A_{p+1}\right(\xi \left)\right)(k+|l\left|\right)=0, all possible solutions of such equations are k=-\left|l\right|. In this case, if \left|n\right|\ne \left|m{}^{\prime }\right|, then

⟨\overline{k},{\epsilon }^{-3p}\overline{\omega }⟩+⟨\tilde{k},\iota ⟩+{\epsilon }^{-3p}\left(\right|n{|}^2+|n{}^{\prime }{|}^2+2{\varphi }_1^0)+2⟨l^{+},\iota ⟩=⟨\overline{k},{\epsilon }^{-3p}\overline{\omega }⟩+{\epsilon }^{-3p}\left(\right|n{|}^2-\left|m{}^{\prime }{|}^2\right)\ne 0.

Note 5.2　Note that the properties of partial zero momentum and partial gauge invariance ensure that the coefficient of {\phi }_1^0 is 0. Therefore, all eigenvalues must not be constantly equal to 0. Similar proofs apply to other cases.

From [14, Lemma 4.6], \mathrm{d}\mathrm{e}\mathrm{t}(⟨k,\omega ⟩I\pm {\mathcal{M}}_n\otimes I_2\pm I_2\otimes {\mathcal{M}}_{n{}^{\prime }}) is a polynomial function of ξ with degree at most 4p. Therefore,

\left|{\mathrm{\partial }}_{\xi }^{4p}\right(\mathrm{d}\mathrm{e}\mathrm{t}(⟨k,\omega ⟩I\pm {\mathcal{M}}_n\otimes I_2\pm I_2\otimes {\mathcal{M}}_{n{}^{\prime }})\left)\right|\ge \frac{1}{2(p+1)(2\pi {)}^{2p}}\left|k\right|\ne 0.

By removing the part of the parameter set with measure at most \mathcal{O}\left({\gamma }^{\frac{1}{4p}}\right), we have

\left|\mathrm{d}\mathrm{e}\mathrm{t}(⟨k,\omega ⟩I\pm {\mathcal{M}}_n\otimes I_2\pm I_2\otimes {\mathcal{M}}_{n{}^{\prime }})\right|\ge \frac{\gamma }{K^{\tau }},k\ne 0,n,n{}^{\prime }\in {\mathcal{L}}_2,

thus ensuring the rationality of each step of iteration. Therefore, (A3) is satisfied.

Hence, for the Hamiltonian function (3.1), by applying Theorem 4.1, we obtain Theorem 1.1.

6 Iteration steps

In the ν-th step of the KAM iteration, consider the Hamiltonian function

H_{\nu }=N_{\nu }+{\mathcal{B}}_{\nu }+{\overline{\mathcal{B}}}_{\nu }+P_{\nu },

where {\mathcal{B}}_{\nu }+{\overline{\mathcal{B}}}_{\nu }+P_{\nu } is defined on D_{{\rho }_{\nu }}(r_{\nu },s_{\nu })\times {\mathcal{O}}_{\nu }, and the Hamiltonian function satisfies assumptions (A1)‒(A6).

Next, the main approach is to find a function F by solving a cohomological equation, where the Hamiltonian flow Hamilton {\mathrm{\Phi }}_v at time 1 is the symplectic map needed for the iteration:

{\mathrm{\Phi }}_v:D_{{\rho }_{\nu }}(r_{\nu +1},s_{\nu +1})\times {\mathcal{O}}_{\nu }\to D_{{\rho }_{\nu }}(r_{\nu },s_{\nu })\times {\mathcal{O}}_{\nu },

with the aim of making the Hamiltonian function increasingly closer to an integrable gauge form. To ensure the rationality of the symplectic transformation, it is necessary to perform a measure digging operation on the parameter set. Consider the new Hamiltonian function after one step of iteration by the symplectic transformation:

H_{\nu +1}=H_{\nu }\circ {\mathrm{\Phi }}_{\nu }=N_{\nu +1}+{\mathcal{B}}_{\nu +1}+{\overline{\mathcal{B}}}_{\nu +1}+P_{\nu +1}.

On the set D_{\rho \nu +1}(r_{\nu +1},s_{\nu +1})\times {\mathcal{O}}_{\nu }, after removing the measure, it still satisfies the aforementioned iteration assumptions (A1)−(A6), and meets the condition:

{\Vert X_{P_{\nu +1}}\Vert }_{D_{{\rho }_{\nu +1}}(r_{\nu +1},s_{\nu +1}),{\mathcal{O}}_{\nu }}={\Vert X_{H_{\nu }\circ {\mathrm{\Phi }}_{\nu }}-X_{N_{\nu +1}+{\mathcal{B}}_{\nu +1}+{\mathcal{B}}_{\nu +1}}\Vert }_{D_{{\rho }_{\nu +1}}(r_{\nu +1},s_{\nu +1}),{\mathcal{O}}_{\nu }}\le {\epsilon }_{\nu +1}.

To simplify the notation, terms without superscripts or subscripts represent items in the v-th step, while items with superscripts or subscripts represent those in the (v+1)-th step. Consider the Hamiltonian function defined on D_{\rho }(r,s)\times \mathcal{O}, where 0 < r₊ < r. Define

s_{+}=\frac{1}{4}s{\epsilon }^{\frac{1}{3}},{\epsilon }_{+}=c{\gamma }^{-(4p+1)}{K}^{(4p+1)(\tau +1)}(r-r_{+}{)}^{-c}{\epsilon }^{\frac{4}{3}}.

Next, we explain how to construct a set {\mathcal{O}}_{+}\subset \mathcal{O} (and quantify the measure of the parameter set removed) and a symplectic transformation \mathrm{\Phi }:D_{+}\times {\mathcal{O}}_{+}=D_{\rho }(r_{+},s_{+})\times {\mathcal{O}}_{+}\to D_{\rho }(r,s)\times \mathcal{O}, ensuring that the transformed Hamiltonian function H_{+}=N_{+}+{\mathcal{B}}_{+}+{\overline{\mathcal{B}}}_{+}+P_{+}\equiv H\circ {\mathrm{\Phi }}_{} still satisfies the iteration assumptions (A1)‒(A6) for the new s₊, ε₊, r₊, and \xi \in {\mathcal{O}}_{+}.

Express the perturbation term P as a Fourier-Taylor series, and let R be the truncation of P with respect to the time Fourier index \left|k\right|\le K. By the method of coefficient comparison, the coefficients of F (which have the same form as R, excluding the resonant terms from R) satisfy the equation

(6.1)\{N+\mathcal{B}+\overline{\mathcal{B}},F\}+R-P_{0000}-⟨\hat{\omega },I⟩-\sum _{n\in {\mathbb{Z}}^2} P_{nn}^{011}{z}_n{\overline{z}}_n-\hat{\mathcal{B}}-\widehat{\overline{\mathcal{B}}}=0,

where

\begin{array}{rl}& \hat{\omega }=\int \frac{\mathrm{\partial }P}{\mathrm{\partial }I}\mathrm{d}\theta {|}_{z=\overline{z}=0,I=0},\hat{\mathcal{B}}=\sum _{n\in {\mathcal{L}}_2} P_{nm}^{020}{z}_n{z}_m,\widehat{\overline{\mathcal{B}}}=\sum _{n\in {\mathcal{L}}_2} P_{nm}^{002}{\overline{z}}_n{\overline{z}}_m,\\ & N_{+}=N+P_{0000}+⟨\hat{\omega },I⟩+\sum _n P_{nn}^{011}{z}_n{\overline{z}}_n,\\ & {\mathcal{B}}_{+}=B+\hat{\mathcal{B}},\overline{{\mathcal{B}}_{+}}=\overline{\mathcal{B}}+\widehat{\overline{\mathcal{B}}}=\overline{\mathcal{B}}+\overline{\hat{\mathcal{B}}},\\ & P_{+}={\int }_0^1 (1-t)\left\{\right\{N+B+B,F\},F\}\circ {\phi }_F^t\mathrm{d}t+{\int }_0^1 \{R,F\}\circ {\phi }_F^t\mathrm{d}t+(P-R)\circ {\phi }_F^1.\end{array}

Proposition 6.1　 F satisfies Eq. (6.1), where the Fourier coefficients of F₀ and F₁ are given by the system of equations:

The Fourier coefficients of F₂ are given by the following two propositions.

Case 1　 n,m\in {\mathbb{Z}}_1^2\setminus {\mathcal{L}}_2.

Proposition 6.2　 F satisfies Eq. (6.1), where the Fourier coefficients of F₂ are given by the system of equations:

Case 2　 n\in {\mathbb{Z}}_1^2\setminus {\mathcal{L}}_2,n{}^{\prime }\in {\mathcal{L}}_2.

Proposition 6.3　 F satisfies Eq. (6.1), where the Fourier coefficients of F₂ are given by the system of equations:

\begin{array}{rl}& \left[\right(⟨k,\omega ⟩+{\mathrm{\Omega }}_n)I_2+{\mathcal{M}}_{n{}^{\prime }}](F_{nn{}^{\prime }}^{k20},F_{nm{}^{\prime }}^{k11}{)}^{\mathrm{T}}=\mathrm{i}(P_{nn{}^{\prime }}^{k20},P_{nm{}^{\prime }}^{k11}{)}^{\mathrm{T}},\\ & \left[\right(⟨k,\omega ⟩+{\mathrm{\Omega }}_n)I_2+{\mathcal{M}}_{n{}^{\prime }}](F_{nn{}^{\prime }}^{k02},F_{m{}^{\prime }n}^{k11}{)}^{\mathrm{T}}=\mathrm{i}(P_{nn{}^{\prime }}^{k02},P_{m{}^{\prime }n}^{k11}{)}^{\mathrm{T}},\\ & \left[\right(⟨k,\omega ⟩-{\mathrm{\Omega }}_n)I_2+{\mathcal{M}}_{n{}^{\prime }}](F_{nn{}^{\prime }}^{k11},F_{nm{}^{\prime }}^{k20}{)}^{\mathrm{T}}=\mathrm{i}(P_{nn{}^{\prime }}^{k11},P_{nm{}^{\prime }}^{k20}{)}^{\mathrm{T}},\\ & \left[\right(⟨k,\omega ⟩+{\mathrm{\Omega }}_n)I_2-{\mathcal{M}}_{n{}^{\prime }}](F_{n{}^{\prime }n}^{k11},F_{m{}^{\prime }n}^{k02}{)}^{\mathrm{T}}=\text{i}(P_{n{}^{\prime }n}^{k11},P_{m{}^{\prime }n}^{k02}{)}^{\mathrm{T}}.\end{array}

Case 3　 n,n{}^{\prime }\in {\mathcal{L}}_2.

Proposition 6.4　 F satisfies Eq. (6.1), where the Fourier coefficients of F₂ are given by the system of equations:

\begin{array}{rl}(⟨k,\omega ⟩I_4-{\mathcal{M}}_n& \otimes I_2+I_2\otimes {\mathcal{M}}_{n{}^{\prime }})(F_{nn{}^{\prime }}^{k11},F_{nm{}^{\prime }}^{k20},F_{mn{}^{\prime }}^{k02},F_{m{}^{\prime }m}^{k11}{)}^{\mathrm{T}}\\ & =\mathrm{i}(P_{nn{}^{\prime }}^{k11},P_{nm{}^{\prime }}^{k20},P_{mn{}^{\prime }}^{k02},P_{m{}^{\prime }m}^{k11}{)}^{\mathrm{T}},\end{array}

\begin{array}{rl}(⟨k,\omega ⟩I_4+{\mathcal{M}}_n& \otimes I_2-I_2\otimes {\mathcal{M}}_{n{}^{\prime }})(F_{n{}^{\prime }n}^{k11},F_{m{}^{\prime }n}^{k02},F_{n{}^{\prime }m}^{k20},F_{mm{}^{\prime }}^{k11}{)}^{\mathrm{T}}\\ & =\mathrm{i}(P_{n{}^{\prime }n}^{k11},P_{m{}^{\prime }n}^{k02},P_{n{}^{\prime }m}^{k20},P_{mm{}^{\prime }}^{k11}{)}^{\mathrm{T}},\end{array}

\begin{array}{rl}(⟨k,\omega ⟩I_4-{\mathcal{M}}_n& \otimes I_2-I_2\otimes {\mathcal{M}}_{n{}^{\prime }})(F_{nn{}^{\prime }}^{k20},F_{nm{}^{\prime }}^{k11},F_{n{}^{\prime }m}^{k11},F_{mm{}^{\prime }}^{k02}{)}^{\mathrm{T}}\\ & =\mathrm{i}(P_{nn{}^{\prime }}^{k20},P_{nm{}^{\prime }}^{k11},P_{n{}^{\prime }m}^{k11},P_{mm{}^{\prime }}^{k02}{)}^{\mathrm{T}},\end{array}

\begin{array}{rl}(⟨k,\omega ⟩I_4+{\mathcal{M}}_n& \otimes I_2+I_2\otimes {\mathcal{M}}_{n{}^{\prime }})(F_{nn{}^{\prime }}^{k02},F_{m{}^{\prime }n}^{k11},F_{mn{}^{\prime }}^{k11},F_{m{}^{\prime }m}^{k20}{)}^{\mathrm{T}}\\ & =\mathrm{i}(P_{nn{}^{\prime }}^{k02},P_{m{}^{\prime }n}^{k11},P_{mn{}^{\prime }}^{k11},P_{m{}^{\prime }m}^{k20}{)}^{\mathrm{T}}.\end{array}

Please refer to [14] for the proof of the foregoing two propositions.

Next, we consider one of the equations as an example (others are similar):

Q_{n{}^{\prime }}^{-1}\left[\right(⟨k,\omega ⟩-{\mathrm{\Omega }}_n)I_2-{\mathcal{M}}_{n{}^{\prime }}]Q_{n{}^{\prime }}{Q}_{n{}^{\prime }}^{-1}(F_{nn{}^{\prime }}^{k20},F_{nm{}^{\prime }}^{k11}{)}^{\mathrm{T}}=\mathrm{i}{Q}_{n{}^{\prime }}^{-1}(P_{nn{}^{\prime }}^{k20},P_{nm{}^{\prime }}^{k11}{)}^{\mathrm{T}},\left|k\right|\le K,

where there exists an invertible matrix Q_{n{}^{\prime }}, satisfying Q_{n{}^{\prime }}^{-1}{\mathcal{M}}_{n{}^{\prime }}{Q}_{n{}^{\prime }}=J_{n{}^{\prime }}, which is a Jordan canonical form, namely

\left[\right(⟨k,\omega ⟩-{\mathrm{\Omega }}_n)I_2-J_{n{}^{\prime }}]({\tilde{F}}_{nn{}^{\prime }}^{k20},{\tilde{F}}_{nm{}^{\prime }}^{k11}{)}^{\mathrm{T}}=\mathrm{i}({\tilde{P}}_{nn{}^{\prime }}^{k20},{\tilde{P}}_{nm{}^{\prime }}^{k11}{)}^{\mathrm{T}},\left|k\right|\le K.

Here, we define

({\tilde{F}}_{nn{}^{\prime }}^{k20},{\tilde{F}}_{nm{}^{\prime }}^{k11}{)}^{\mathrm{T}}=Q_{n{}^{\prime }}^{-1}(F_{nn{}^{\prime }}^{k20},F_{nm{}^{\prime }}^{k11}{)}^{\mathrm{T}},

({\tilde{P}}_{nn{}^{\prime }}^{k20},{\tilde{P}}_{nm{}^{\prime }}^{k11}{)}^{\mathrm{T}}=Q_{n{}^{\prime }}^{-1}(P_{nn{}^{\prime }}^{k20},P_{nm{}^{\prime }}^{k11}{)}^{\mathrm{T}}.

Based on the different forms of the second-order Jordan canonical form, we have two cases:

(⟨k,\omega ⟩-{\mathrm{\Omega }}_n-{\mu }_1)\left({\tilde{F}}_{nn{}^{\prime }}^{k20}\right)=\mathrm{i}\left({\tilde{P}}_{nn{}^{\prime }}^{k20}\right),(⟨k,\omega ⟩-{\mathrm{\Omega }}_n-{\mu }_2)\left({\tilde{F}}_{nm{}^{\prime }}^{k11}\right)=\mathrm{i}\left({\tilde{P}}_{nm{}^{\prime }}^{k11}\right),

or

\left(\begin{array}{cc}⟨k,\omega ⟩-{\mathrm{\Omega }}_n-\mu & -1\\ 0& ⟨k,\omega ⟩-{\mathrm{\Omega }}_n-\mu \end{array}\right)\left(\begin{array}{c}{\tilde{F}}_{nn{}^{\prime }}^{k20}\\ {\tilde{F}}_{nm{}^{\prime }}^{k11}\end{array}\right)=\mathrm{i}\left(\begin{array}{c}{\tilde{P}}_{nn{}^{\prime }}^{k20}\\ {\tilde{P}}_{nm{}^{\prime }}^{k11}\end{array}\right).

The estimation of (F_{nn{}^{\prime }}^{k20},F_{nm{}^{\prime }}^{k11}{)}^{\mathrm{T}}, n\in {\mathbb{Z}}_1^2\setminus {\mathcal{L}}_2,n{}^{\prime }\in {\mathcal{L}}_2, is straightforward.

\left(\begin{array}{c}{F}_{nn{}^{\prime }}^{k20}\\ F_{nm{}^{\prime }}^{k11}\end{array}\right)=\mathrm{i}{Q}_{n{}^{\prime }}\left(\begin{array}{c}{\tilde{P}}_{nn{}^{\prime }}^{k20}\\ {\tilde{P}}_{nm{}^{\prime }}^{k11}\end{array}\right)\frac{1}{⟨k,\omega ⟩-{\mathrm{\Omega }}_n-\mu }+\mathrm{i}{Q}_{n{}^{\prime }}\left(\begin{array}{c}{\tilde{P}}_{nm{}^{\prime }}^{k11}\\ 0\end{array}\right)\frac{1}{(⟨k,\omega ⟩-{\mathrm{\Omega }}_n-\mu {)}^2}.

Under the assumption

{\epsilon }_{\nu +1}=c{\gamma }^{-(4p+1)}(r_{\nu }-r_{\nu +1}{)}^{-c}{K}_{\nu }^{(4p+1)(\tau +1)}{\epsilon }_{\nu }^{\frac{4}{3}},

we have:

\begin{array}{rl}& |\left(\begin{array}{c}{F}_{nn{}^{\prime }}^{k20}\\ F_{nm{}^{\prime }}^{k11}\end{array}\right)|\le c\epsilon \frac{K^{2\tau }}{{\gamma }^2}{\mathrm{e}}^{-\rho |n+n{}^{\prime }|}{\mathrm{e}}^{-\left|k\right|r},\\ & \Vert \left(\begin{array}{c}{F}_{nn{}^{\prime }}^{k20}\\ F_{nm{}^{\prime }}^{k11}\end{array}\right)\Vert \le c{\epsilon }_{\nu +1}\frac{K_{\nu }^{(4p+1)(\tau +1)}}{{\gamma }^{(4p+1)}}\le {\epsilon }_{\nu +1}^{\frac{1}{3}}.\end{array}

Due to the small divisor assumption, we have the following estimates:

At this point, the required symplectic transformation has been found and is reasonable. For the new canonical form and the properties of the new perturbation terms after one step of symplectic transformation, please refer to [15]. With this, one iteration step is completed.

Here is the definition of the parameters required for each iteration step. For any s, ε, r, y, and for all \nu ⩾1, define

\begin{array}{rl}& r_{\nu +1}=r\left(1-\sum _{i=2}^{\nu +2} 2^{-i}\right),{\epsilon }_{\nu +1}=c{\gamma }^{-(4p+1)}(r_{\nu }-r_{\nu +1}{)}^{-c}{K}_{\nu }^{(4p+1)(\tau +1)}{\epsilon }_{\nu }^{\frac{4}{3}},\\ & {\eta }_{\nu +1}={\epsilon }_{\nu +1}^{\frac{1}{3}},L_{\nu +1}=L_{\nu }+{\epsilon }_{\nu },s_{\nu +1}=2^{-2}{\eta }_{\nu }{s}_{\nu }=2^{-2(\nu +1)}{\left(\prod _{i=0}^{\nu } {\epsilon }_i\right)}^{\frac{1}{3}}{s}_0,\\ & {\rho }_{\nu +1}=\rho \left(1-\sum _{i=2}^{\nu +2} 2^{-i}\right),K_{\nu +1}=c\left(\frac{1}{{\rho }_v-{\rho }_{v+1}}ln\frac{1}{{\epsilon }_{v+1}}\right).\end{array}

Here, c is a constant, and the parameters r₀, ε₀, L₀, s₀ and K₀ are defined as r, ε, L, s, and \mathrm{l}\mathrm{n}\frac{1}{\epsilon }, respectively. For the iteration lemma and convergence, please refer to [15].

Finally, the most important measure estimate is provided.

For convenience of notation, let {\mathcal{O}}_{-1}=\mathcal{O},K_{-1}=0. Then, in the ν-th step of the KAM iteration, it is necessary to remove the following parameter sets to ensure smooth iteration

where

where

Note 6.1　In the (ν+1)th step of the KAM iteration, when \left|k\right|\le K_{\nu }, the small divisor condition is automatically satisfied, so it is only necessary to remove the above resonance sets {\mathcal{R}}^{\nu +1}.

Proposition 6.5　We have

Note 6.2　Proposition 6.5 requires the use of property (A6). The main method is to use techniques of repeated limits to handle infinitely many non-resonance conditions as a finite set, thereby obtaining the small measure estimates removed in the proposition. This ultimately ensures convergence of the iteration on an almost full measure set of parameters. The following proposition provides specific estimates.

After infinitely many steps of KAM iteration, removing the measure of parameter sets, we have the following proposition.

Proposition 6.6　Let \tau >4p(4p+1)(12+b+1), after an infinite number of steps of KAM iteration process, the accumulated measure that needs to be removed is

Thus, two main parts of the proof of Theorem 4.1 have been completed: First, the construction of the iterative procedure, where at each step of iteration, the non-integrable part of the Hamiltonian function is eliminated through coordinate transformations using the cohomological equation, and the newly generated perturbation term is of higher order infinitesimal compared to the previous step’s perturbation term; second, the construction of the convergence domain, where in each iteration process, the values of variables that may cause the iteration not to converge are “removed” from the domain of action variables, ultimately demonstrating convergence of the iteration on a set of parameters with a finite measure, and providing measure estimates. By applying this infinite-dimensional KAM theorem, Theorem 1.1 is obtained.

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