Abstract
In this paper, we prove the reducibility for some linear quasi-periodic Hamiltonian derivative wave and half-wave equations under the Brjuno–Rüssmann non-resonance conditions. This is an extension of previous results of reducibility on Hamiltonian PDEs that required stronger (Diophantine) non-resonance conditions.
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1 Introduction and Main Result
The reducibility theory of linear quasi-periodic systems is the generalization of the classical Floquet theory for linear periodic systems. It is important both in the linear problems (spectral analysis of operator, growth of Sobolev norms) and in the non-linear case(linear stability analysis of quasi-periodic solutions of non-linear systems). The first reducibility result via Kolmogorov–Arnold–Moser (KAM) theory was due to Bogoljubov, Mitropoliskii and Samoilenko [11], Dinaburg and Sinai [17] for finite degrees of freedom systems. Since then KAM theory has been a powerful tool to study reducibility theory. In the late 1980s and early 1990s, KAM theory was extended to non-linear partial differential equations (PDEs) by Kuksin [33] and Wayne [48]. See also [35, 42, 43] for further developments. As a corollary, these results imply the reducibility of the variational equations for quasi-periodic solutions of non-linear PDEs. In fact, “reducibility is not only an important outcome of KAM but also an essential ingredient in the proof” [20].
The first pure reducibility result for linear quasi-periodic PDEs was given by Bambusi and Graffi [5]. They proved the reducibility of linear Schrödinger equations with unbounded perturbations. Eliasson and Kuksin [19] investigated the reducibility of higher dimensional linear quasi-periodic Schrödinger equations. Combining the pseudo-differential calculus, Baldi, Berti and Montalto [1, 2] obtained the reducibility of quasi-linear forced perturbations of Airy equation and quasi-linear KdV equation. Thereafter, these results are developed and extended widely. One could refer to [3, 4, 6,7,8,9, 28,29,30, 36, 37, 39] and the references therein.
Consider a linear quasi-periodic PDE of the form
where A is a positive self-adjoint operator and P is a operator-valued function with the basic frequencies \(\omega .\) It is well known that KAM reducibility requires a lower bound on small divisors of the form
where \(k\cdot \omega =\sum ^n_{i=1} k_i \omega _i \) and \(\{\lambda _i\}\) are the eigenvalues of the operator A. In all the above-mentioned papers, the lower bound of Diophantine type was used. Namely, the following non-resonance conditions holds: \( | k\cdot \omega + \lambda _i(\omega )-\lambda _i(\omega )|\ge \frac{\gamma }{|k|^\tau },\) where the constants \(\gamma>0,\,\, \tau >n-1.\) On the other hand, thanks to the pioneering works of Brjuno [12], the Diophantine conditions can be weakened to the Brjuno conditions. To make it applicable in KAM scheme, Rüssmann [44, 45] introduced the notion of an approximation function to characterize the Brjuno conditions. Under such Brjuno–Rüssmann type conditions, Pöschel [40] proved the persistence of elliptic lower dimensional tori in finite dimensional Hamiltonian systems. In [41], Pöschel also proved the existence of infinite dimensional invariant tori in infinite dimensional Hamiltonian systems of the form \(H=\omega \cdot I+P(\theta ,I).\) Later on, Xu and You [49] and Chavaudret and Marmi [14] proved the reducibility of linear ODEs with almost periodic coefficients and quasi-periodic cocycles under such Brjuno–Rüssmann type conditions, respectively. See also [31, 46, 47] for nonlinear forced ODEs. We also mention some Brjuno type quasi-periodic results of Corsi and Gentile [15] and Gentile [27] for forced non-Hamiltonian ODEs without using approximation function.
To the best of our knowledge, there has been no Brjuno–Rüssmann type results in KAM theory for PDEs. In this paper, we establish a reducibility theorem for some linear Hamiltonian PDEs under Brjuno-Rüssmann non-resonance conditions. More precisely, we consider the following linear quasi-periodic derivative wave equations
and linear quasi-periodic half-wave equations
under Dirichlet boundary conditions, where the Fourier multiplier \(\textbf{D}_m:=\sqrt{-\partial _{xx}+m}.\) The basic fequencies \(\omega \) of the potential V satisfy the Brjuno–Rüssmann non-resonance conditions. The wave Eq. (1.3) covers the variational equation around any small amplitude quasi-periodic solutions of nonlinear Hamiltonian derivative wave equation \(\partial _{tt}u-\partial _{xx}u+mu+ f(\textbf{D}_mu)=0,\) where \(f(z)=az^3+O(z^5),\,a\ne 0.\) Quasi-periodic solutions with Diophantine frequencies of this nonlinear wave equation under periodic boundary conditions have been obtained in [10]. The half-wave Eq. (1.4) is an important class of PDEs arising in various physical problems [13, 18, 24, 32, 38]. There are two main difficulties when studying the reducibility theory of the Eqs. (1.3) and (1.4). The first one is the weak dispersion relation since the eigenvalues \(\lambda _j\sim j,\,j\rightarrow \infty .\) The second one is the bad smoothness of the perturbations. To overcome this, we introduce a simplified version of Töplitz–Lipschitz functions and Töplitz–Lipschitz matrices, which were first proposed by Eliasson and Kuksin [20] in KAM theory for the higher dimensional Schrödinger equations. Such simplified form is more suitable to the Eqs. (1.3) and (1.4) and it was also used in [25, 26]. Different from that in [25, 26], we characterize the Töplitz–Lipschitz functions in a way of semi-norm. We also mention the quasi-Töplitz functions introduced in [10] for nonlinear Hamiltonian derivative wave equations, which is also an improved version of Eliasson–Kuksin’s Töplitz–Lipschitz functions. Comparing to the quasi-Töplitz functions, our simplified form is more easy to handle. For further work on the reduction of linear operators involving weak dispersion relations, please refer to references [21,22,23].
To state our main results, we introduce some definitions and assumptions on the potentials V in the Eqs. (1.3) and (1.4).
Definition 1.1
(Approximation function, [40, 45]). A non-decreasing function
is called an approximation function, if
and
in addition, the normalization \(\Delta (0)=1\) is imposed for definition.
Remark 1.1
Below we list three typical approximation functions: \(\Delta _1=\exp (t^\alpha /\alpha ),\,\,0<\alpha <1,\) \(\Delta _2=\exp \left( \frac{t}{1+\log ^\alpha (1+t)}\right) ,\,\,\alpha >1\) and \( \Delta _3=\exp \left( \frac{t}{\log ^\alpha t}\right) ,\,\,\alpha >1.\)
Definition 1.2
(Brjuno–Rüssmann frequency) Let \(\Delta \) be an approximation function. A vector \(\omega \in \mathbb {R}^n\) is called Brjuno–Rüssmann frequency vector if it satisfies
for some constant \(\gamma >0.\)
Assumption 1
Suppose the function \(V:\mathbb {T}^n\times [0, \pi ]\rightarrow \mathbb {R}\) is real analytic in \((\theta ,x).\) For \(\theta \in \mathbb {T}^n,\) \(V(\theta , \cdot )\) is a \(2\pi -\)periodic, even function \(V(\theta , x)=V(\theta , -x).\) Then it can be written as
Moreover, suppose for all \(\theta ,\) the function \(V(\theta , \cdot )\) extends to a complex analytic function on a strip \(|Im x|<2a\) for some \(a>0.\) For all x, the function \(V(\cdot , x)\) extends to a complex analytic function on a strip on \(|Im \theta |<2r\) for some \(r>0.\) Then there is a positive constant \(C_V>0\) such that for \(p\ge 0,\)
where the norm \( \Vert \cdot \Vert _{D(2r)}\) is defined in Sect. 2.
Let \(\phi _j(x)=\sqrt{\frac{2}{\pi }}\sin jx,\,j\ge 1\) be the normalized Dirichlet eigenfunctions of the operator \(D^2_m:=-\partial _{xx}+m\) associated to the eigenvalues \(\lambda _j^2=j^2+m,\,j\ge 1\). We consider the Eqs. (1.3) and (1.4) in the following function space
Our main result is stated as follows.
Theorem 1.1
Let \(m\ge 0.\) Under the Assumption 1 on the potential functions V, there is \(\epsilon _0\) so that for all \(0<\epsilon <\epsilon _0\) there exists \(\mathcal {O}_\epsilon \subseteq [0, 2\pi )^{n}\) of positive Lebesgue measure such that for all \(\omega \in \mathcal {O}_\epsilon \) satisfying Brjuno–Rüssmann non-resonance conditions, the above linear quasi-periodic wave Eq. (1.3) and half-wave Eq. (1.4) reduce to the linear equations with constant coefficients with respect to the time variable.
In Sect. 5, we prove this theorem by the reducibility Theorem 4.1.
As a corollary of Theorem 1.1, we have the following conclusion concerning the solutions of the Eqs. (1.3) and (1.4):
Corollary 1.1
Let the initial data \(u_0\in \mathcal {H}^{a,p}_0,\) \(v_0\in \mathcal {H}^{a,p-1 }_0.\) Under the Assumption 1, there is \(\epsilon _0\) so that for all \(0<\epsilon <\epsilon _0\) and \(\omega \in \mathcal {O}_\epsilon \),
(i) there exists a unique solution \((u(t,x), u_t(t,x))\in \mathcal {H}^{a,p}_0\times \mathcal {H}^{a,p-1 }_0\) of the wave Eq. (1.3) with \((u(0,x), u_t(0,x))=(u_0, v_0).\) Moreover, u(t, x) is almost-periodic in time and stable, i.e.,
\( \forall t \in \mathbb {R},\) for some constant \(C=C(a,p,\omega )>0.\)
(ii) there exists a unique solution \(u(t,x)\in \mathcal {H}^{a,p}_0\) of the half-wave Eq. (1.4) with \(u(0,x)=u_0.\) Moreover, u(t, x) is almost-periodic in time and stable, i.e.,
for some constant \(C=C(a,p,\omega )>0.\)
Remark 1.2
More recently, using the Renormalization Group method under Brjuno-type conditions without employing an approximation function, Corsi et al. [16] have constructed almost-periodic solutions with Gevrey regularity for the NLS equation with a convolution potential of arbitrarily high regularity.
2 Functional Setting
Let \(\mathcal {O} \subset \mathbb {R}^n\) be a parameter set of positive Lebesgue measure. Throughout the paper, for any real or complex valued function depending on parameters \(\xi \in \mathcal {O},\) its derivatives with respect to \(\xi \) are understood in the sense of Whitney. We denote by \(C^1_W(\mathcal {O})\) the class of \(C^1\) Whitney differentiable functions on \(\mathcal {O}.\)
Suppose \(f\in C^1_W(\mathcal {O}),\) we define its norm as
where \(|\cdot |\) denotes the sup-norm of complex vectors.
Given an n-torus \(\mathbb {T}^n=\mathbb {R}^n/{(2\pi \mathbb {Z})^n}\) and its complex neighborhood
Consider a real analytic function \(f(\theta ;\xi )\) on \(\theta \in D(r).\) It is also \(C^1_W\) on \(\xi \in \mathcal {O}.\) Its Fourier expansion reads \(f(\theta ;\xi )=\sum _{k\in \mathbb {Z}^n}\widehat{f}(k;\xi )\textrm{e}^{\textrm{i}k\cdot \theta },\) then we define its norm as
where \(k\cdot \theta =\sum ^{n}_{i=1}k_i\theta _i\) and \(|k|=\sum ^{n}_{i=1}|k_i|\).
Let \(K>0.\) For \(f(\theta ;\xi )\) above, its \(K-\)order Fourier truncation \(\mathcal {T}_Kf\) is defined as
The remainder \(\mathcal {R}_Kf\) of f is defined by \((\mathcal {R}_Kf)(\theta ):=f(\theta )-\mathcal {T}_Kf(\theta ).\) Suppose \(0<2\sigma <r,\) we have the following estimate for \(\mathcal {R}_Kf:\)
The average [f] of f on \(\mathbb {T}^n\) is defined as
Let \(a, p> 0\), we introduce the Banach space \(\ell ^{a,p}\) of all real or complex sequences \(z=(z_j)_{j\in \mathbb {Z}}\) with
Given \(r, s>0,\) we define the phase space
and a complex neighborhood
of \(\mathcal {T}^{n}_0:=\mathbb {T}^{n}\times \{I=0\}\times \{z=0\}\times \{\bar{z}=0\}\) in \(\mathcal {P}^{a,p}_{\mathbb {C}}:=\mathbb {C}^n \times \mathbb {C}^n\times \ell ^{a,p}\times \ell ^{a,p}.\)
Consider a real analytic function \(f(\theta ,I, z,\bar{z}; \xi )\) on D(r, s), which is also \(C^1_W\) on \(\xi \in \mathcal {O}.\) Its Taylor–Fourier expansion reads
where we use the multi-index notations \(l=(l_j)^n_{j=1},\) \(\alpha =(\alpha _j)_{j\ge 1},\) \(\beta =(\beta _j)_{j\ge 1}\) with \(l_j, \alpha _j,\beta _j\in \mathbb {N}.\) \(\alpha \) and \(\beta \) have only finitely many nonzero components. \(I^lz^{\alpha }\bar{z}^{\beta }=(\prod ^n_{i=1}I^{l_i}_i )(\prod _{j\in \mathbb {Z}}z^{\alpha _j}_j\bar{z}^{\beta _j}_j).\)
We define the majorant of f as
and the norm of f as
Consider an infinite dimensional dynamical system on D(r, s) :
where the vector field
Suppose vector field \(X(w;\xi )\) is real analytic on D(r, s) and \(C^1_W\) smooth on \(\mathcal {O},\) we define the weighted norm of X as follows
3 Töplitz–Lipschitz Functions
3.1 Definitions
In this section, we introduce a class of real analytic functions with exponentially off-diagonal decay.
Definition 3.1
Let \(r, s, \rho >0.\) Suppose \(P(\theta ,z,\bar{z}; \xi )\) is real analytic on \((\theta ,z,\bar{z})\in D(r,s)\) and \(C^1_W-\)smooth on parameters \(\xi \in \mathcal {O}.\) We say that P is Töplitz–Lipschitz and write \(P\in \mathcal {T}^\rho _{ D(r,s)\times \mathcal {O}}\) if
where the semi-norm \(\langle P\rangle _{\rho ,D(r,s)\times \mathcal {O}}\) is the smallest non-negative real number that satisfies the following conditions
(T1) Exponentially off-diagonal decay.
(T2) Asymptotically Töplitz. The limits
exist and are finite for all \(i,j\in \mathbb {Z}.\)
(T3) Lipschitz at infinity. For sufficiently large \(|t|, t\in \mathbb {Z},\) the following hold.
Remark 3.1
By the definition of \(\langle P\rangle _{\rho ,D(r,s)\times \mathcal {O}}\), it is not difficult to verify that
-
\(\langle P\rangle _{\rho ,D(r,s)\times \mathcal {O}}\ge 0;\)
-
\(\langle \lambda P\rangle _{\rho ,D(r,s)\times \mathcal {O}}=|\lambda |\langle P\rangle _{\rho ,D(r,s)\times \mathcal {O}}\) for all \(\lambda \in \mathbb {C};\)
-
\(\langle P+F\rangle _{\rho ,D(r,s)\times \mathcal {O}}\le \langle P\rangle _{\rho ,D(r,s)\times \mathcal {O}}+ \langle F\rangle _{\rho ,D(r,s)\times \mathcal {O}}.\)
Note that \(\langle P\rangle _{\rho ,D(r,s)\times \mathcal {O}}= 0\) could not imply \(P=0.\) This means \(\langle \cdot \rangle _{\rho ,D(r,s)\times \mathcal {O}}\) is only a semi-norm.
Remark 3.2
From (T1) and (T3), the limits in (T3) satisfy
Remark 3.3
By the definition of the semi-norm \(\langle \cdot \rangle _{\rho ,D(r,s)\times \mathcal {O}}\), it is not difficult to verify that the following conclusions hold:
- (1):
-
\(\langle P\rangle _{\rho ,D(r',s')\times \mathcal {O}}\le \langle P\rangle _{\rho ,D(r,s)\times \mathcal {O}}\) if \(0<r'\le r,\, 0<s'\le s;\)
- (2):
-
\(\langle P\rangle _{\rho ',D(r,s)\times \mathcal {O}}\le \langle P\rangle _{\rho ,D(r,s)\times \mathcal {O}}\) if \(0< \rho '\le \rho ;\)
- (3):
-
Let \(K>0,\) then the Fourier truncation \(\mathcal {T}_KP\) of P satisfies
$$\begin{aligned} \langle \mathcal {T}_KP\rangle _{\rho ,D(r,s)\times \mathcal {O}}\le \langle P\rangle _{\rho ,D(r,s)\times \mathcal {O}} \end{aligned}$$and the remainder \(\mathcal {R}_KP\) of P satisfies
$$\begin{aligned} \langle \mathcal {R}_KP\rangle _{\rho ,D(r',s)\times \mathcal {O}}\le \textrm{e}^{-K(r-r')}\langle P\rangle _{\rho ,D(r,s)\times \mathcal {O}} \end{aligned}$$if \(0< r'\le r.\)
Definition 3.2
Let \(\ell ^{a,p}_{0}\) be the unilateral infinite sequences space defined by
Given a real analytic function \(P(\theta , z, \bar{z})\) with \((z, \bar{z})\in \ell ^{a,p}_{0}\times \ell ^{a,p}_{0},\) we lift it from \(\ell ^{a,p}_{0}\times \ell ^{a,p}_{0}\) to \(\ell ^{a,p}\times \ell ^{a,p} \) by \(\widetilde{P}(\theta , \tilde{z}, \bar{\tilde{z}})= P(\theta , z, \bar{z}),\) where \((\tilde{z}, \bar{\tilde{z}})\in \ell ^{a,p}\times \ell ^{a,p}\) and \(\tilde{z}=z_j, \bar{\tilde{z}}=\bar{z}_j\) for all \( j\ge 1.\)
We say that the function P is Töplitz–Lipschitz and write \(P\in \mathcal {T}^\rho _{ D(r,s)\times \mathcal {O}}\) if \(\widetilde{P}(\theta , \tilde{z}, \bar{\tilde{z}})\) is Töplitz–Lipschitz and define
Below we focus on a class of quadratic functions on \((z, \bar{z})\) of the form
We study the Töplitz–Lipschitz property for these functions under the action of the Poisson bracket, the flow of linear Hamiltonian system and the canonical transformation.
Proposition 3.1
(Poisson bracket). Let \(0<\delta <{\min \{\rho ,1\}}\). Suppose the quadratic functions \(R,\,F\in \mathcal {T}^\rho _{ D(r,s)\times \mathcal {O}},\) then \(\{R,F\}\in \mathcal {T}^{\rho -\delta }_{ D(r,s)\times \mathcal {O}}\) and there exists a constant \(C>0\) so that
Proof
The Poisson bracket \(\{R, F\}\) reads
In what follows, it remains to analyze the second derivative \(\frac{\partial ^2 \{R, F\}}{\partial z_i \partial \bar{z}_j}\) with respect to \(z_i, \bar{z}_j,\) and the other second derivatives could be similarly done.
Since the functions R and F are both quadratic on \((z,\bar{z}),\) their third derivatives vanish. Then we have
\(\bullet \) We first verify the property (T1) for \(\frac{\partial ^2 \{R, F\}}{\partial z_i \partial \bar{z}_j}.\) It suffices to consider the sums \(\sum _{k\ge 1}\frac{\partial ^2 R}{\partial z_k \partial \bar{z}_j} \frac{\partial ^2 F}{\partial z_i \partial \bar{z}_k}\) and \(\sum _{k\ge 1}\frac{\partial ^2 R}{\partial z_i \partial z_k} \frac{\partial ^2 F}{\partial \bar{z}_k \partial \bar{z}_j}\) in (3.14), and the others can be similarly done.
Since the functions R and F satisfy the property (T1), then we have
and
here we use the inequality \(\sum _{k}\textrm{e}^{-\delta (|i-k|+|k-j|)}\le C\delta ^{-1} \) ( see Lemma 7.5, Appendix).
\(\bullet \) We then verify the property (T2) for \(\frac{\partial ^2 \{R, F\}}{\partial z_i \partial \bar{z}_j}.\) From the above analysis, we know that the functional series \(\sum _{k\ge 1}\frac{\partial ^2 R}{\partial z_k \partial \bar{z}_j} \frac{\partial ^2 F}{\partial z_i \partial \bar{z}_k}\) and \(\sum _{k\ge 1}\frac{\partial ^2 R}{\partial z_i \partial z_k} \frac{\partial ^2 F}{\partial \bar{z}_k \partial \bar{z}_j}\) converge uniformly on \(D(r,s)\times \mathcal {O}.\) Since the limits \( \lim _{t\rightarrow \infty } \frac{\partial ^2 P}{\partial z_{i+t} \partial \bar{z}_{j+t}}\), \(\lim _{t\rightarrow \infty } \frac{\partial ^2 P}{\partial z_{i+t} \partial z_{j-t}}\) and \(\lim _{t\rightarrow \infty } \frac{\partial ^2 P}{\partial \bar{z}_{i+t} \partial z_{j-t}}\) exist and are finite, then the limits
and
also exist and are finite. This implies the property (T2) holds for \(\frac{\partial ^2 \{R, F\}}{\partial z_i \partial \bar{z}_j}.\)
\(\bullet \) Finally, we verify the property (T3) for \(\frac{\partial ^2 \{R, F\}}{\partial z_i \partial \bar{z}_j}.\) For the sake of convenience, we introduce the notations
and
In view of \(R,\,F\in \mathcal {T}^\rho _{ D(r,s)\times \mathcal {O}}\) and thanks to the difference equality
and the inequality in Lemma 7.5, we have
and
These imply that
\(\square \)
3.2 Töplitz–Lipschitz Matrices
Denote by \(\mathcal {M}_{2}(\mathbb {C})\) the space of \(2\times 2\) complex matrices. Let \(\Vert \cdot \Vert \) be any sub-multiplicative norm on \(\mathcal {M}_{2}(\mathbb {C}).\) Consider a bilateral infinite dimensional \(\mathcal {M}_{2}(\mathbb {C})-\)valued matrix
The matrix multiplication is defined by \((AB)_{ij}=\sum _{k\in \mathbb {Z}} A_{ik}B_{kj}.\)
Now we consider the matrices depend on \((\theta , \xi )\in D(r)\times \mathcal {O}.\)
Definition 3.3
(Matrices with Töplitz–Lipschitz property) Let \(r, \rho >0.\) We say that a matrix \(A=A(\theta , \xi )\) on \(D(r)\times \mathcal {O}\) is Töplitz–Lipschitz and write \(A\in \mathfrak {M}^{\rho }_{r,\mathcal {O}}\) if \(\langle \langle A\rangle \rangle _{\rho ,r,\mathcal {O}}<\infty , \) where the norm \(\langle \langle A\rangle \rangle _{\rho ,r,\mathcal {O}}\) is defined by the following conditions:
(\(\textbf{T1}^{\prime }\)) Exponentially off-diagonal decay
(\(\textbf{T2}^{\prime }\)) Asymptotically Töplitz The limits
exist and are finite for all \(i,j\in \mathbb {Z}.\)
(\(\textbf{T3}^{\prime }\)) Lipschitz at infinity For sufficiently large \(|t|,\,\,t\in \mathbb {Z},\) the following hold.
Definition 3.4
Given a unilateral infinite dimensional \(\mathcal {M}_{2}(\mathbb {C})-\)valued matrix
we lift it from \(\mathbb {N}\times \mathbb {N}\) to \(\mathbb {Z}\times \mathbb {Z} \) by
We say that A is Töplitz–Lipschitz and write \(A\in \mathfrak {M}^{\rho }_{r,\mathcal {O}}\) if \(\tilde{A}\) is Töplitz–Lipschitz and define
The following conclusion indicate that \(\mathfrak {M}^{\rho }_{r,\mathcal {O}}\) is an algebra. This important property will be applied to Proposition 3.4.
Proposition 3.2
Let \(0<\delta < \rho .\) Suppose the matrices \(A,B\in \mathfrak {M}^{\rho }_{r,\mathcal {O}}.\) Then their product \(AB\in \mathfrak {M}^{\rho -\delta }_{r,\mathcal {O}}\) and there exists a constant \(C>0\) so that
The proof is given in Sect. 7.2, Appendix.
3.3 Flow of Linear Hamiltonian System
In this section, we study the Hamiltonian flow generated by a quadratic Töplitz–Lipschitz function \(F(\theta ,z,\bar{z};\xi )\in \mathcal {T}^\rho _{ D(r,s)\times \mathcal {O}}.\)
In the sequel, we use the notations \(Z=(Z_j)^T_{j\in \mathbb {Z}}\) with \(Z_j=(z_j, \bar{z}_j)^T.\) The Hessian \(\partial ^2_{Z}F\) of F with respect to Z reads
where
Denote \(A=J\partial ^2_{Z}F,\) where
then
By the definitions of Töplitz–Lipschitz function and Töplitz–Lipschitz matrix, they have the following relation.
Lemma 3.3
Let \(\rho >0.\) Suppose \(F(\theta , z, \bar{z},; \xi )\) is a quadratic function on \(D(r,s)\times \mathcal {O}.\) Then \(F\in \mathcal {T}^\rho _{ D(r,s)\times \mathcal {O}}\) if and only if \(A=J\partial ^2_{Z}F\in \mathfrak {M}^{\rho }_{r,\mathcal {O}}.\) Moreover,
The Hamiltonian equation associated to the quadratic function F reads
Under the new notation Z, the quadratic function \(F\in \mathcal {T}^\rho _{ D(r,s)\times \mathcal {O}}\) can be rewritten as
and the Eq. (3.28) reads
The Jacobian \(\partial _{Z^0} Z\) (the derivative of Z(t) with respect to \(Z^0\)) is
Proposition 3.4
Let \(0<\delta <{\min \{\rho ,1\}}\) and \(0<\sigma <r/3.\) Suppose \(Cs\sigma \le \ln 2\) and \(Cs^2 \le 2\) and the quadratic function \(F\in \mathcal {T}^\rho _{ D(r,s)\times \mathcal {O}}\) and
Then the solution \((\theta (t), I(t),Z(t))\) of the Eq. (3.30) with initial condition \((\theta ^0,I^0,Z^0)\in D(r-\sigma ,\frac{s}{4})\) satisfies \((\theta (t), I(t), Z(t))\) \(\in D(r,\frac{s}{2})\) for all \(0\le t\le 1.\) Moreover, the Jacobian \(\partial _{Z^0} Z(t)\) satisfies
where the notation Id is the identity mapping.
Proof
Since \(\dot{\theta }(t)=0,\) then \(\theta (t)\equiv \theta ^0\in D(r-\sigma )\) remains unchanged.
Consider the equation for Z :
It is a linear system with constant coefficients, thus its solution is
Thus thanks to \(Cs\sigma \le \ln 2\), for all \(0\le t \le 1,\)
Consider the equation for I. By (3.30) and (3.34), we have
The integral form of the above Eq. (3.35) is
Then thanks to \(Cs^2 \le 2,\) for all \(0\le t \le 1,\)
Thus the flow \(X^t_{F}\) exists for all \(0\le t\le 1\) and it maps the domain \(D(r-\sigma ,\frac{s}{4})\) to \(D(r,\frac{s}{2})\). Denote the solution \((\theta (t), I(t), z(t), \bar{z}(t))=X^t_{F}(\theta ^0, I^0, z^0, \bar{z}^0),\) then for \(0 \le t\le 1\) and \((\theta ^0, I^0, z^0, \bar{z}^0)\in D(r-\sigma ,\frac{s}{4}),\) the solution \((\theta (t), I(t), z(t), \bar{z}(t))\in D(r,\frac{s}{2}).\)
Now we prove the estimate (3.32). Rewrite the solution Z(t) in (3.34) as
where
By Proposition 3.2 and Lemma 3.3, for all \(0\le t \le 1,\)
This completes the proof of the estimate (3.32). \(\square \)
Proposition 3.5
(Canonical transformation). Let \(0<\delta <\min \{\rho /3,1\},\) \(0<\sigma <r\) and \(R,\,F\in \mathcal {T}^\rho _{ D(r,s)\times \mathcal {O}},\) where the Hamiltonian F is a quadratic function. Assume that the Hamiltonian F satisfies (3.31). Then the composition \(R\circ X^1_F\in \mathcal {T}^{\rho -3\delta }_{ D(r-\sigma ,s/4)\times \mathcal {O}}\) and there exists a constant \(C>0\) so that
Proof
By Proposition 3.4, the time-1 mapping \(X^1_F\) maps \((\theta ^0,I^0,Z^0)\in D(r-\sigma ,\frac{s}{4})\) to \((\theta , I, Z):=X^1_F(\theta ^0,I^0,Z^0)\in D(r,\frac{s}{2}).\)
Since the mapping Z is linear in \(Z^0,\) the Hessian \(\partial ^2_{Z^0} Z =0.\) Then the Hessian \(\partial ^2_{Z^0}(R\circ X^1_F)\) of \(R\circ X^1_F\) with respect to \(Z^0\) becomes
Note that \(\langle \langle J^T(\partial _{Z^0} Z)^TJ\rangle \rangle _{\rho ,r}=\langle \langle \partial _{Z^0} Z\rangle \rangle _{\rho ,r}\) and \(J^T = J^{-1} = -J\), then by Lemma 3.3 and Proposition 3.4, we have
\(\square \)
4 A Reducibility Theorem Under Brjuno Condition
Consider the following quadratic Hamiltonian with time quasi-periodic perturbation:
where \((z,\bar{z})\in \ell ^{a,p}_{0}\times \ell ^{a,p}_{0},\) the space \(\ell ^{a,p}_{0}\) is the unilateral infinite sequences space defined in (3.11). The forcing frequency vector \(\omega \in [0, 2\pi )^{n}\) and the normal frequencies \(\Omega _j\in \mathbb {R}\) for all \(j\ge 1.\) Then the associated linear Hamiltonian system reads
Introducing the angle variables \(\theta =\omega t\in \mathbb {T}^n,\) and the auxiliary action variables \(I\in \mathbb {R}^n,\) then we obtain an autonomous Hamiltonian system
on the phase space \(\mathcal {P}^{a,p}_0:=\mathbb {T}^n \times \mathbb {R}^n\times \ell ^{a,p}_0\times \ell ^{a,p}_0\) with respect to the symplectic form
The new Hamiltonian is
Given \(s,r>0,\) in the following, we investigate Hamiltonian (4.4) on the domain \(D(r,s) \subseteq \mathcal {P}^{a,p}_{0,\mathbb {C}}.\) The forcing frequency \(\omega \in [0, 2\pi )^{n}\) will play the role of parameters. Suppose \(H(\theta ,I, z,\bar{z};\omega )\) in (4.4) is real analytic on \((\theta ,I, z,\bar{z};\omega )\) and \(C^1_W-\)smooth in compact subset \(\mathcal {O}\subseteq [0, 2\pi )^{n}\) with positive Lebesgue measure. Furthermore, suppose Hamiltonian (4.4) satisfies the following assumptions.
(A1) Asymptotics of normal frequencies:
where \( \breve{\Omega }_j\in C^1_W(\mathcal {O})\) and there exist positive constants \(A_0,\) such that \(\sup _{j\ge 1,\omega \in \mathcal {O}}|\breve{\Omega }_j|\le A_0\). \(\sup _{j\ge 1}\sup _{\omega \in \mathcal {O}}|\partial _\omega \breve{\Omega }_j|\le \varepsilon _0\).
(A2) Non-resonance conditions: There exist a constant \(0<\gamma \le 1\) and some fixed approximation function \(\Delta \) such that uniformly on \(\mathcal {O},\) for all \((k,l)\in \mathbb {Z}^n\times \mathbb {Z}^\infty {\setminus }\{0\},\)
where \(|k|=|k_1|+\cdots +|k_n|\), \(|l|=\sum _{j}|l_j|.\)
(A3) Regularity: The Hamiltonian vector field \(X_P=(0, -P_\theta , \textrm{i}P_{\bar{z}}, -\textrm{i}P_z)^T\) of perturbation P defines a map
\(X_P(\cdot ;\omega )\) is real analytic in D(r, s) for each \(\omega \in \mathcal {O},\) and \(P(\chi ;\cdot )\) is \(C^1_W-\)smooth in \(\mathcal {O}\) for each \(\chi \in D(r,s).\)
(A4) Töplitz–Lipschitz property: \(\breve{\Omega }:=diag(\breve{\Omega }_j)_{j\ge 1}\in \mathfrak {M}^{\rho }_{r,\mathcal {O}}\) and \(P\in \mathcal {T}^\rho _{ D(r,s)\times \mathcal {O}}\) for some \(\rho >0.\)
Denote
Theorem 4.1
Let \(\Delta \) be an approximation function such that
If the Hamiltonian \(H=N+P\) in (4.4) satisfies the above assumptions (A1)–(A4) and there exists \(0< \varepsilon _0 < \min \{\frac{\gamma }{4}(\sqrt{\Delta (1)}-1), (C_*\gamma 2^5)^{\frac{3}{2}},\frac{1}{12n}\}\) so that
Then there exist
- (i):
-
a Cantor subset \(\mathcal {O}_\gamma \subset \mathcal {O}\) with Lebesgue measure \({{\,\textrm{mes}\,}}(\mathcal {O}\setminus \mathcal {O}_\gamma )=O(\sqrt{\gamma })\) as \(\gamma \rightarrow 0\);
- (ii):
-
a \(C^1_W-\)smooth family of real analytic, symplectic coordinate transformations \({\Phi =\Phi _\omega }: \mathcal {P}^{a,0}_{0}\times \mathcal {O}_\gamma \rightarrow \mathcal {P}^{a,0}_{0}\) of the form
$$\begin{aligned} \Phi _{\omega }\left( \begin{array}{c} \theta \\ I\\ Z \\ \end{array} \right) = \left( \begin{array}{c} \theta \\ I+\frac{1}{2}Z^TM_{\omega }(\theta )Z\\ L_{\omega }(\theta )Z \\ \end{array} \right) \end{aligned}$$(4.9)where \(Z=(Z_j)^T_{j\ge 1}\) with \(Z_j=(z_j, \bar{z}_j)^T.\) \(M_{\omega }(\theta )\) and \(L_{\omega }(\theta )\) are linear bounded operators on \(\ell ^{a,p}_{0} \times \ell ^{a,p}_{0}\) for all \(p\ge 0\), and \(L_{\omega }(\theta )\) is also invertible;
- (iii):
-
a \(C^1_W-\)smooth family of new normal forms
$$\begin{aligned} N^{\infty }=\sum _{j=1}^n\omega _jI_j+\sum _{j\ge 1}\Omega ^{\infty }_j z_j\bar{z}_j \end{aligned}$$(4.10)such that on \(\mathcal {P}^{a,0}_{0}\times \mathcal {O}_\gamma ,\)
$$\begin{aligned} H\circ \Phi =N^{\infty }. \end{aligned}$$Moreover the new normal frequencies are close to the original ones
$$\begin{aligned} |\Omega ^{\infty }-\Omega |_{\mathcal {O}_\gamma } \le c \varepsilon , \end{aligned}$$and the the new frequencies satisfy a non-resonant condition: for all \(\omega \in {\mathcal {O}_\gamma },\)
$$\begin{aligned} | k\cdot \omega |\ge & \frac{\gamma }{2\Delta (|k|)},\quad \forall k\ne 0,\\ | k\cdot \omega +l\cdot \Omega ^{\infty }(\omega )|\ge & \frac{\gamma }{2\Delta (|k|)},\quad \forall k\in \mathbb {Z}^n,\quad |l|=2. \end{aligned}$$
5 Applications to Some Linear Hamiltonian PDEs
We give the proof of Theorem 1.1 by Theorem 4.1.
5.1 The Hamiltonian Derivative Wave Equations
We consider the wave Eq. (1.3). Let
Then the Eq. (1.3) is written as a non-autonomous Hamiltonian equation
with the Hamiltonian
Recall the function space \(\mathcal {H}^{a,p}_0\) in (1.10). Through the inverse discrete Fourier transform \(\mathcal {S}:\ell ^{a,p}_0\rightarrow \mathcal {H}^{a,p}_0\), the space \(\mathcal {H}^{a,p}_0\) can be identified with the space \(\ell ^{a,p}_0.\)
We expand \(w(t, x),\,\, \bar{w}(t, x)\) on the eigenfunctions
with \(q=(q_j)_{j\ge 1},\, \bar{q}=(\bar{q}_j)_{j\ge 1} \in \ell ^{a,p}_0.\) Then the Eq. (4.3) becomes
where
Now we introduce the angle variables \(\theta =\omega t\in \mathbb {T}^n,\) the auxiliary action variables \(I\in \mathbb {R}^n\) and the complex coordinates \(z=(z_j)_{j\ge 1},\,\, \bar{z}=(\bar{z}_j)_{j\ge 1}\) via letting \(z_j = -q_j,\,\, \bar{z}_j = -\bar{q}_j.\) Then we obtain an autonomous Hamiltonian system
on the phase space \(\mathcal {P}^{a,p}_0\) with respect to the symplectic form
The Hamiltonian associated to the system (5.4) is
where
In the following, we check that the Hamiltonian (5.5) satisfies the assumptions (A1)–(A4). Let r be that in Assumption 1.1 and \(s>0\) be a suitable positive number. Take \(\varepsilon _0=(2^{p+1}+2^{4}+ \frac{18 n}{r})C_V\epsilon >0.\)
- (1):
-
Verifying the assumption (A1). Since \(\lambda _j =\sqrt{j^2+m}=j+\frac{m}{2j}-\frac{m^2}{8j^3}+\cdots ,\) then we take \(\Omega _j=j+\breve{\Omega }_j=j+O(\frac{1}{j}).\) Note that \(\breve{\Omega }_j\) does not depend on \(\omega \in [0, 2\pi )^{n},\) thus \(\partial _\omega \breve{\Omega }_j=0\) and \( \breve{\Omega }_j\in C^1_W([0, 2\pi )^{n}).\) Take \(A_0=1+m.\) Since \(\breve{\Omega }_j=O(\frac{1}{j})\) and \(\partial _\omega \breve{\Omega }_j=0,\) then for all \(j\ge 1\) and \(\omega \in [0, 2\pi )^{n},\) we have \(|\breve{\Omega }_j|\le A_0\) and \(|\partial _\omega \breve{\Omega }_j|\le \varepsilon _0.\)
- (2):
-
Verifying the assumption (A2). Take the vector \(v=({{\,\textrm{sgn}\,}}(k_1),\ldots ,{{\,\textrm{sgn}\,}}(k_n))\) then \(k\cdot v=|k|.\) Let \(\omega =\omega _\mu =\mu v+w\) with \(\mu \in \mathbb {R},\) \(w\in v^{\perp }.\) Consider the function \(f(\mu )=k\cdot \omega _\mu + l\cdot \Omega =|k|\mu +k\cdot w + l\cdot \Omega .\) Thanks to \(\partial _\omega \Omega =0,\) we have
$$\begin{aligned} | f'(\mu )|=|k|. \end{aligned}$$By Lemma 7.6 in Appendix, we have
$$\begin{aligned} {{\,\textrm{mes}\,}}\{\mu :\mu v+w\in [0, 2\pi )^{n}, |f(\mu )|\le \delta \}\le \frac{4\delta }{|k|}. \end{aligned}$$It follows that the measure
$$\begin{aligned}&{{\,\textrm{mes}\,}}\left\{ \omega \in [0, 2\pi )^{n} :|k\cdot \omega + l\cdot \Omega |\le \frac{\gamma }{\Delta (|k|)},|l|=0,2\right\} \nonumber \\&\quad \le {{\,\textrm{diam}\,}}^{n-1}([0, 2\pi )^{n}){{\,\textrm{mes}\,}}\left\{ \mu :\mu v+w\in [0, 2\pi )^{n}, |f(\mu )|\le \frac{\gamma }{\Delta (|k|)}\right\} \nonumber \\&\quad \le (2\pi )^{n(n-1)}\frac{4\gamma }{|k|\Delta (|k|)}. \end{aligned}$$(5.7)Thus there is a subset \(\mathcal {O}\subset [0, 2\pi )^{n}\) of positive Lebesgue measure with \({{\,\textrm{mes}\,}}\mathcal {O}\ge (2\pi )^{n}(1-O(\gamma ))\) such that the assumption(A2) holds on \(\mathcal {O}.\)
- (3):
-
Verifying the assumption (A3). The perturbation P in (5.6) reads
$$\begin{aligned} P(\theta , z, \bar{z})=\frac{\epsilon }{2}\sum \limits _{ij\ge 1}p^{20}_{ij}(\theta ) z_iz_j + \epsilon \sum \limits _{ij\ge 1}p^{11}_{ij}(\theta ) z_i\bar{z}_j+\frac{\epsilon }{2}\sum \limits _{ij\ge 1}p^{02}_{ij}(\theta ) \bar{z}_i\bar{z}_j , \end{aligned}$$where
$$\begin{aligned} p^{20}_{ij}(\theta )=p^{11}_{ij}(\theta )=p^{02}_{ij}(\theta )&=\int ^{\pi }_{0} V(\theta ,x)\phi _i(x)\phi _j(x)dx \nonumber \\&= {\left\{ \begin{array}{ll} \frac{1}{2}(\widetilde{V}_{i-j}(\theta )- \widetilde{V}_{i+j}(\theta )), \quad i>j, \\ \widetilde{V}_{0}(\theta ) -\frac{1}{2} \widetilde{V}_{2j}(\theta ), \quad i=j, \\ \frac{1}{2}(\widetilde{V}_{j-i}(\theta )- \widetilde{V}_{i+j}(\theta )), \quad i<j. \end{array}\right. } \end{aligned}$$(5.8)Now we investigate the regularity of the perturbation vector field\(X_P=(0, -\frac{\partial P}{\partial \theta }, \textrm{i}\frac{\partial P}{\partial \bar{z}}, -\textrm{i}\frac{\partial P}{\partial z}).\) Note that the vector field \(X_P\) does not depend on \(\omega .\) For the above \(r, s>0,\) we estimate the vector field norm
$$\begin{aligned}&\Vert X_P\Vert _{s;D(r,s)\times \mathcal {O}}\\&=\frac{1}{s^2}\sum \limits ^\text {n}_{h=1}\left\| \frac{\partial P}{\partial \theta _h}\right\| _{D(r,s)\times \mathcal {O}} +\frac{1}{s}\sup \limits _{\Vert z\Vert _{a,p}<s, \Vert \bar{z}\Vert _{a,p}<s}\sum \limits ^{\infty }_{i=1}i^{p}\textrm{e}^{ai }\left( \left\| \frac{\partial P}{\partial \bar{z}_i}\right\| _{D(r)\times \mathcal {O}} +\left\| \frac{\partial P}{\partial z_i}\right\| _{D(r)\times \mathcal {O}}\right) . \end{aligned}$$\(\bullet \) We first estimate the sum
$$\begin{aligned}&\sum \limits ^\text {n}_{h=1}\left\| \frac{\partial P}{\partial \theta _h}\right\| _{D(r,s)\times \mathcal {O}}\\&\quad =\epsilon \sup \limits _{\Vert z\Vert _{a,p}<s,\Vert \bar{z}\Vert _{a,p}<s}\sum \limits ^\text {n}_{h=1} \sum \limits _{i, j\ge 1}\left( \frac{1}{2}\Vert \frac{\partial p^{20}_{ij}}{\partial \theta _h}\Vert _{D(r)}|z_i||z_j|\right. \\&\qquad \left. +\Vert \frac{\partial p^{11}_{ij}}{\partial \theta _h}\Vert _{D(r)}|z_i||\bar{z}_j|+\frac{1}{2}\Vert \frac{\partial p^{02}_{ij}}{\partial \theta _h}\Vert _{D(r)}|\bar{z}_i||\bar{z}_j|\right) \\&\quad \le \frac{n \epsilon }{r}\sup \limits _{\Vert z\Vert _{a,p}<s, \Vert \bar{z}\Vert _{a,p}<s} \left( \sum \limits _{i, j\ge 1}\Vert p^{20}_{ij}\Vert _{D(2r)}|z_i||z_j| \right. \\&\qquad \left. +\sum \limits _{i, j\ge 1}\Vert p^{11}_{ij}\Vert _{D(2r)}|z_i||\bar{z}_j|+\sum \limits _{i, j\ge 1}\Vert p^{02}_{ij}\Vert _{D(2r)}|\bar{z}_i||\bar{z}_j|\right) . \end{aligned}$$For this purpose, it suffices to estimate each of three sums on the last line:
$$\begin{aligned}&\sum \limits _{i, j\ge 1}\Vert p^{11}_{ij}\Vert _{D(2r)}|z_i||\bar{z}_j|\\&\quad =\sum \limits _{j\ge 1} \Vert p^{11}_{jj}\Vert _{D(2r)}|z_j||\bar{z}_j| +\sum \limits _{j\ge 1} \sum \limits _{1\le i\le j-1} \Vert p^{11}_{ij}\Vert _{D(2r)}|z_i||\bar{z}_j| +\sum \limits _{j\ge 1} \sum \limits _{i\ge j+1} \Vert p^{11}_{ij}\Vert _{D(2r)}|z_i||\bar{z}_j|\\&\quad \le \sum \limits _{j\ge 1} (\Vert \widetilde{V}_{0}(\theta )\Vert _{D(2r)}+\Vert \widetilde{V}_{2j}(\theta )\Vert _{D(2r)})|z_j||\bar{z}_j|\\&\qquad +\sum \limits _{j\ge 1} \sum \limits _{1\le i\le j-1} (\Vert \widetilde{V}_{j-i}\Vert _{D(2r)}+\Vert \widetilde{V}_{j+i}\Vert _{D(2r)})|z_i||\bar{z}_j|\\&\qquad +\sum \limits _{j\ge 1} \sum \limits _{i\ge j+1} (\Vert \widetilde{V}_{i-j}\Vert _{D(2r)}+\Vert \widetilde{V}_{j+i}\Vert _{D(2r)})|z_i||\bar{z}_j|\\&\quad \le 6C_V\Vert z\Vert _{a,p}\Vert \bar{z}\Vert _{a,p}. \end{aligned}$$Similarly, we have
$$\begin{aligned} \sum \limits _{i, j\ge 1}\Vert p^{20}_{ij}\Vert _{D(2r)}|z_i||z_j|\le 6C_V\Vert z\Vert _{a,p}\Vert z\Vert _{a,p} \end{aligned}$$and
$$\begin{aligned} \sum \limits _{i, j\ge 1}\Vert p^{02}_{ij}\Vert _{D(2r)}|\bar{z}_i||\bar{z}_j|\le 6C_V\Vert \bar{z}\Vert _{a,p}\Vert \bar{z}\Vert _{a,p}. \end{aligned}$$This shows that
$$\begin{aligned} \frac{1}{s^2}\sum \limits ^\text {n}_{h=1}\Vert \frac{\partial P}{\partial \theta _h}\Vert _{D(r,s)\times \mathcal {O}}\le \frac{18 n}{r}C_V \epsilon . \end{aligned}$$(5.9)\(\bullet \) We turn to the estimate for
$$\begin{aligned} \frac{1}{s}\sup \limits _{\Vert z\Vert _{a,p}<s,\Vert \bar{z}\Vert _{a,p}<s}\sum \limits ^{\infty }_{i=1}i^{p}\textrm{e}^{ai }\left( \left\| \frac{\partial P}{\partial \bar{z}_i}\right\| _{D(r)\times \mathcal {O}} +\left\| \frac{\partial P}{\partial z_i}\right\| _{D(r)\times \mathcal {O}}\right) . \end{aligned}$$It suffices to consider
$$\begin{aligned} \sum \limits ^{\infty }_{i=1}i^{p}\textrm{e}^{ai }\left\| \frac{\partial P}{\partial z_i}\right\| _{D(r)\times \mathcal {O}} =\epsilon \sum \limits ^{\infty }_{i=1}\sum \limits ^{\infty }_{j=1}i^{p}\textrm{e}^{ai }\Vert p^{11}_{ij}\Vert _{D(r)}|z_j|+\epsilon \sum \limits ^{\infty }_{i=1}\sum \limits ^{\infty }_{j=1}i^{p}\textrm{e}^{ai }\Vert p^{02}_{ij}\Vert _{D(r)}|\bar{z}_j|. \end{aligned}$$By (5.8),
$$\begin{aligned}&\sum \limits ^{\infty }_{i=1}\sum \limits ^{\infty }_{j=1}i^{p}\textrm{e}^{ai }\Vert p^{11}_{ij}\Vert _{D(r)}|z_j|\\&\quad =\sum \limits _{j\ge 1}j^{p}\textrm{e}^{aj } \Vert \widetilde{V}_{0}(\theta ) -\frac{1}{2} \widetilde{V}_{2j}(\theta )\Vert _{D(2r)}|z_j|\,\,\,\cdots (*1)\\&\qquad +\sum \limits _{j\ge 1} \sum \limits _{1\le i\le j-1} i^{p}\textrm{e}^{ai }\Vert \frac{1}{2}(\widetilde{V}_{j-i}(\theta )- \widetilde{V}_{i+j}(\theta ))\Vert _{D(2r)}|z_j|\,\,\,\cdots (*2)\\&\qquad +\sum \limits _{j\ge 1} \sum \limits _{i\ge j+1}i^{p}\textrm{e}^{ai } \Vert \frac{1}{2}(\widetilde{V}_{i-j}(\theta )- \widetilde{V}_{i+j}(\theta ))\Vert _{D(2r)}|z_j|\,\,\,\cdots (*3),\\ \end{aligned}$$where
$$\begin{aligned} (*1)&=\sum \limits _{j\ge 1}j^{p}\textrm{e}^{aj } \Vert \widetilde{V}_{0}(\theta ) -\frac{1}{2} \widetilde{V}_{2j}(\theta )\Vert _{D(2r)}|z_j| \le 2\Vert V\Vert _{D(2r),b,p}\Vert z\Vert _{a,p}.\\ (*2)&=\sum \limits _{j\ge 1} \sum \limits _{1\le i\le j-1} i^{p}\textrm{e}^{ai }\Vert \frac{1}{2}(\widetilde{V}_{j-i}(\theta )- \widetilde{V}_{i+j}(\theta ))\Vert _{D(2r)}|z_j|\\&\le \sum \limits _{j\ge 1} \sum \limits _{1\le i\le j-1} i^{p}\textrm{e}^{ai }\frac{1}{2} \Vert V\Vert _{D(2r),b,p} (j-i)^{-p} \textrm{e}^{-b(j-i)} |z_j| \\&\quad + \sum \limits _{j\ge 1} \sum \limits _{1\le i\le j-1} i^{p}\textrm{e}^{ai }\frac{1}{2} \Vert V\Vert _{D(2r),b,p} (j+i)^{-p} \textrm{e}^{-b(j+i)} |z_j|\\&\le \frac{1}{2} \Vert V\Vert _{D(2r),b,p} \sum \limits _{j\ge 1} \sum \limits _{1\le i\le j-1} \left( \frac{i}{j-i}\right) ^p \textrm{e}^{ai } \textrm{e}^{-b(j-i)} |z_j| \\&\quad +\frac{1}{2} \Vert V\Vert _{D(2r),b,p} \sum \limits _{j\ge 1} \sum \limits _{1\le i\le j-1} \left( \frac{i}{j+i}\right) ^p \textrm{e}^{ai } \textrm{e}^{-b(j+i)} |z_j|\\&\le \frac{1}{2} \Vert V\Vert _{D(2r),b,p} \sum \limits _{j\ge 1} |z_j| j^p 2\textrm{e}^{aj} +\frac{1}{2} \Vert V\Vert _{D(2r),b,p} \sum \limits _{j\ge 1} 2 |z_j| \\&\le 2C_V\Vert z\Vert _{a,p}.\\ (*3)&=\sum \limits _{j\ge 1} \sum \limits _{i\ge j+1}i^{p}\textrm{e}^{ai } \Vert \frac{1}{2}(\widetilde{V}_{i-j}(\theta )- \widetilde{V}_{i+j}(\theta ))\Vert _{D(2r)}|z_j|\\&\le \sum \limits _{j\ge 1} \sum \limits _{i\ge j+1} i^{p}\textrm{e}^{ai }\frac{1}{2} \Vert V\Vert _{D(2r),b,p} (i-j)^{-p} \textrm{e}^{-b(i-j)} |z_j| \\&\quad + \sum \limits _{j\ge 1} \sum \limits _{i\ge j+1} i^{p}\textrm{e}^{ai }\frac{1}{2} \Vert V\Vert _{D(2r),b,p} (j+i)^{-p} \textrm{e}^{-b(j+i)} |z_j|\\&\le \frac{1}{2} \Vert V\Vert _{D(2r),b,p} \sum \limits _{j\ge 1} \sum \limits _{i\ge j+1} \left( \frac{i}{i-j}\right) ^p \textrm{e}^{ai } \textrm{e}^{-b(i-j)} |z_j| \\&\quad +\frac{1}{2} \Vert V\Vert _{D(2r),b,p} \sum \limits _{j\ge 1} \sum \limits _{i\ge j+1} \left( \frac{i}{j+i}\right) ^p \textrm{e}^{ai } \textrm{e}^{-b(j+i)} |z_j|\\&\le (2^p +2)C_V\Vert z\Vert _{a,p}. \end{aligned}$$Then
$$\begin{aligned} \sum \limits ^{\infty }_{i=1}\sum \limits ^{\infty }_{j=1}i^{p}\textrm{e}^{ai }\Vert p^{11}_{ij}\Vert _{D(r)}|z_j|\le (*1)+(*2)+(*3) \le (2^p +6)C_V\Vert z\Vert _{a,p}. \end{aligned}$$By the similar argument, we get
$$\begin{aligned} \sum \limits ^{\infty }_{i=1}\sum \limits ^{\infty }_{j=1}i^{p}\textrm{e}^{ai }\Vert p^{02}_{ij}\Vert _{D(r)}|\bar{z}_j|\le (2^p +6)C_V\Vert \bar{z}\Vert _{a,p}. \end{aligned}$$It follows that
$$\begin{aligned} \frac{1}{s}\sup \limits _{\Vert z\Vert _{a,p}<s,\Vert \bar{z}\Vert _{a,p}<s}\sum \limits ^{\infty }_{i=1}i^{p}\textrm{e}^{ai }\left( \Vert \frac{\partial P}{\partial \bar{z}_i}\Vert _{D(r)\times \mathcal {O}} +\Vert \frac{\partial P}{\partial z_i}\Vert _{D(r)\times \mathcal {O}}\right) \le 2(2^p +6)C_V\epsilon . \end{aligned}$$(5.10)We conclude from (5.9) and (5.10) that
$$\begin{aligned}&\Vert X_P\Vert _{s;D(r,s)\times \mathcal {O}}\\&\quad =\frac{1}{s^2}\sum \limits ^\text {n}_{h=1}\Vert \frac{\partial P}{\partial \theta _h}\Vert _{D(r,s)\times \mathcal {O}} +\frac{1}{s}\sup \limits _{\Vert z\Vert _{a,p}<s, \Vert \bar{z}\Vert _{a,p}<s}\sum \limits ^{\infty }_{i=1}i^{p}\textrm{e}^{ai }\left( \Vert \frac{\partial P}{\partial \bar{z}_i}\Vert _{D(r)\times \mathcal {O}} +\Vert \frac{\partial P}{\partial z_i}\Vert _{D(r)\times \mathcal {O}}\right) \\&\quad \le (2^{p+1}+12+ \frac{18 n}{r})C_V\epsilon \le \varepsilon _0. \end{aligned}$$Thus we complete the verification of the regularity for \(X_P.\)
- (4):
-
Verifying the assumption (A4). \(\bullet \) We verify \(\breve{\Omega }:=diag(\breve{\Omega }_j)_{j\ge 1}\) satisfies Töplitz–Lipschitz property. During the verification of the assumption (A1), we have obtained \(|\breve{\Omega }_j|\le \frac{C_0}{j},\) where \(C_0\) is a constant depending on m. It is evident that \(\lim _{t\rightarrow \infty }\breve{\Omega }_{j+t}=0\) and
$$\begin{aligned} & \left\| \lim _{t\rightarrow \infty }\breve{\Omega }_{j+t}\right\| _{\mathcal {O}}\le C_0.\\ & \left\| \breve{\Omega }_{j+t}- \lim _{t\rightarrow \infty }\breve{\Omega }_{j+t}\right\| _{\mathcal {O}}=\left\| \breve{\Omega }_{j+t}- \lim _{t\rightarrow \infty }\breve{\Omega }_{j+t}\right\| _{\mathcal {O}}\le \frac{C_0}{|j+t|}\le \frac{C_0}{|t|}. \end{aligned}$$\(\bullet \) Taking \(\rho =2a,\) we verify the perturbation \(P\in \mathcal {T}^\rho _{ D(r,s)\times \mathcal {O}}.\) We first consider \(\frac{\partial ^2 P}{\partial z_i \partial \bar{z}_j}.\) By (5.8), we have for \(t\ge 1,\)
$$\begin{aligned} \frac{\partial ^2 P}{\partial z_{i+t} \partial \bar{z}_{j+t}}=\epsilon p^{11}_{i+t,j+t}(\theta ) = {\left\{ \begin{array}{ll} \frac{\epsilon }{2}(\widetilde{V}_{i-j}(\theta )- \widetilde{V}_{i+j+2t}(\theta )), \quad i>j ,\\ \epsilon \widetilde{V}_{0}(\theta ) -\frac{\epsilon }{2} \widetilde{V}_{2j+2t}(\theta ), \quad i=j , \\ \frac{\epsilon }{2}(\widetilde{V}_{j-i}(\theta )- \widetilde{V}_{i+j+2t}(\theta )), \quad i<j. \end{array}\right. } \end{aligned}$$Due to \( \Vert \widetilde{V}_{j}\Vert _{D(2r)}\le C_V \textrm{e}^{-2aj}, j\ge 1, \) the limit \(\lim _{t\rightarrow \infty }\frac{\partial ^2 P}{\partial z_{i+t} \partial \bar{z}_{j+t}}\) exists and
$$\begin{aligned} \lim \limits _{t\rightarrow \infty }\frac{\partial ^2 P}{\partial z_{i+t} \partial \bar{z}_{j+t}} = {\left\{ \begin{array}{ll} \frac{\epsilon }{2}\widetilde{V}_{i-j}(\theta ), \quad i>j ,\\ \epsilon \widetilde{V}_{0}(\theta ) , \quad i=j , \\ \frac{\epsilon }{2}\widetilde{V}_{j-i}(\theta ), \quad i<j. \end{array}\right. } \end{aligned}$$Moreover,
$$\begin{aligned} \left\| \lim \limits _{t\rightarrow \infty } \frac{\partial ^2 P}{\partial z_{i+t} \partial \bar{z}_{j+t}}\right\| _{D{(r,s)}\times \mathcal {O}} \le \epsilon \left\| \widetilde{V}_{|i-j|}(\theta )\right\| _{D{(r,s)}\times \mathcal {O}} \le \varepsilon _0 \textrm{e}^{-\rho |i-j|}. \end{aligned}$$Thanks to the exponentially decay of \(\widetilde{V}_{j},\) we also have
$$\begin{aligned}&\left\| \frac{\partial ^2 P}{\partial z_{i+t} \partial \bar{z}_{j+t}}-\lim \limits _{t\rightarrow \infty }\frac{\partial ^2 P}{\partial z_{i+t} \partial \bar{z}_{j+t}}\right\| _{D{(r,s)}\times \mathcal {O}}\\&\quad = {\left\{ \begin{array}{ll} \frac{\epsilon }{2}\left\| \widetilde{V}_{i+j+2t}(\theta ))\right\| _{D{(r,s)}\times \mathcal {O}}, \quad i>j ,\\ \frac{\epsilon }{2} \left\| \widetilde{V}_{2j+2t}(\theta )\right\| _{D{(r,s)}\times \mathcal {O}}, \quad i=j , \\ \frac{\epsilon }{2}\left\| \widetilde{V}_{i+j+2t}(\theta ))\right\| _{D{(r,s)}\times \mathcal {O}}, \quad i<j, \end{array}\right. }\\&\quad \le \frac{\epsilon }{2} \Vert V\Vert _{D(2r),b,p} \textrm{e}^{-2a(i+j+2t)} \le \frac{\varepsilon _0}{t}\textrm{e}^{-\rho |i-j|}. \end{aligned}$$where we use the inequality \(\textrm{e}^{-2a(i+j+2t)}=\textrm{e}^{-2a(i+j)}\textrm{e}^{-2 t}\le \frac{1}{t}\textrm{e}^{-2a|i-j|}.\) As to the second derivative \(\frac{\partial ^2 P}{\partial z_i \partial z_j}=\frac{\epsilon }{2} p^{20}_{ij}(\theta ),\) we consider the lift \(\widetilde{P}(\theta , \tilde{z}, \bar{\tilde{z}})= P(\theta , z, \bar{z}),\) where \((\tilde{z}, \bar{\tilde{z}})\in \ell ^{a,p}\times \ell ^{a,p}\) and \(\tilde{z}=z_j, \bar{\tilde{z}}=\bar{z}_j\) when \( j\ge 1.\) (recall the Definition 3.2). Then
$$\begin{aligned} \frac{\partial ^2 \widetilde{P}}{\partial \tilde{z}_i \partial \tilde{z}_j}= {\left\{ \begin{array}{ll} \frac{\partial ^2 P}{\partial z_i \partial z_j}, i\ge 1,j\ge 1,\\ 0,\,\, \hbox {otherwise}.\\ \end{array}\right. } \end{aligned}$$(5.11)When |t| is sufficiently large, we have either \(i+t<0\) or \(j-t<0,\) then \( \frac{\partial ^2 \widetilde{P}}{\partial \tilde{z}_{i+t} \partial \tilde{z}_{j-t}}=0\) and thus the limit \(\lim _{t\rightarrow \infty }\frac{\partial ^2 \widetilde{P}}{\partial \tilde{z}_{i+t} \partial \tilde{z}_{j-t}}=0.\) It is obvious that
$$\begin{aligned} \left\| \lim \limits _{t\rightarrow \infty } \frac{\partial ^2 \widetilde{P}}{\partial \tilde{z}_{i+t} \partial \tilde{z}_{j-t}}\right\| _{D{(r,s)}\times \mathcal {O}} \le \varepsilon _0 \textrm{e}^{-\rho |i+j|} \end{aligned}$$and
$$\begin{aligned} \left\| \frac{\partial ^2 \widetilde{P}}{\partial \tilde{z}_{i+t} \partial \tilde{z}_{j-t}}-\lim \limits _{t\rightarrow \infty }\frac{\partial ^2 \widetilde{P}}{\partial \tilde{z}_{i+t} \partial \tilde{z}_{j-t}}\right\| _{D{(r,s)}\times \mathcal {O}} \le \frac{\varepsilon _0}{t}\textrm{e}^{-\rho |i+j|}. \end{aligned}$$Similar argument also applies to the second derivative \(\frac{\partial ^2 P}{\partial \bar{z}_i \partial \bar{z}_j}.\) It follows that \(P\in \mathcal {T}^\rho _{ D(r,s)\times \mathcal {O}}\) and \(\langle P\rangle _{\rho ,D(r,s)\times \mathcal {O}}\le \varepsilon _0.\)
5.2 The Half-Wave Equations
Denote the inner product \(\langle u,v\rangle =Re\int ^\pi _0u(x)\overline{v(x)}dx.\) The half-wave Eq. (1.4) can be written as
where the Hamiltonian
We expand u(t, x) on the eigenfunctions
(see (1.10) on the space \(\mathcal {H}^{a,p}_0\)) where \(q=(q_j)_{j\ge 1} \in \ell ^{a,p}_0.\) Then the Eq. (5.12) becomes
where
To rewrite the above equation as an autonomous Hamiltonian system, we introduce the angle variables \(\theta =\omega t\in \mathbb {T}^n,\) the action variables \(I\in \mathbb {R}^n\) and the complex coordinates \(z=(z_j)_{j\ge 1},\,\, \bar{z}=(\bar{z}_j)_{j\ge 1}\) through
Then we obtain an autonomous Hamiltonian system
on the phase space \(\mathcal {P}^{a,p}_0\) with respect to the symplectic form
The new Hamiltonian associated to the system (5.14) is
where
The next is the verification of the assumptions (A1)–(A4) for the Hamiltonian (5.15). Let r be that in Assumption 1.1 and \(s>0\) be a suitable positive number. Take \(\varepsilon _0=(2^{p+1}+2^{4}+ \frac{n}{2r})C_V\epsilon >0.\)
- (1):
-
Verifying the assumption (A1). Since \(\lambda _j =j,\) then we take \(\Omega _j=j+\breve{\Omega }_j\) with \(\breve{\Omega }_j=0,\) thus \( \breve{\Omega }_j\in C^1_W([0, 2\pi )^{n}).\) Let \(A_0=1.\) It is obvious that for all \(j\ge 1\) and \(\omega \in [0, 2\pi )^{n},\) \(|\breve{\Omega }_j|\le A_0\) and \(|\partial _\omega \breve{\Omega }_j|\le \varepsilon _0.\)
- (2):
-
Verifying the assumption (A2). Following the verification of the assumption (A2), we can also prove that there is a subset \(\mathcal {O}\subset [0, 2\pi )^{n}\) of positive Lebesgue measure with \({{\,\textrm{mes}\,}}\mathcal {O}\ge (2\pi )^{n}(1-O(\gamma ))\) such that the assumption (A2) holds for (5.15) on \(\mathcal {O}.\)
- (3):
-
Verifying the assumption (A3). The perturbation P in (5.15) reads
$$\begin{aligned} P=\varepsilon \sum \limits _{ij\ge 1}p_{ij}(\theta ) z_i\bar{z}_j, \end{aligned}$$(5.16)where
$$\begin{aligned} p_{ij}(\theta )&:=\int ^{\pi }_{0} V(\theta ,x)\phi _i(x)\phi _j(x)dx \\&= {\left\{ \begin{array}{ll} \frac{1}{2}(\widetilde{V}_{i-j}(\theta )- \widetilde{V}_{i+j}(\theta )), \quad i>j, \\ \widetilde{V}_{0}(\theta ) -\frac{1}{2} \widetilde{V}_{2j}(\theta ), \quad i=j , \\ \frac{1}{2}(\widetilde{V}_{j-i}(\theta )- \widetilde{V}_{i+j}(\theta )), \quad i<j. \end{array}\right. } \end{aligned}$$Following the arguments in the verification of the assumption (A3) for the wave Eq. (1.3), one can prove that
$$\begin{aligned} \Vert X_P\Vert _{s;D(r,s)\times \mathcal {O}} \le (2^{p+1}+12+\frac{n}{2r})\Vert V\Vert _{D(2r),b,p}\epsilon \le \varepsilon _0. \end{aligned}$$This shows the regularity of Hamiltonian vector field \(X_P.\)
- (4):
-
Verifying the assumption (A4). Let \(\rho =2a.\) Thanks to \(\breve{\Omega }_j\equiv 0,\) it is obvious that \(\breve{\Omega }:=diag(\breve{\Omega }_j)_{j\ge 1}\in \mathfrak {M}^{\rho }_{r,\mathcal {O}}.\) Now we verify \(P\in \mathcal {T}^\rho _{ D(r,s)\times \mathcal {O}}.\) By (5.16), we have
$$\begin{aligned} \frac{\partial ^2 P}{\partial z_i \partial \bar{z}_j}=\epsilon p_{ij}(\theta )\,\, \hbox {and}\,\, \frac{\partial ^2 P}{\partial z_i \partial z_j}=0=\frac{\partial ^2 P}{\partial \bar{z}_i \partial \bar{z}_j}. \end{aligned}$$Following the arguments in verifying the assumption (A4), we have the limit \(\lim _{t\rightarrow \infty }\frac{\partial ^2 P}{\partial z_{i+t} \partial \bar{z}_{j+t}}\) exists. Moreover,
$$\begin{aligned} \left\| \lim \limits _{t\rightarrow \infty } \frac{\partial ^2 P}{\partial z_{i+t} \partial \bar{z}_{j+t}}\right\| _{D{(r,s)}\times \mathcal {O}} \le \varepsilon _0 \textrm{e}^{-\rho |i-j|} \end{aligned}$$and
$$\begin{aligned} \left\| \frac{\partial ^2 P}{\partial z_{i+t} \partial \bar{z}_{j+t}}-\lim \limits _{t\rightarrow \infty }\frac{\partial ^2 P}{\partial z_{i+t} \partial \bar{z}_{j+t}}\right\| _{D{(r,s)}\times \mathcal {O}} \le \frac{\epsilon }{2} \Vert V\Vert _{D(2r),b,p} \textrm{e}^{-b(i+j+2t)} \le \frac{\varepsilon _0}{t}\textrm{e}^{-\rho |i-j|}. \end{aligned}$$This together with \(\frac{\partial ^2 P}{\partial z_i \partial z_j}=0=\frac{\partial ^2 P}{\partial \bar{z}_i \partial \bar{z}_j}\) shows that the perturbation \(P\in \mathcal {T}^\rho _{ D(r,s)\times \mathcal {O}}\) and \(\langle P\rangle _{\rho ,D(r,s)\times \mathcal {O}}\le \varepsilon _0.\)
5.3 Proof of Corollaries 1.1 and‘1.2
Below, we provide the proof for Corollaries 1.1 and 1.2, focusing on the case of the half-wave equation. The same argument applies to the derivative wave equation.
From Theorem 4.1, in the new coordinates \((\theta ^\infty , I^\infty , z^\infty , \bar{z}^\infty ) = \Phi _\omega ^{-1}(\theta , I, z, \bar{z})\), the dynamics are linear with \(I^\infty \) invariant:
As (1.4) is equivalent to system (5.14), the solutions u(t, x) of (1.4) with initial data \(u_0(x) = \sum _{j \ge 1} z_j(0) \phi _j(x)\) read
with
and
Thus,
where
Therefore, the solutions are almost-periodic in time with a non-resonant frequency vector \((\omega , \Omega ^\infty _1,\Omega ^\infty _2,\ldots )\). Furthermore, we observe that \(\psi _j(\omega t, x) e^{i \Omega _j^\infty t}\) solves (1.4) if and only if \(k \cdot \omega + \Omega _j^\infty \) is an eigenvalue of the operator \(K_2\) (above Corollary 1.2). This demonstrates that the spectrum of the Floquet operator \(K_2\) equals \(\{k \cdot \omega + \Omega _j^\infty : k \in \mathbb {Z}^n, j \ge 1\}\), thereby proving Corollary 1.2.
For Corollary 1.1, the key point is that when \( V \) is real analytic and satisfies (1.9), the perturbation \( P \) in (5.15) satisfies Assumption (A3) for all \( p \ge 0 \). That is, \( X_{P} \) maps smoothly from \( \mathcal {P}^{a,p} \) into itself. Therefore, Theorem 4.1 applies for all \( p \ge 2 \), and by (4.9), the canonical transformation \( \Phi \) is close to the identity in the \( \mathcal {P}^{a,p} \)-norm. Since in the new variables, \( (\theta ^\infty , I^\infty , z^\infty , \bar{z}^\infty ) = \Phi _\omega ^{-1}(\theta , I, z, \bar{z}) \), the modulus of \( z^\infty _j \) is invariant. We deduce that there exists a constant \( C \) such that
which in turn implies that
6 Proof of the Reducibility Theorem 4.1
6.1 Basic Strategy
The reducibility Theorem 4.1 is proved by KAM method. We construct a sequence of Hamiltonian \(H=N+P\) of the form (4.4). Suppose the perturbation \(P=O(\varepsilon ),\) then we construct a symplectic coordinate transformation \(\Phi \) such that it transforms \(H=N+P\) into a new Hamiltonian \(H_+=H\circ \Phi =N_++P_+\) with new normal form \(N_+\) and a smaller perturbation \(P_+=O(\varepsilon ^\kappa ),\,\, 1<\kappa <2,\) than the old perturbation P.
The above transformation \(\Phi \) is constructed via the flow \(X^t_F\) generated by a quadratic Hamiltonian F. Taking \(\Phi =X^1_{F}\) and denoting \(R=\mathcal {T}_KP\), then
The new normal form is defined as \(N_+=N+\hat{N}.\) This leads to the following homological equation
where the unknowns are F and \(\hat{N}.\) We solve this homological equation in the next section.
6.2 Solving the Homological Equation
Consider the homological equation
on \(D(r,s)\times \mathcal {O},\) where
with the fixed tangential frequencies \(\omega (\xi )\in \mathbb {R}^n.\) The normal frequencies \(\Omega _j(\xi )\in \mathbb {R},\,j\ge 1\) satisfy (4.5). The Hamiltonian R is a quadratic on \((z,\bar{z})\) of the form
It does not depend on the action variables I and satisfies \(R=\mathcal {T}_KR.\) We define its mean value [R] with respect to \(\theta \) by
In the following, we use the notations
Proposition 6.1
Let \(\gamma >0 \) and \(0<5\sigma <r.\) Suppose N and R satisfy the above conditions (A1)–(A2), then the homological Eq. (6.1) has the unique solutions F and \(\widehat{N}\) satisfying \([F]=0\) and the estimates
where the constant \(\mathcal {C}_0\) depends only on \(A_0.\)
Proof
We look for a Hamiltonian F of the form
Denote \(\omega \cdot \nabla f(\theta ):=\sum ^n_{b=1}\omega _b \frac{\partial f}{\partial \theta _b}.\) We take \(\hat{N}=[R].\) By the comparison of coefficients, the homological Eq. (6.1) is equivalent to the following scalar form: For all \(i,j\ge 1,\)
and
here \(\delta _{ij}=1, \hbox {if}\ i=j,\hbox {and}\ 0, \hbox {otherwise}.\)
Consider the Eq. (6.7). For \(i=j,\) the Eq. (6.7) becomes
then by Fourier expansion,
and we obtain the form solution
For \(i\ne j,\) by Fourier expansion, the Eq. (6.7) becomes
and we obtain the form solution
Now we give the estimate for \( F^{11}_{ij}.\) Denote \(S_{ij}=k\cdot \omega + \Omega _{i}- \Omega _{j}.\) For all \(1\le a\le n,\)
where the constant \(\mathcal {C}_0=\mathcal {C}_0(E,L)\) depends only on E and L.
Then
Similarly, we have
Note that the derivative
then
Similarly,
For each \(1\le b\le n,\) by (6.10), the norm of the derivative \(\frac{\partial F^{11}_{ij}}{\partial \theta _b}\) is
Similarly, we have
and
It follows that
From (6.13), (6.14) and (6.15), we obtain the estimate for the Hamiltonian vector field \(X_F:\)
The estimates of \(X_{\hat{N}}\) follow from the observation that \(\hat{N}_{z\bar{z}}\) is the diagonal of the mean value of \(R_{z\bar{z}}.\) \(\square \)
The above lemma implies the estimate for the Jacobian \(DX_F:\)
Now we verify the Töplitz–Lipschitz property of the solutions of homological Eq. (6.1).
Proposition 6.2
Suppose N and R satisfy the above conditions (A1)–(A2) and \(R\in \mathcal {T}^\rho _{D(r,s)\times \mathcal {O}},\) then there exists a constant \(C:= 5+4\mathcal {C}_0\) such that for any \(0<\sigma <r,\) the solutions F and \(\hat{N}\) of homological Eq. (6.1) are Töplitz–Lipschitz on \(D(r,s)\times \mathcal {O},\) i.e., \(F\in \mathcal {T}^\rho _{D(r-\sigma ,s)\times \mathcal {O}},\,\,\hat{N}\in \mathcal {T}^\rho _{D(r,s)\times \mathcal {O}},\) and
Proof
The estimation of \(\hat{N}\) follows from the observation that \(\hat{N}_{z\bar{z}}\) is the diagonal of the mean value of \(R_{z\bar{z}}.\) In the following, we prove the estimation (6.17).
From (6.10) in the proof of Lemma 6.1, the second derivative of F w.r.t. \( z_i,\,\,\bar{z}_j\) is
\(\bullet \) We first verify the exponentially off-diagonal decay of \(\frac{\partial ^2 F}{\partial z_i \partial \bar{z}_j}.\)
Since \(R\in \mathcal {T}^\rho _{D(r,s)\times \mathcal {O}},\) we have
Then
\(\bullet \)We then check the asymptotically Töplitz property of \(\frac{\partial ^2 F}{\partial z_i \partial \bar{z}_j}.\)
Since \(\Omega _j=j+\breve{\Omega }_j,\,\,j\ge 1,\) and \(\langle \langle \breve{\Omega }\rangle \rangle _{\rho ,r,\mathcal {O}}<\varepsilon _0,\) the limits \(\lim _{t\rightarrow \infty }\breve{\Omega }_{j+t}\) exist and satisfy
Note that
then for all i, j the limits \(\Omega _{i,j,\infty }:=\lim _{t\rightarrow \infty }(\Omega _{i+t}-\Omega _{j+t})\) exist and satisfy the non-resonance conditions
Denote \(S_{ij,\infty }:=k\cdot \omega + \Omega _{i,j,\infty }.\) For \(1\le a\le n, \)
where the constant \(\mathcal {C}_0=\mathcal {C}_0(E,L).\)
Since \(R\in \mathcal {T}^\rho _{D(r,s)\times \mathcal {O}},\) the limit \(R^{11}_{ij,\infty }:=\lim _{t\rightarrow \infty }R^{11}_{i+t,j+t}.\) exists. Consider a similar equation to the Eq. (6.7):
By the non-resonance conditions (6.21), the solution \(F^{11}_{ij,\infty }\) of the above equation exists:
Moreover, similar to the estimate for \(\Vert F^{11}_{ij}\Vert _{D(r-\sigma )\times \mathcal {O}}\) in (6.11), we obtain
thus
\(\bullet \) Finally, we check that \(\frac{\partial ^2 F}{\partial z_i \partial \bar{z}_j}\) is Lipschitz at infinity.
By (6.10) and (6.22), we write the difference \(F^{11}_{i+t,j+t}-F^{11}_{ij,\infty }\) as
where
For \( a=1,\ldots , n,\) the Whitney derivatives of \(\mathcal {F}_{k,ij}(\xi )\) with respect to \(\xi _a\) are
In view of \(\langle \langle \breve{\Omega }\rangle \rangle _{\rho ,r,\mathcal {O}}<\varepsilon _0,\) we have
and for \( a=1,\ldots , n,\)
It follows that
and
Therefore,
This together with \(R\in \mathcal {T}^\rho _{D(r,s)\times \mathcal {O}}\) shows that
Similarly, we have
and
Thus we complete the proof of the estimation (6.17). \(\square \)
6.3 KAM Iteration and Convergence
Let \(C_*\) be a constant that is twice the maximum of all implicit constants used during the KAM step, and it depends only on \(n, A_0\) and \(\rho _0.\)
We take the Hamiltonian \(H=N+P\) in (4.4) as the initial Hamiltonian \(H_0=N_0+P_0\). Similarly, we set other initial quantities as those in Sect. 4. Namely, we set \(r_0=r,\, s_0=s,\, \gamma _0=\gamma ,\, \rho _0=\rho ,\, K_0=K,\,\, \mathcal {O}_{0}=\mathcal {O}.\)
For \(\nu \ge 0\),
Denote
Given \(\sigma >0\) with \(6\sigma <r_0.\) There exists a non-increasing positive sequence
such that
and
see Appendix for the proof. For such a fixed sequence \(\{\sigma _\nu \}\), we define
and
then
The order \(K_\nu \) of Fourier truncation is defined implicitly by
Finally, we set
and denote the domain \(D_\nu =D(r_\nu , s_\nu ).\)
Remark 6.1
Recall that the non-resonance conditions in our KAM iterative steps are of Brjuno-type and are given by a class of approximation functions \(\Delta (t)\). This differs from the usual Diophantine non-resonance conditions, which are given by an explicit power function \(t^\tau \). Thus, some iterative parameters such as perturbation parameters \(\varepsilon _\nu \) and \(K_\nu \) cannot be constructed explicitly but rather implicitly.
Below we provide some heuristic considerations about the construction of \(\varepsilon _\nu \) and \(K_\nu \). Now for some iterative sequences, we drop the index \(\nu \) and write ‘+’ for ‘\(\nu +1\)’ to simplify notation. Suppose a Hamiltonian \(H = N + P\) on D(r, s), where the perturbation P is of size \(\varepsilon \) under the norm \(``[\cdot ]"\) as defined in (3.7). From (6.46), after one iteration step, the new perturbation \(P_+\) on \(D(r_+, s_+)\) is of the form
where \(\kappa = 4/3\) and \(\Gamma (\sigma ) = \Gamma _{23}(\sigma )\). To ensure the iterative scheme follows a Newton-like form, the size \(\varepsilon _+\) of the new perturbation \(P_+\) will be of the form \(\varepsilon _+ \sim \Gamma \varepsilon ^{\kappa }\) with \(\Gamma \sim \Gamma (\sigma )\). Therefore, it is necessary to set up
i.e., \(\textrm{e}^{-K\sigma } \le \delta ^2 \Gamma (\sigma ) \varepsilon ^{1/3}\). Since \(\varepsilon \ll \delta \), we let \(\textrm{e}^{-K\sigma } \sim \Gamma \varepsilon ^{1/2}\), which leads us to define the sequence \(K_\nu \) implicitly as in (6.28). Note that \(\varepsilon _+ \sim \Gamma \varepsilon ^{\kappa }\) gives the sequence \(\varepsilon _\nu \) in (6.26) and (6.27). From (6.27), the definition of the quantity \(\Xi (\sigma )\) in (6.24) is natural.
Lemma 6.3
(Iterative Lemma). Let \(0<\varepsilon _0<\min \{(C_*\gamma _02^5)^{\frac{3}{2}}, \delta ^{12}_{0}, (\gamma _{0}\delta _{0})^{9/2}, {\frac{1}{12n}}\}.\) Given a sequence of parameter domains
Suppose for \(\ell =0,1,\ldots ,\nu ,\) the Hamiltonian \(H_\ell =N_\ell +P_\ell \) are regular on \(D_\ell \times \mathcal {O}_\ell ,\) where the normal forms
with \(\Omega _{\ell ,j}(\omega )=j+\widetilde{\Omega }_{\ell ,j}(\omega )\) satisfies
on \(\mathcal {O}_\ell ,\) and the perturbation \(P_\ell \) satisfies
Then there exists a Whitney smooth family of real analytic symplectic transformations \(\Phi _{\nu +1}: D_{\nu +1} \times \mathcal {O}_{\nu } \rightarrow D_\nu \) satisfying
and a closed subset of \(\mathcal {O}_\nu :\)
where
such that \(\Phi _{\nu +1}\) transforms \(H_\nu \) into
and on the domain \(D_{\nu +1} \times \mathcal {O}_{\nu +1},\) \(N_{\nu +1}\) and \(P_{\nu +1}\) satisfy the conditions \((6.29)_{\nu +1},\) \((6.30)_{\nu +1},\) \((6.31)_{\nu +1}\) and \((6.32)_{\nu +1}.\)
Proof
\(\blacklozenge \) The construction of symplectic transformation \(\Phi _{\nu +1}.\)
Let \(R_{\nu }=T_{K_\nu }P_{\nu }\) be the Fourier truncation of order \(K_{\nu }\) of \(P_{\nu }.\) Using the inequalities
and by Propositions 6.1 and 6.2, under the non-resonance conditions \((6.31)_{\nu },\) the homological equation
has a set of unique solutions \(F=F_{\nu }\) and \(\hat{N}=\hat{N}_{\nu }\) satisfying the estimates
and
Since \(\Gamma _{12}\le \Gamma _{13}\) by the definition and \(\Gamma _{13}\le \sigma \Gamma _{23}\) by Lemma 7.1 in Appendix, we have
Then by Lemma 3.4, the flow \(X^t_{F_{\nu }}\) generated by the Hamiltonian vector field \(X_{F_{\nu }}\) exists on \(D(r_{\nu }-\sigma _{\nu },\frac{s_{\nu }}{4})\) for all \(0\le t\le 1.\) Taking \(\Phi _{\nu +1}=X^1_{F_{\nu }},\) it maps \(D(r_{\nu }-\sigma _{\nu },\frac{s_{\nu }}{4})\) into \( D(r_{\nu },\frac{s_{\nu }}{2}).\)
Now we prove the estimate (6.33). Since \(\varepsilon _\nu \ll 1\) and \(\gamma _{\nu }\) and \(\sigma _\nu \) are both bounded sequences, it follows from (6.4), (6.25) and (6.28) that
By the Cauchy estimate and (6.16), using the same approach as for (6.41), we obtain the estimate
\(\blacklozenge \) The new Hamiltonian \(H_{\nu +1}.\)
Using the Taylor formula together with the homological Eq. (6.35), we define the new Hamiltonian
where
with \(\widehat{R}_{\nu }(t)=(1-t)\hat{N}_{\nu }+tR_\nu .\)
\(\bullet \) The estimation for \(P_{\nu +1}.\)
We first consider the estimation for \(\Vert X_{P_{\nu +1}}\Vert _{s_{\nu +1};D_{\nu +1}\times \mathcal {O}_{\nu +1}}.\) Note that
Then using the classical estimates for the pull-back of a vector field and the Lie bracket of two vector fields (see Sect. 3 in [42]), and by (2.1) and (6.3), we obtain the estimate
We then consider the estimation for \(\langle P_{\nu +1}\rangle ^{\rho _{\nu +1}}_{D_{\nu +1}\times \mathcal {O}_{\nu +1}}.\) Using Remarks 3.1 and 3.3(3), Propositions 3.1 and 3.5, we have:
It follows from (4.7), (6.44), (6.45), (6.26) and (6.28) that
\(\bullet \) The new frequency and non-resonance condition.
In the new normal form \(N_{\nu +1},\) the frequencies \(\Omega _{\nu +1,j}=j+\breve{\Omega }_{\nu +1,j}=\Omega _{\nu ,j}+\widehat{\Omega }_{\nu ,j},\) where \(\widehat{\Omega }_{\nu ,j}=\frac{\partial ^2 \widehat{N}_{\nu }}{\partial z_j \partial \bar{z}_j}.\) Thus
Recall the proof of Proposition 6.1, \(\hat{N}_\nu \) is the average of \(R_\nu =\mathcal {T}_K P_\nu \) from the perturbation \(P_\nu \) with respect to \(\theta \), i.e., \(\hat{N}_\nu = [R_\nu ]\). So, by Remark 3.3 (3),
Then following the definition of the semi-norm \(\langle \hat{N}_{\nu } \rangle _{\rho _{\nu }, D_{\nu } \times \mathcal {O}_{\nu }}\), we have
These imply
Therefore,
\(\blacklozenge \)Finally, we consider the construction of \(\mathcal {O}_{\nu +1}\). It suffices to verify
By the definition of \(\gamma _\nu ,\) \(\Gamma _\nu \) and \(K_\nu ,\) we have
This implies \(\gamma _{\nu }-\gamma _{\nu +1}\ge 2 \varepsilon _\nu \Delta (|k|) \) for all \(0<|k|\le K_\nu ,\) thus
Then after removing the resonance zones for \(K_\nu <|k|\le K_{\nu +1},\) we get a closed set \(\mathcal {O}_{\nu +1}\subseteq \mathcal {O}_{\nu }\) with the desired properties. \(\square \)
The Convergence Proof.
By the iterative Lemma 6.3, we obtain a sequence of decreasing domains \(D_\nu \times \mathcal {O}_\nu \) and symplectic transformations \(\Phi ^{\nu }=\Phi _{1}\circ \Phi _{2} \circ \cdots \circ \Phi _{\nu }:D_\nu \times \mathcal {O}_{\nu -1} \rightarrow D_{\nu -1},\,\nu \ge 1.\) Then by (6.33) and following the arguments in [42], the sequence \(\Phi ^{\nu }\) of symplectic transformations converge uniformly on \(D(r/2) \times \mathcal {O}_\gamma \) to a real analytic torus embedding \(\Phi :\mathbb {T}^n\rightarrow \mathcal {P}^{a,p},\) for which we also need to verify
- (a):
-
the symplectic coordinate transformation \(\Phi \) is of the form given in (4.9);
- (b):
-
the new Hamiltonian eventually reduces to the new normal form, i.e., \(P^\infty =0;\)
- (c):
-
the symplectic coordinate transformation \(\Phi \), which is defined by Theorem 4.1 on each \(\mathcal {P}^{a,p}\), extends to \(\mathcal {P}^{a,0}.\)
In fact, by (3.34) and (3.36) in Section 3.36, the the symplectic coordinate transformation \(\Phi _{\nu }\) at the \(\nu \)th-step has the form the form
In particular, the linear operator \(L_{\nu }(\theta )=\textrm{e}^{JA_{\nu }(\theta )}\) is invertible. Then property (a) is satisfied at each step, and thus we can iterate the process. It follows that the limiting transformation \(\Phi =\Phi _{1}\circ \Phi _{2} \circ \cdots \) also satisfies the property (a). Similar to the initial Hamiltonian, the transformed Hamiltonian is linear in I and quadratic in Z, which implies that the new Hamiltonian eventually reduces to the new normal form, i.e., \(P^\infty =0\).
Since \(\Phi ^{(Z)}\) is a linear symplectomorphism, then following Prop.1.3 [34] by duality, it extends on \(\ell ^{a,p}\times \ell ^{a,p}\) for all \(p\in [-2, 2]\) and thus the conclusion (c) holds if we take \(p=0\).
The sequence of closed subset \(\mathcal {O}_\nu \) converges to a closed set
By the construction of \(\gamma _\nu \) and \(|\Omega _{\nu +1} - \Omega _{\nu }|_{\mathcal {O}}=|\widehat{\Omega }_{\nu }|_{\mathcal {O}}\le \varepsilon _{\nu },\) we have \(|\Omega ^{\infty } - \Omega |_{\mathcal {O}}\le \varepsilon ^{1/2}_{0},\) and thus for all \(\omega \in \mathcal {O}_\gamma ,\)
The measure estimate of \(\mathcal {O} \setminus \mathcal {O}_\gamma \) of bad frequencies is given in the next section.
6.4 Measure Estimate
In this subsection, we complete the Lebesgue measure estimate of the parameter set \(\mathcal {O} \setminus \mathcal {O}_\gamma .\) In the process of constructing iterative sequences, we obtain a decreasing sequence of closed sets \(\mathcal {O}_0\supset \mathcal {O}_1\supset \cdots \) such that \(\mathcal {O}_\gamma =\bigcap _{\nu \ge 0}\mathcal {O}_\nu \) and
where \(\mathcal {R}^{\nu }_{k}, \mathcal {R}^{+,\nu }_{kij}, \mathcal {R}^{-,\nu }_{kij}\) are defined in (6.34).
Below we only consider the most difficult resonance set \(\mathcal {R}^{-,\nu }_{kij}(\gamma _{\nu }).\) The proof for other resonance sets \(\mathcal {R}^{\nu }_{k},\,\,\mathcal {R}^{+,\nu }_{kij}\) are more simple, and thus omitted.
Since \(\Omega _{\nu ,j}=j+\breve{\Omega }_{\nu ,j},\) then by (6.30), there is a constant \(A_1>0\) such that \(|\Omega _{\nu ,i}-\Omega _{\nu ,j}|\ge A_1|i-j|\). Denote \(A_2=(1+2A_1+2A_0)/ A_1.\) Note that when \(|i-j|>A_2|k|,\)
thus in this case there is no small divisor, and in the following it remains to consider the case of \(|i-j|\le A_2|k|.\)
Denote
and introduce the following resonant sets
Lemma 6.4
For \(i,j\ge 1\) with \(|i-j|\le A_2|k|\), there exist \(i',\,\, j'\ge 1\) satisfying \(i'\le 2A_2|k|,\,\, j'\le 2A_2|k|\) and \(t\ge 1\) such that \(i=i'+ t,\,\, j=j'+ t.\) Consequently,
Proof
Without loss of generalization, we assume \(j>i.\) For given \(i,\,\, j,\) choosing a \(t_0\ge 1\) such that \(0\le i-t_0\le A_2|k|.\) Let \(i'=i-t_0\) and \(j'=i'+j-i=j-t_0\), then
It follows that (6.51) holds. \(\square \)
Lemma 6.5
For fixed \(k, i', j',\)
Proof
For \(\omega \in \bigcup _{t>\sqrt{\frac{\Delta (|k|)}{\gamma }}}\mathcal {R}^{-,\nu }_{k,i'+t,j'+t}(\gamma _{\nu }),\) suppose \(\omega \in \mathcal {R}^{-,\nu }_{k,i'+t_0,j'+t_0}(\gamma _{\nu })\) for some \(t_0>\sqrt{\frac{\Delta (|k|)}{\gamma }}.\)
From the Töplitz–Lipschitz property of \(P_\nu \) and \(\breve{\Omega }_\nu \), we conclude that
Then
Thus
We give the estimate of \(\mathcal {Q}^{\nu }_{k,i',j',\infty }.\) Taking the vector \(v=|k|({{\,\textrm{sgn}\,}}(k_1),\ldots ,{{\,\textrm{sgn}\,}}(k_n)),\) then \(k\cdot v=|k|^2.\) Let \(\omega ={\omega _\mu }={\mu } v+w\) with \({\mu }\in \mathbb {R},\) \(w\in v^{\perp }.\) Let
Due to \(\sup _{\omega \in \mathcal {O}}|\lim _{t\rightarrow \infty }\partial _\omega \widetilde{\Omega }_{\nu ,i+t}|\le 3\varepsilon _0\) and \(\varepsilon _0\le \frac{1}{12n},\) the derivative
Then by Lemma 7.6, one has
It follows that, by Fubini’s theorem,
Similarly, for the resonant set \(\mathcal {R}^{-,\nu }_{k,i'+t,j'+t},\) following the argument of estimating \({{\,\textrm{mes}\,}}\left( \mathcal {Q}^{\nu }_{k,i',j',\infty }\right) ,\) we have
Then
Using (6.53) and (6.55), we complete the proof. \(\square \)
Finally, we give the estimate of \({{\,\textrm{mes}\,}}\left( \mathcal {O} {\setminus } \mathcal {O}_\gamma \right) .\)
Lemma 6.6
Let \(\Delta \) be an approximation function satisfying (4.8),i.e., \( \sum _{k\in \mathbb {Z}^n}\frac{1}{\sqrt{\Delta (|k|)}}< \infty .\) Then the total measure of resonant set should be excluded during the KAM iteration is
where the implicit constants in “O" depend only on \(n, A_2, B_0, \Delta \) and are made explicit in the proof.
Proof
By Lemma 6.5,
Then
Consequently, the measure of the set \(\mathcal {O} {\setminus } \mathcal {O}_\gamma \) is
\(\square \)
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Acknowledgements
The author wishes to thank Prof. Jiansheng Geng for valuable comments and suggestions. We also sincerely thank the anonymous reviewers for their valuable suggestions and constructive comments. Their thorough reviews and insightful feedback have significantly enhanced the quality and clarity of our manuscript. We deeply appreciate their time and effort in helping us improve our work. The research was supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11901291) and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20190395).
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Appendix
Appendix
1.1 Some Properties of Approximation Functions
Lemma 7.1
For all integers \(k\ge 1,\) \(l\ge 0,\)
where \(\Gamma _{k}(\sigma )=\Gamma _{k3}(\sigma )=\sup _{t\ge 0}(1+t)^k\Delta ^3(t)\textrm{e}^{-t\sigma }.\)
Proof
Let
Its derivative
If \(\sigma (1+t)\le 1,\) then
It follows that \((1+t)^k\Delta ^3(t)\textrm{e}^{-t\sigma }\) arrive at its supremum at some point \(t_*\) with \(\sigma (1+t_*)\ge 1.\) Therefore, for all \(l\ge 0,\)
\(\square \)
Recall \(\Xi \) defined in (6.24):
where \(\Gamma (\sigma )=\Gamma _{2}(\sigma )=\Gamma _{23}(\sigma )=\sup _{t\ge 0}(1+t)^2\Delta ^3(t)\textrm{e}^{-t\sigma }.\)
Lemma 7.2
The \(\Xi \) defined in (6.24) is finite for all \(\sigma >0.\) In particular, let \(T>0,\) if
then
Proof
Let \(\delta (t)=\log (1+t)^2\Delta ^3(t)\) and
for \(\nu \ge 0.\) By condition (1.5) and the hypotheses, \(\sigma _0 \ge \sigma _1 \ge \cdots >0\) and
Since \(\delta (t)-\sigma _\nu t\le 0\) for \(t\ge t_\nu ,\) then by condition (1.5) again the supremum of \(\delta (t)-\sigma _\nu t\) is obtained on the interval \([0, t_\nu ]\) and thus smaller than \(\delta (t_\nu ).\) It follows that
in view of the definition of \(\sigma _\nu \) and hence by \(\kappa _\nu t_\nu = T,\)
\(\square \)
Lemma 7.3
provided that \(t^n/\Delta (t)\) as \(t\rightarrow 0.\)
Proof
Note that
Let \(V_n(t)={{\,\textrm{card}\,}}\{k\in \mathbb {N}^n: |k|\le t\}.\) Then By the monotonicity of approximation functions the sum above may be written as a Stieltjes integral
by partial integration. From the proof of Lemma 8.3 in [40],
this prove the lemma. \(\square \)
Lemma 7.4
There are approximation functions \(\Delta \) such that
for all sufficiently large n with some constant K.
Proof
For \(t\le n, \)
for all \(n\ge 1.\) Hence
for every approximation function \(\Delta .\)
For \(t\ge n, \)
Let \(\varphi \) be given by \(\varphi (s)=\log ^2s,\) and define \(\Delta \) by stipulating that \(t\mapsto s=\log \sqrt{\Delta (t)}\) is the inverse function of \(s\mapsto t=s\varphi (s),\) at least for large t and s respectively. Let \(s_n=n/\varphi (n)\) Since
by the monotonicity of \(\varphi ,\) then
For all large n and \(s\ge s_n,\)
Thus, for all large n,
here \(A_n=(1+h_n)^{1+h_n+1/n}.\) The final estimate follows, since \(A_n\rightarrow 1\) as \(n\rightarrow \infty \) and \(nh_n \log n= 4n \log \log n.\) \(\square \)
1.2 Proof of Proposition 3.2
Proof
We only give the proof for the estimate of \((AB)^{(11)}_{ij}\) and \((AB)^{(12)}_{ij}\), the proofs for the estimates of \((AB)^{(21)}_{ij}\) and \((AB)^{(22)}_{ij}\) are similar.
By the matrix multiplication, we have
and
\(\bullet \)Verifying the property (\({\textbf {T1}}^{\prime }\)). In view of \(A,B\in \mathfrak {M}^{\rho }_r\) and the inequality in Lemma 7.5, we have
and
\(\bullet \)Verifying the property (\({\textbf {T2}}^{\prime }\)). In view of \(A, B\in \mathfrak {M}^{\rho }_r,\) then following the verification of Property (\({\textbf {T1}}^{\prime }\)), we have
and
These imply the property (T2) holds.
\(\bullet \)Verifying the property (\({\textbf {T3}}^{\prime }\)). Denote \(A^{11}_{i,j,\infty }:=\lim _{t\rightarrow \infty }A^{11}_{i+t,j+t}\) and \( A^{12}_{i,j,\infty }:=\lim _{t\rightarrow \infty }A^{12}_{i+t,j-t}.\) Similarly for other terms.
Then by the difference equality (3.15) and the inequality in Lemma 7.5, we have
and
\(\square \)
1.3 Some Technical Lemmas
Lemma 7.5
Let \(0<\delta \le 1.\)
where C is a positive constant that does not depend on \(\delta \) and i, j.
Proof
Without loss of generality, we assume \(i \ge j.\)
Since the function \(f(x) = (x+1) \textrm{e}^{-\delta x}, \, (x \ge 0)\) reaches its maximum at \(x = \frac{1}{\delta } - 1\), then if \(0 < \delta \le 1,\)
It follows that the inequality holds. \(\square \)
Lemma 7.6
[45] Let \(f:[a,b]\rightarrow \mathbb {R}\) be a \(q-\)times continuously differentiable function satisfying
for some \(q\in \mathbb {N}\) and \(\beta >0.\) Then we have the estimate
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Lou, Z. Reducibility of Linear Quasi-periodic Hamiltonian Derivative Wave Equations and Half-Wave Equations Under the Brjuno Conditions. J Dyn Diff Equat (2024). https://doi.org/10.1007/s10884-024-10390-7
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DOI: https://doi.org/10.1007/s10884-024-10390-7