Almost global solutions to two classes of 1-d Hamiltonian Derivative Nonlinear Schr\"odinger equations

TL;DR

This paper proves almost global existence for two classes of 1-d Hamiltonian Derivative Nonlinear Schr"odinger equations with unbounded nonlinearities, using Birkhoff normal forms and symmetry properties under periodic boundary conditions.

Contribution

It develops a method to establish long-time solutions for Hamiltonian DNLS equations with unbounded nonlinearities by constructing Birkhoff normal forms and exploiting symmetry.

Findings

Solutions remain small over long time intervals proportional to \\varepsilon^{-r_*}

Most potentials lead to almost global solutions for small initial data

Method handles unbounded nonlinearities in Hamiltonian systems

Abstract

Consider two kinds of 1-d Hamiltonian Derivative Nonlinear Schr\"odinger (DNLS) equations with respect to different symplectic forms under periodic boundary conditions. The nonlinearities of these equations depend not only on $(x,\psi,\bar{\psi})$ but also on $(\psi_x,\bar{\psi}_x)$, which means the nonlinearities of these equations are unbounded. Suppose that the nonlinearities depend on the space-variable $x$ periodically. Under some assumptions, for most potentials of these two kinds of Hamiltonian DNLS equations, if the initial value is smaller than $\varepsilon\ll1$ in $p$-Sobolev norm, then the corresponding solution to these equations is also smaller than $2\varepsilon$ during a time interval $(-c\varepsilon^{-r_*},c\varepsilon^{-r_*})$(for any given positive $r_*$). The main methods are constructing Birkhoff normal forms to two kinds of Hamiltonian systems which have unbounded nonlinearities and using the special symmetry of the unbounded nonlinearities of Hamiltonian functions to obtain a long time estimate of the solution in $p$-Sobolev norm.

Equations

$$\mathbf{i}{\psi }_t$$

$$\psi (t,x)=\psi (t,x+2\pi ).$$

$$F(x+2\pi ,\psi ,\bar{\psi },{\partial }_x\psi ,{\partial }_x\bar{\psi })=F(x,\psi ,\bar{\psi },{\partial }_x\psi ,{\partial }_x\bar{\psi })\text{mbox}andF(x,0,0,0,0)=0.$$

$$\mathbf{i}{\psi }_t-{\psi }_{xx}+V(x)\psi +\frac{\partial H}{\partial \bar{\psi }}(\psi ,\bar{\psi })=0,$$

$$\mathbf{i}{\psi }_t-{\psi }_{xx}+v|\psi {|}^2\psi +\frac{\partial H}{\partial \bar{\psi }}(\psi ,\bar{\psi })=0,$$

$$\Vert \psi (t){\Vert }_{H^p}<C\epsilon ,\text{mbox}forany|t|\le {\epsilon }^{-B}.$$

$$\mathbf{i}{\psi }_t={\partial }_{xx}\psi +V\ast \psi +(\frac{\partial f(\psi ,\bar{\psi })}{\partial \bar{\psi }}{)}_x,x\in [0,2\pi ],$$

$$\mathbf{i}{\psi }_t={\partial }_{xx}\psi +V\ast \psi +\frac{1}{2}{\partial }_x{\partial }_{\bar{\psi }}F(x,\psi ,\bar{\psi })+{\partial }_{\psi \bar{\psi }}F(x,\psi ,\bar{\psi }){\psi }_x,$$

$$\Theta^{0}_{m}:=\bigg{\{}V(x)\in L^{2}([0,2\pi],\mathbb{R})\ \bigg{|}\ \widehat{V}_{j}\cdot\max\{1,|j|^{m}\}\in[-\frac{1}{2},\frac{1}{2}],\ \widehat{V}_{j}=\widehat{V}_{-j},\ \forall\ j\in\mathbb{Z}\bigg{\}}$$

$${\bf i}{\psi}_{t}=\partial_{xx}\psi+V*\psi+{\bf i}\partial_{x}\big{(}\partial_{\bar{\psi}}F(x,\psi,\bar{\psi})\big{)},$$

$$\Theta^{1}_{m}:=\left\{V\in L^{2}([0,2\pi],\mathbb{C})\bigg{|}\ \widehat{V}_{j}\cdot\max\{1,|j|^{m}\}\in[-\frac{1}{2},\frac{1}{2}],\ \forall\ j\in\mathbb{Z}\setminus\{0\},\ \widehat{V}_{0}=0\right\}.$$

$$\left\{\begin{array}[]{ll}\dot{u}_{j}=-{\bf i}\mbox{sgn}^{\theta}(j)\cdot\partial_{\bar{u}_{j}}H^{w_{\theta}}(u,\bar{u}),\\ \\ \dot{\bar{u}}_{j}=\ \ {\bf i}\mbox{sgn}^{\theta}(j)\cdot\partial_{u_{j}}H^{w_{\theta}}(u,\bar{u}),\end{array}\right.\quad j\in\mathbb{Z}^{*}:=\left\{\begin{array}[]{ll}\mathbb{Z}\ \mbox{or}\ \mathbb{Z}\backslash\{0\}&\mbox{when }\theta=0\\ \\ \mathbb{Z}\backslash\{0\}&\mbox{when }\theta=1\end{array}\right.$$

$$H^{w_{\theta }}(u,\bar{u})=H_0^{w_{\theta }}+P^{w_{\theta }}(u,\bar{u}),(u,\bar{u})\in {\mathcal{H}}^p,\theta \in \{0,1\}$$

$$w_{\theta}:=\left\{\begin{array}[]{lll}&{\bf i}\sum_{j\in\mathbb{Z}^{*}}\mathrm{d}u_{j}\wedge\mathrm{d}\bar{u}_{j}&\theta=0,\\ \\ &{\bf i}\sum_{j\in\mathbb{Z}^{*}}\mbox{sgn}(j)\mathrm{d}u_{j}\wedge\mathrm{d}\bar{u}_{j}&\theta=1,\end{array}\right.$$

$$H^{(r,w_{\theta })}=H^{w_{\theta }}\circ {\mathcal{T}}_{w_{\theta }}^{(r)}=H_0^{w_{\theta }}+\underset{P^{(r,w_{\theta })}}{\underbrace{Z^{(r,w_{\theta })}+{\mathcal{R}}^{N(r,w_{\theta })}+{\mathcal{R}}^{T(r,w_{\theta })}}}$$

$$\frac{d\Vert u{\Vert }_p^2}{dt}=\{\Vert u{\Vert }_p^2,H^{(r,w_{\theta })}(u,\bar{u}){\}}_{w_{\theta }}.$$

$$|\{\Vert u{\Vert }_p^2,{\mathcal{R}}^{N(r,w_{\theta })}(u,\bar{u})+{\mathcal{R}}^{T(r,w_{\theta })}(u,\bar{u}){\}}_{w_{\theta }}|\prec R^{r+1}$$

$$|\{\Vert u{\Vert }_p^2,Z^{(r,w_{\theta })}(u,\bar{u}){\}}_{w_{\theta }}|\prec R^{r+1}$$

$$\{u_j{\bar{u}}_{-j}+{\bar{u}}_j{u}_{-j},\Vert u{\Vert }_p^2{\}}_{w_0}=0$$

$$\{u_j{\bar{u}}_{-j}+{\bar{u}}_j{u}_{-j},\Vert u{\Vert }_p^2{\}}_{w_1}\ne 0.$$

$$H^{p}([0,2\pi],\mathbb{C}):=\left\{\ \psi\in L^{2}([0,2\pi],\mathbb{C})\ \bigg{|}\ \frac{\partial^{r}\psi}{\partial x^{r}}\in L^{2}([0,2\pi],\mathbb{C}),\ \forall\ 0\leq r\leq p\ \right\}$$

$$⟨\zeta ,\eta ⟩:=\mathbf{Re}{\int }_0^{2\pi }\zeta \cdot \bar{\eta }\mathrm{d}x,\text{mbox}forany\zeta ,\eta \in L^2([0,2\pi ],\mathbb{C}).$$

$$\dot{\xi }=A\xi +f(\xi )$$

$$\alpha (A\xi +f(\xi ),\eta )=-⟨dH(\xi ),\eta ⟩\text{mbox}forany\xi \in O_p,\eta \in T{O}_p(T{O}_p\text{mbox}isthetangentspaceof{O}_p).$$

$$A:H^p([0,2\pi ],\mathbb{C})\times H^p([0,2\pi ],\mathbb{C})\to H^{p-d_A}([0,2\pi ],\mathbb{C})\times H^{p-d_A}([0,2\pi ],\mathbb{C})$$

$$f:H^p([0,2\pi ],\mathbb{C})\times H^p([0,2\pi ],\mathbb{C})\to H^{p-d_f}([0,2\pi ],\mathbb{C})\times H^{p-d_f}([0,2\pi ],\mathbb{C}).$$

$$\mathbf{i}{\psi }_t={\partial }_{xx}\psi +V\ast \psi +\mathbf{i}f(x,\psi ,\bar{\psi },{\psi }_x,{\bar{\psi }}_x),\psi \in H^p([0,2\pi ],\mathbb{C})$$

$$\psi (x,t)=\psi (x+2\pi ,t),$$

$$\Theta^{0}_{m}:=\left\{V(x)=\sum_{j\in\mathbb{Z}}\widehat{V}_{j}e^{{\bf i}jx}\in L^{2}([0,2\pi],\mathbb{R})\ \bigg{|}\ v^{w_{0}}_{j}:=\widehat{V}_{j}\langle j\rangle^{m}\in[-\frac{1}{2},\frac{1}{2}],\ v^{w_{0}}_{j}=v^{w_{0}}_{-j},\ \forall\ j\in\mathbb{Z}\right\}$$

$$f(x,\psi ,\bar{\psi },{\psi }_x,{\bar{\psi }}_x)=\frac{1}{2}{\partial }_x{\partial }_{\bar{\psi }}F(x,\psi ,\bar{\psi })+{\partial }_{\psi \bar{\psi }}F(x,\psi ,\bar{\psi }){\psi }_x.$$

Full text

Keyword. Derivative Nonlinear Schrödinger (DNLS) equations, Hamiltonian systems, unbounded, long time stability, momentum, Birkhoff normal form

AMS subject classifications. 37K55, 37J40

1 Introduction

It is very interesting to research the behavior of the solution in high-index Sobolev norm to nonlinear evolution equations with derivative in nonlinearities during a long time interval.

Consider a nonlinear Schrödinger equation

[TABLE]

under periodic boundary condition

[TABLE]

Suppose that $F$ satisfies

[TABLE]

Under this assumption $\psi=0$ is an equilibrium solution to equation (1.1). I am interested in the behavior of solutions around $\psi=0$ during a long time interval.

If $F$ only depends on $(x,\psi,\bar{\psi})$ and vanishes at order $n+1$ about $(\psi,\bar{\psi})$ at the origin ($n$ is a positive integer), local existence theory implies that when initial value $\|\psi(x,0)\|_{H^{p}}\leq\varepsilon\ll 1$ the corresponding solution $\psi(x,t)$ exists at least over an interval $(-c\varepsilon^{-n},c\varepsilon^{-n})$ and $\|\psi(x,t)\|_{H^{p}}$ stays bounded on such an interval. The problem that I am interested in is that construct almost global solutions when $F$ depends on $(\psi_{x},\bar{\psi}_{x})$. An almost global solution means that for any given $k>0$, when the initial value $\psi(x,0)$ is smaller than $0<\varepsilon\ll 1$, solution $\psi(x,t)$ is also small in a high index Sobolev norm for any $t\in(-c\varepsilon^{-k},c\varepsilon^{-k})$ (refer [31]).

When investigation concerns equation (1.1) on a compact manifold, no dispersion is available. Nevertheless, two ways may be used to obtain solutions, defined on time-intervals larger than the one given by local existence theory. The first one is using KAM theory to get small amplitude periodic or quasi-periodic (hence global) solutions. A lot of work have been devoted to these questions and readers refer [5, 6, 8, 9, 11, 13, 21, 39, 24, 28, 29, 30, 32, 33, 34, 35].

The second approach concerns the construction of almost global $H^{p}$-small solutions for (1.1) on compact manifold. Use Birkhoff normal form method to improve the order of normal form and then exploit integral principle to get almost global solutions. When nonlinearity $F$ to equation (1.1) depends only on $(\psi,\bar{\psi})$, small initial data give rise to global solutions and keep uniform control of the $p$-Sobolev norm of solutions ($p$ large enough), over time-intervals of length $\varepsilon^{-k}$, for any given positive $k$. This has been initiated by Bourgain [10], who stated results of almost global existence and uniform control to equation

[TABLE]

for any typical (with large probability) $V(x)$. Bourgain in [11] stated that for any small typical initial value to equation

[TABLE]

the solution $\psi$ will satisfy

[TABLE]

More results may be found in the book of Bourgain [12]. Almost global solutions for Hamiltonian Semi-linear Klein-Gordon Equations (without derivative in nonlinearity) on spheres and Zoll manifolds have been obtained by Bambusi, Delort, Grébert and Szeftel in [3]. Berti and Delort in [7] give almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions.

Bambusi and Grébert [4] (see also Bambusi [1] and Grébert [25]) prove an abstract Birkhoff normal form theorem for Hamiltonian partial differential equations and apply this theorem to semi-linear equations: nonlinear wave equation, nonlinear Schrödinger equation on the $d$-dimensional ($d\geq 1$) torus with nonlinearities satisfying a property-tame modulus. In a non-resonant case they deduce that any small amplitude solution remains very close to a torus for a very long time. In [2] Bambusi researches the NLW equation with nonlinearity function depending on $x$ periodically.

Faou and Grébert in [26] consider a general class of infinite dimensional reversible differential systems and prove that if the $p$-Sobolev norm of initial data is smaller than $\varepsilon$ ( $0<\varepsilon$ small enough) then the solution is bounded by $2\varepsilon$ during time of order $\varepsilon^{-r}$ with $r$ arbitrary. This theorem applies to a class of reversible semi-linear PDEs including nonlinear Schrödinger equation on the $d$-dimensional torus and a class of coupled NLS equations which is reversible but not Hamiltonian. Feola and Iandoli in [23] give the long time existence for a large class of fully nonlinear, reversible and parity preserving Schrödinger equations on the one dimensional torus.

Delort and Szeftel in [17], [18], Delort in [19], [20] research semi-linear Klein-Gordon equation (with derivative in nonlinearity) on spheres and Zoll manifold, quasi-linear Klein-Gordon equation on tori and $\mathbb{S}^{1}$, and obtain that when the initial value is small than $\varepsilon>0$, the corresponding solution exists when $|t|\leq\varepsilon^{-r}.$

Given a DNLS equation

[TABLE]

Yuan and Zhang in [37] obtain that for most $V$ the solution to (1.4) is still smaller than $2\varepsilon$ among time $|t|\leq\varepsilon^{-r}$ (for any given positive $r$), if the initial value is smaller than $\varepsilon\ll 1$. The nonlinearity in (1.4) does not directly depend on space variable $x$. In [38] Yuan and Zhang research the long time behavior of the solution to the perturbed KdV equation the nonlinearity of which is trigonometric polynomial about $x$.

In this paper, I focus on the behavior of solutions during a long time interval to two types of Hamiltonian Derivative Nonlinear Schrödinger (DNLS) equations which depend on $x$ periodically. One type is of the following form

[TABLE]

where $V$ belongs to

[TABLE]

and the other type is as follow

[TABLE]

where $V$ belongs to

[TABLE]

Under some assumptions, (1.5) becomes into a Hamiltonian equation with respect to a symplectic form $w^{0}:=J_{0}d\psi\wedge d\bar{\psi},$ J_{0}^{-1}:=\footnotesize{\big{(}\begin{array}[]{cc}0&-{\bf i}\\ {\bf i}&0\\ \end{array}\big{)}} and (1.6) is Hamiltonian under a symplectic form $w^{1}:=J_{1}d\psi\wedge d\bar{\psi},$ (J_{1})^{-1}:=\footnotesize{\big{(}\begin{array}[]{cc}0&\partial_{x}\\ \partial_{x}&0\\ \end{array}\big{)}}.

When $\partial_{\bar{\psi}}F(x,0,0)=0$ and $\partial^{2}_{\bar{\psi}\psi}F(x,0,0)=0,$ $\psi=0$ is an equilibrium point of equations (1.5) and (1.6). In order to get the almost global solution around the origin to (1.5) and (1.6), it is required to research the behavior of solutions around $\psi=0$ during a long time interval.

The result in [37] holds ture for Hamiltonian DNLS equation with nonlinearity independent of $x$. In other words the momentum of the corresponding Hamiltonian function equals to zero. This property is important in proving the long time stability result. In [38] one researches an unbounded perturbed KdV equation the nonlinearity of which is a trigonometric polynomial about $\sin kx$ and $\cos kx$ ($|k|\leq M$), i.e., the momentum of the corresponding Hamiltonian function are bounded. But generally, the sets of the momentum of Hamiltonian functions to equation (1.5) and (1.6) under Fourier transformation may be unbounded. Even if assume that for any $(\psi,\bar{\psi})$ around origin $F\in H^{\beta}([0,2\pi],\mathbb{R})$ ($\beta$ big enough), the corresponding nonlinear vector field of equations (1.5) and (1.6) are sitll unbounded. Denote the part of $F$, the momentum of which is bigger than $\delta>0$, as $F_{1}$. Even if $\delta$ is very large, the Hamiltonian vecotr field of $F_{1}$ in equations (1.5) and (1.6) are still unbounded. The results and methods in [37] and [38] do not work to equations (1.5) and (1.6), directly. In [14] one consider quasi-linear Klein-Gordon equation on $S^{1}$. The nonlinearities are polynomials and smooth depend on $x$. Their methods are not suitable to DNLS equations (1.5) and (1.6). In [23] they consider the reversible and parity preserving Schrödinger equation. It is necessary to construct a long time stability theory to solutions of Hamiltonian DNLS equations (1.5) and (1.6) around the origin.

Under Fourier transformation, equations (1.5) and (1.6) are transformed into two types of Hamiltonian systems $\theta\in\{0,1\}$

[TABLE]

with Hamiltonian function

[TABLE]

under symplectic form

[TABLE]

where $H_{0}^{w_{\theta}}:=\sum_{j\in\mathbb{Z}^{\theta}}\omega^{w_{\theta}}_{j}|u_{j}|^{2}$ and \omega^{w_{\theta}}_{j}:=\left\{\begin{array}[]{lll}(-j^{2}+\hat{V}_{j})&\theta=0\\ \mbox{sgn}(j)(-j^{2}+\hat{V}_{j})&\theta=1\\ \end{array}\right.. $P^{w_{\theta}}(u,\bar{u})$ is a power series having $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{\theta}>0$ (refer definitions • ‣ 3.3 and 3.4 in section 3). Note that the coefficients of $P^{w_{\theta}}(u,\bar{u})$ are not bounded. This leads to the Hamiltonian vector field of $P^{w_{\theta}}(u,\bar{u})$ being unbounded. See Proposition 4.1 in section 4.

The problem of finding almost global solutions around the origin to equations (1.5) and (1.6) is changed into considering a long time stability of solutions around equilibrium point $(u,\bar{u})=0$ of (1.7).

In section 3 Theorem 3 states that under some assumptions, the solution to the two type of Hamiltonian systems which have $(\beta,\theta)$-type symmetric coefficients $(\theta\in\{0,1\}$) is still smaller than $2\varepsilon$ during a time interval $(-\varepsilon^{-r_{*}},\varepsilon^{-r_{*}})$, if its initial value is smaller than $\varepsilon\ll 1$.

The idea of proving Theorem 3 is to combine Birkhoff normal form method with the property of $(\beta,\theta)$-type symmetric coefficients used to obtain energy inequalities.

Let us introduce the important steps in proving Theorem 3.

First step: construct a coordination transformation ${\cal T}_{w_{\theta}}^{(r)}$ under which the Hamiltonian function $H^{w_{\theta}}(u,\bar{u})$ in (1.8) can be transformed into a new Hamiltonian function

[TABLE]

with a high degree $(\theta,\gamma,\alpha,N)$-normal form ${Z}^{(r,w_{\theta})}$ (see definition 5.1). Because the system (1.7) is in an infinite dimension, one can only get a partial normal form. ${\cal R}^{N(r,w_{\theta})}$ is at least 3 order about $(u_{j},\bar{u}_{j})_{|j|>N}$ ($N$ is large enough) and ${\cal R}^{T(r,w_{\theta})}$ has a zero of high order about $(u,\bar{u})$. The Hamiltonian vector field of ${P^{(r,w_{\theta})}}$ is still unbounded. The construction of ${\cal T}_{w_{\theta}}^{(r)}$ is from solving Homological equation (refer Lemma 5.2). Because the perturbation in equation (1.7) is unbounded, a strong non resonant condition to frequencies $\{\omega^{w_{\theta}}_{j}(V)\}$ is needed to keep the transformation ${\cal T}_{w_{\theta}}^{(r)}$ bounded. This condition will effort the estimate of sets of potential $V(x)$ and the expression of $(\theta,\gamma,\alpha,N)$-normal form. Moreover, if $P^{w_{\theta}}(u,\bar{u})$ has $(\beta,\theta)$-type symmetric coefficients, then $P^{(r,w_{\theta})}(u,\bar{u})$ is still of $(\beta,\theta)$-type symmetric coefficients.

Second step: The solution to the new Hamiltonian system satisfies

[TABLE]

From above equation, it is obvious that estimating $\{\|u\|^{2}_{p},H^{(r,w_{\theta})}(u,\bar{u})\}_{w_{\theta}}$ is the key to get a long time behavior of the solution. For a general function $f(u,\bar{u})$ with unbounded coefficients, $\{\|u\|^{2}_{p},f(u,\bar{u})\}_{w_{\theta}}$ is not bounded even if $\|u\|_{p}$ is small enough. Fortunately, $P^{(r,w_{\theta})}(u,\bar{u})$ has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{\theta}>0$. Studying the Possion bracket of $\|u\|_{p}^{2}$ and $P^{(r,w_{\theta})}(u,\bar{u})$ with $(\beta,\theta)$-type symmetric coefficients is an important problem in this paper. Proposition 4.2 in section 4 and Lemma 5.1 in section 5 state that

[TABLE]

and

[TABLE]

hold true for any $\|u\|_{p}\leq R\ll 1$ and large enough $N$. With the help of (1.11), (1.12) and (1.10), the long time behavior of solution to the new Hamiltonian system can be obtained.

Since the two Hamiltonian DNLS equations have some difference, there still are some differences in the results of existence of almost global solution. The main difference is the sets of the potentials. From Lemma 7.1, there exist positive measure subsets $\tilde{\Theta}^{\theta}_{m}\subset\Theta^{\theta}_{m}$ ($\theta\in\{0,1\}$) such that when $V\in\tilde{\Theta}^{\theta}_{m}$, frequencies $\{\omega^{w_{\theta}}_{j}(V)\}$ are $(\theta,\gamma,\alpha,N)$-non resonant (see definition 5.1). When $V\in\Theta_{m}^{0}$, its Fourier coefficients satisfy $\hat{V}_{j}=\hat{V}_{-j}\in\mathbb{R},$ for any $j\in\mathbb{Z},$ which makes the corresponding frequencies satisfying $\omega^{w_{0}}_{j}=\omega^{w_{0}}_{-j}$; while $V\in\Theta_{m}^{1}$, $\hat{V}_{j}$ does not always equal to $\hat{V}_{-j}$. Thus $\omega^{w_{1}}_{j}$ is not related to $\omega^{w_{1}}_{-j}$ for any $j\in\mathbb{N}$. The potential sets to equations (1.5) and (1.6) are different, because (1.5) and (1.6) have different symplectic forms and nonlinearities. To be specific, from the definitions of symplectic structures, the following equations hold true for any $j\in\mathbb{Z}\setminus\{0\}$

[TABLE]

and

[TABLE]

In order to make $|\{H^{w_{\theta}}(u,\bar{u}),\|u\|_{p}^{2}\}_{w_{\theta}}|$ being high order small as $\|u\|_{p}$ is small, when $\theta=0$, from (1.13) the terms depending on \big{(}(u_{j}\bar{u}_{-j}+\bar{u}_{j}u_{-j})\big{)}_{j\in\mathbb{Z}} in $H^{w_{\theta}}(u,\bar{u})$ will not need to be eliminated by symplectic transformations; when $\theta=1$, from (1.14) it needs to eliminate the terms depending on \big{(}(u_{j}\bar{u}_{-j}+\bar{u}_{j}u_{-j})\big{)}_{j\in\mathbb{Z}} in $H^{w_{\theta}}(u,\bar{u})$. Therefore, it needs more parameters in the case $\theta=1$ than in the case $\theta=0$ and the sets of potential $V(x)$ are different to equations (1.5) and (1.6).

The paper is organized as follows: The section 2 of this paper is devoted to introduction of two types of Hamiltonian DNLS equations with respect to different symplectic forms. There are many differences between these two types of equations (see Remark 2.1). Then I give the main results in this paper, the existence of global solutions with small initial values to these two types of DNLS equations (See Theorem 1 and Theorem 2).

In the third section I present a definition of $(\beta,\theta)$-type symmetric($\theta\in\{0,1\}$). Using this definition, one can describe the coefficients of nonlinearities of two types of Hamiltonian DNLS equations under Fourier transformation. The long time stability result to infinite dimensional Hamiltonian systems owing $(\beta,\theta)$-type symmetric coefficients is given in Theorem 3. Theorem 1 and Theorem 2 follow from Theorem 3.

In the fourth section I give two main estimates. One is the estimate of the ${\cal H}^{p-1}$ norm of Hamiltonian vector field of a polynomial $f^{w_{\theta}}(u,\bar{u})$ with $(\beta,\theta)$-type symmetric coefficients under symplectic form $w_{\theta}$ ($\theta\in\{0,1\}$). This estimate is given in Proposition 4.1. It is easy to found that the Hamiltonian vector field of $f^{w_{\theta}}(u,\bar{u})$ is unbounded. Proposition 4.2 states that $|\{f^{w_{\theta}}(u,\bar{u}),\|u\|_{p}^{2}\}_{w_{\theta}}|$ is small when $\|u\|_{p}$ is small enough and $f^{w_{\theta}}(u,\bar{u})$ has $(\beta,\theta)$-type symmetric coefficients. Even if the set of momentum of $f^{w_{\theta}}(u,\bar{u})$ is unbounded, the result still holds true. The property of having $(\beta,\theta)$-type symmetric coefficients is invariant under some operators, such as truncated operators $\Gamma^{N}_{\leq 2}$ and $\Gamma^{N}_{>2}$ defined in (4.4) and (4.5). See Corollary 1.

In the fifth section, in order to improve the order of Birkhoff normal forms of Hamiltonian systems under two different symplectic forms, I will find suitable bounded symplectic transformations (See Theorem 4). These transformations are constructed by solving Homological equations. Since the nonlinear vector fields of Hamiltonian systems are unbounded (see Proposition 4.1), a stronger non-resonant condition (see definition 5.1) is needed. Under these transformations, new Hamiltonian systems are obtained. The nonlinearities of the new Hamiltonian functions still have $(\beta,\theta)$-type symmetric coefficients (See Lemma 5.3). Although the high order normal forms $Z^{(r,w_{\theta})}(u,\bar{u})\ (\theta\in\{0,1\})$ in the new Hamiltonian functions are not standard Birkhoff normal forms, from Lemma 5.1 $\{\|u\|_{p}^{2},Z^{(r,w_{\theta})}(u,\bar{u})\}_{w_{\theta}}$ is high order small when $\|u\|_{p}$ is small enough. The detail of the proof of Theorem 4 is listed in Appendix.

In the sixth section Theorem 3 is proved by applying Theorem 4, Proposition 4.2, Corollary 1 and Lemma 5.1, .

In the seventh section the proofs of Theorem 1 and Theorem 2 are given. Using Lemma 7.1, there exists a positive measure subset $\tilde{\Theta}_{m}^{\theta}\subset\Theta_{m}^{\theta}$ ($\theta\in\{0,1\}$) such that when $V\in\tilde{\Theta}_{m}^{\theta}$ the eigenvalues of linear operator $\partial_{xx}+V(x)\ast$ are stronger non resonant.

2 Hamiltonian DNLS equations and main results

2.1 Hamiltonian DNLS equations

Let

[TABLE]

be a p-Sobolev space. The inner product of the space $L^{2}([0,2\pi],\mathbb{C})$ is defined as

[TABLE]

The important definition of Hamiltonian PDEs is introduced in [32]. I list it as following. Consider an evolution equation

[TABLE]

defined in symplectic Hilbert scales $(\{H^{p}([0,2\pi],\mathbb{C})\times H^{p}([0,2\pi],\mathbb{C})\},\alpha)$, where $\alpha$ is a non-degenerate closed 2-form. Equation (2.1) is called a Hamiltonian equation, if there exists a Hamiltonian function $H(\xi)$ defined in a domain $O_{p}\subset H^{p}([0,2\pi],\mathbb{C})\times H^{p}([0,2\pi],\mathbb{C})$ making

[TABLE]

The dual space and the tangent space of $H^{p}([0,2\pi],\mathbb{C})\times H^{p}([0,2\pi],\mathbb{C})$ are isometry to $H^{p}([0,2\pi],\mathbb{C})\times H^{p}([0,2\pi],\mathbb{C})$, without confusion I denote them in the same signal in the following content.

Denote $d_{A}$ as the order of the linear operator

[TABLE]

and $d_{f}$ as the order of the mapping

[TABLE]

When the nonlinearity of a partial differential equation includes derivative, the corresponding order of the nonlinear vector field is positive. Otherwise, the order is non-positive. The following notations “bounded” and “unbounded” are given by the signs of the order of the vector field, and readers can refer [28], [37], [38]. For the sake of reference I list the definitions again.

Definition 2.1**.**

If $d_{f}\leq 0$, call $f$ in (2.1) bounded; If $d_{f}>0$, $f$ is called unbounded. Moreover, If $d_{A}-1=d_{f}>0$, call $f$ critical unbounded.

In this paper, I focus on two kinds of Hamiltonian Derivative Nonlinear Schrödinger (DNLS) equations.

Type I–DNLS equation has the following form

[TABLE]

under periodic boundary condition

[TABLE]

where $V$ belongs to

[TABLE]

with $m>1/2$, $\langle j\rangle:=\max\{1,\ |j|\}$ and $\bar{\psi}$ is the complex conjugate of $\psi.$

Suppose that there exists a function $F(x,\psi,\bar{\psi})$ such that

[TABLE]

Moreover, $F$ satisfies assumptions as follows.

${\bf A_{1}}:$

$F(x,\xi,\eta)$ is analytic about $(\xi,\eta)$ in a neighborhood of the origin and satisfies

[TABLE]

and $F(x,\psi,\bar{\psi})$ vanishes at least at order 2 in $(\psi,\bar{\psi})$ at the origin.

${\bf A_{2}}$

For any fixed $(\psi,\bar{\psi})$ a neighborhood of the origin, $F\in H^{\beta+1}([0,2\pi],\mathbb{C})$ ($\beta$ is a big enough positive real number) satisfies

[TABLE]

Then (2.2) becomes a Hamiltonian PDE with a real value Hamiltonian function ***Since $F(x,\psi,\bar{\psi})$ satisfies assumptions ${\bf A_{1}}$-${\bf A_{2}}$, the following equation holds true for any $\psi\in H^{p}([0,2\pi],\mathbb{C})$ fulfilling $\psi(x+2\pi,t)=\psi(x,t)$

$\displaystyle 0=\int_{0}^{2\pi}\frac{dF}{dx}(x,\psi,\bar{\psi})dx=\int_{0}^{2\pi}\partial_{x}F(x,\psi,\overline{\psi})+\partial_{\psi}F(x,\psi,\overline{\psi}){\psi}_{x}+\partial_{\bar{\psi}}F(x,\psi,\overline{\psi})\bar{\psi}_{x}dx,$

i.e.,

$\displaystyle\int_{0}^{2\pi}\frac{1}{2}\partial_{x}F(x,\psi,\overline{\psi})+\partial_{\psi}F(x,\psi,\overline{\psi}){\psi}_{x}dx=-\int_{0}^{2\pi}\frac{1}{2}\partial_{x}F(x,\psi,\overline{\psi})-\partial_{\bar{\psi}}F(x,\psi,\overline{\psi})\bar{\psi}_{x}dx.$

(2.7)

From (2.7) and assumptions ${\bf A_{1}}$-${\bf A_{2}}$, it follows

$\displaystyle\overline{\int_{0}^{2\pi}{\bf i}\frac{1}{2}\partial_{x}F(x,\psi,\overline{\psi})+{\bf i}\partial_{\psi}F(x,\psi,\overline{\psi})\psi_{x}dx}$ $\displaystyle=$ $\displaystyle\int_{0}^{2\pi}(-{\bf i})\frac{1}{2}\partial_{x}F(x,\psi,\overline{\psi})-{\bf i}\partial_{\bar{\psi}}F(x,\psi,\overline{\psi})\bar{\psi}_{x}dx$

$\displaystyle=$ $\displaystyle\int_{0}^{2\pi}{\bf i}\frac{1}{2}\partial_{x}F(x,\psi,\overline{\psi})+{\bf i}\partial_{\psi}F(x,\psi,\overline{\psi})\psi_{x}dx,$

which means that the Hamiltonian function $H_{(\ref{2.23})}(\psi,\bar{\psi})$ is real.

[TABLE]

on symplectic space $(H^{p}([0,2\pi],\mathbb{C})\times H^{p}([0,2\pi],\mathbb{C}),$ $w^{0})$, where

[TABLE]

The corresponding Hamiltonian vector of $H_{(\ref{2.23})}(\psi,\bar{\psi})$ under symplectic form $w^{0}$ is

[TABLE]

and the equation (2.2) can be written as

[TABLE]

Type II–DNLS equation has the form as following

[TABLE]

defined on

[TABLE]

under periodic boundary condition

[TABLE]

The potential $V$ belongs to

[TABLE]

with $m>1/2.$

If equation (2.11) satisfies the following assumptions:

${\bf B_{1}}$

$F(x,\xi,\eta)$ is analytic at the origin about $(\xi,\eta)\in H_{0}^{p}([0,2\pi],\mathbb{C})\times H_{0}^{p}([0,2\pi],\mathbb{C})$ and vanishes at least at order 2 in $(\psi,\bar{\psi})$ at origin. For any $\psi\in H_{0}^{p}([0,2\pi],\mathbb{C})$, it holds

[TABLE]

${\bf B_{2}}$

For any fixed $(\psi,\bar{\psi})$ in a neighborhood of the origin, $F\in H^{\beta+1}([0,2\pi],\mathbb{C})$ ($\beta$ is a big enough positive real number) satisfies

[TABLE]

then equation (2.11) becomes into a Hamiltonian PDE with a real Hamiltonian

[TABLE]

under symplectic space $(H_{0}^{p}([0,2\pi],\mathbb{C})\times H_{0}^{p}([0,2\pi],\mathbb{C}),w^{1})$, where

[TABLE]

is a symplectic from ($w^{1}$ is a non-degenerate closed two form in space $H_{0}^{p}([0,2\pi],\mathbb{C})\times H_{0}^{p}([0,2\pi],\mathbb{C})$).

The Hamiltonian vector $X^{w^{1}}_{H_{(\ref{NLS-1})}}$ of $H_{(\ref{NLS-1})}(\psi,\bar{\psi})$ equals to

[TABLE]

Equation (2.11) can be written as follow

[TABLE]

The DNLS equation researched in [37] is a special case of equation (2.11), i.e., $F(x,\psi,\bar{\psi})$ is independent of $x$.

Remark 2.1**.**

There are some differences between type I-DNLS equations and type II-DNLS equations:

• •

Nonlinearities of these two kinds of DNLS equations are different. It is an essential difference.

• •

Symplectic spaces are different. Type I-DNLS equation is defined in $(H^{p}([0,2\pi],\mathbb{C})\times H^{p}([0,2\pi],\mathbb{C}),$ $w^{1})$ and type II-DNLS equation is defined in $(H_{0}^{p}([0,2\pi],\mathbb{C})\times H_{0}^{p}([0,2\pi],\mathbb{C}),$ $w^{0})$. $w^{1}$ and $w^{0}$ are also different.

• •

The Potential $V$ in type I-DNLS equation belongs to $\Theta^{0}_{m}$ and the one in type II-DNLS equation belongs to $\Theta^{1}_{m}$. $\Theta^{1}_{m}$ is different from $\Theta^{0}_{m}$. When $V\in\Theta^{0}_{m}$ it fulfills $\hat{V}_{j}=\hat{V}_{-j}\in\mathbb{R}$ which means $\overline{V(x)}=V(x)$; while $V\in\Theta^{1}_{m}$, $V$ is a complex valued potential. The potential $V$ will directly determine the eigenvalues of the linear operator $-\partial_{xx}+V(x)\ast$. It is clear that when $V\in\Theta^{0}_{m}$ the corresponding eigenvalues $\omega_{j}$ and $\omega_{-j}$ of the linear operator $-\partial_{xx}+V(x)\ast$ are resonant, when $V\in\Theta^{1}_{m}$, they are independent. The measures of $\Theta^{0}_{m}$ and $\Theta^{1}_{m}$ are defined as follows

[TABLE]

and

[TABLE]

where “meas” means the Lebesgue measure.

Remark 2.2**.**

• •

If type II-DNLS equation (2.11) is defined in space $H^{p}([0,2\pi],\mathbb{C})$, it is easy to verify that for any solution $\psi(x,t)$ to equation (2.11) the quantity $\int_{0}^{2\pi}\psi(x,t)dx$ is a constant for any $t\in\mathbb{R}$. Set

[TABLE]

If $\psi\in H^{p}([0,2\pi],\mathbb{C}),$ then $\phi\in H_{0}^{p}([0,2\pi],\mathbb{C}).$ When $G(x,\phi,\bar{\phi}):=F(x,\phi+c,\overline{\phi+c})$ satisfies ${\bf B_{1}}$ and ${\bf B_{2}}$, then equation (2.11) becomes into a Hamiltonian equation under symplectic form $w^{1}$ about $(\phi,\bar{\phi})$.

• •

From Proposition 4.1 in section 4, the nonlinearities of type I-DNLS equation (2.2) and type II-DNLS equation (2.11) are unbounded.

2.2 Main result

The long time behavior of the solutions around equilibrium point to type I and type II-DNLS Hamiltonian equations are given in this subsection.

Theorem 1**.**

Suppose that the equation (2.2) satisfies assumptions ${\bf A_{1}}$-${\bf A_{2}}$. For any integer $r_{*}>1$, there exist an almost full measure set $\widetilde{\Theta}^{0}_{m}\subset\Theta^{0}_{m}$ and $p_{*}>0$ such that for any fixed $V\in\widetilde{\Theta}^{0}_{m}$ and any $p$ fulfilling $(\beta-4)/2>p>p_{*}$, if the initial data of the solution to (2.2) satisfies

[TABLE]

then one has

[TABLE]

Theorem 2**.**

Suppose that equation (2.11) fulfills assumptions ${\bf B_{1}}$-${\bf B_{2}}$. For any integer $r_{*}>1$, there exist a positive $p_{*}$ and an almost full measure set $\widetilde{\Theta}^{1}_{m}\subset\Theta^{1}_{m}$ ($m>\frac{1}{2}$) such that for any fixed $V\in\widetilde{\Theta}^{1}_{m}$ and any $p$ fulfilling $(\beta-4)/2>p>p_{*}$, the solution to (2.11) satisfies

[TABLE]

if the initial value fulfills

[TABLE]

Remark 2.3**.**

As type I and type II-DNLS equations have many differences (Readers can refer Remark 2.1), the proofs of Theorem 1 and Theorem 2 still have some differences.

Remark 2.4**.**

The DNLS equations researched in our paper are not always invariant under gauge transformation. For example, take

[TABLE]

in equation (2.11), which fulfills assumption ${\bf B_{1}}$. It is easy to check that equation (2.11) with $F$ in (2.16) is not invariant under the transformation $\phi=e^{{\bf i}\theta}\psi$, $\theta\in\mathbb{R}$.

3 Long time stability result to infinite dimension Hamiltonian systems with $(\beta,\theta)$-type symmetric coefficients ($\theta\in\{0,1\}$)

3.1 $(\beta,\theta)$-type symmetric coefficients ($\theta\in\{0,1\}$)

Under Fourier transformation, Hamiltonian DNLS equations with respect to periodic boundary condition can be transformed into two classes of infinite dimension Hamiltonian systems with “unbounded” nonlinearities. In this section, I will introduce long time stability results to two classes of infinite dimension Hamiltonian systems with “unbounded” nonlinearities. First, give some notations and annotations. In this paper, ${\mathbb{Z}}^{*}$ means ${\mathbb{Z}}$ or ${\mathbb{Z}}\setminus\{0\}.$ Denote weighted Hilbert spaces

[TABLE]

and {\mathcal{H}^{p}}(\mathbb{Z}^{*},\mathbb{C}):=\{(u,v)\in\ell^{2}_{p}(\mathbb{Z}^{*},\mathbb{C})\times\ell^{2}_{p}(\mathbb{Z}^{*},\mathbb{C})\ \big{|}\ v=\bar{u}\ \} with norm

[TABLE]

Let the neighborhood of the origin with a radius $R$ be noted by

[TABLE]

Definition 3.1**.**

For any fixed $l,\ k\in\mathbb{N}^{\mathbb{Z}^{*}},$ call the integer

[TABLE]

be the momentum of the ordered vector $(l,k)$ and denote it as ${\cal M}(l,k).$

Readers can refer this definition in [37] and [38].

Remark 3.1**.**

If ${\cal M}(l,k)=i$, from the definition of momentum, it holds that

[TABLE]

Definition 3.2**.**

Call a power series

[TABLE]

have symmetric coefficients, if for any $l,k$ fulfilling $|l+k|=t$ and ${\cal M}(l,k)=i$, the coefficient holds

[TABLE]

Moreover, fixed $\beta>0$, call $f(u,\bar{u})$ has $\beta$-bounded coefficients bounded by $C>0$, if

[TABLE]

for any $l,k$ satisfying $|l+k|=t$ and ${\cal M}(l,k)=i$.

Remark 3.2**.**

A power series $f:{\cal H}^{p}(\mathbb{Z}^{*},\mathbb{C})\rightarrow\mathbb{C}$ is of symmetric coefficients, if and only if $f$ satisfies

[TABLE]

Hence, a real-value Hamiltonian function has symmetric coefficients.

Now define two kinds of power series with “unbounded” special symmetric coefficients.

Definition 3.3**.**

Given $\beta>0$ and $C_{f}>0$, call a power series

[TABLE]

have $(\beta,0)$-type symmetric coefficients*, if its coefficients have the following form*

[TABLE]

where

[TABLE]

and for any $(l^{0},k^{0},i^{0})\in{\cal A}_{f^{i}_{t,lk}}$, the followings hold true

[TABLE]

Moreover, call $f(u,\bar{u})$ have $(\beta,0)$-type symmetric coefficients semi-bounded by $C_{f}$, if $f(u,\bar{u})$ have $(\beta,0)$-type symmetric coefficients and there exists a constant $C_{f}>0$ such that for any $l,k\in\mathbb{N}^{\mathbb{Z}^{*}}$ with $|l+k|=t$ and ${\cal M}(l,k)=i$, the following inequality holds true

[TABLE]

Suppose that Type I-DNLS equation satisfies assumption ${\bf A_{1}}$-${\bf A_{2}}$. Under Fourier transformation, there exists a constant $C>0$ such that the Hamiltonian function of Type I-DNLS equation have $(\beta,0)$-type symmetric coefficients semi-bounded by $C$. See section 7 for details. This symmetric property is invariant under a symplectic transformation. Refer Lemma 5.3 in section 5.

Definition 3.4**.**

Given $\beta>0$ and $C_{g}>0$, call a power series

[TABLE]

have $(\beta,1)$-type symmetric coefficients*, if for any $l,k\in\mathbb{N}^{\mathbb{Z}^{*}}$ with $|l+k|=t$ and ${\cal M}(l,k)=i\in M_{g_{t}}\subset\mathbb{Z}$, its coefficient has the following form*

[TABLE]

and satisfies

[TABLE]

Moreover, call $g(u,\bar{u})$ have $(\beta,1)$-type symmetric coefficients semi-bounded by $C_{g}$, if $g(u,\bar{u})$ have $(\beta,1)$-type symmetric coefficients and there exists a constant $G_{g}>0$ such that

[TABLE]

for any $l,k$ fulfilling $|l+k|=t$ and ${\cal M}(l,k)=i.$

Remark 3.3**.**

Suppose a power series $f^{w_{\theta}}(u,\bar{u})$ is of $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{\theta}>0$ ($\theta\in\{0,1\}$). Thence,

• •

the “semi-bounded” does not means the coefficients of $f^{w_{\theta}}(u,\bar{u})$ are bounded, even if $f^{w_{\theta}}(u,\bar{u})$ is an $r$-degree polynomial;

• •

the momentum set $M_{f^{w_{\theta}}_{t}}$ is symmetric, i.e., if $i\in M_{f^{w_{\theta}}_{t}}$, then $-i\in M_{f^{w_{\theta}}_{t}}$. And the number of elements in $M_{f^{w_{\theta}}_{t}}$ satisfies

[TABLE]

So does the power series having $\beta$-bounded symmetric coefficients.

• •

when $\theta=0$ for any given $(l,k)\in\mathbb{N}^{\mathbb{Z}^{*}}\times\mathbb{N}^{\mathbb{Z}^{*}}$ and $i\in\mathbb{Z}$,

[TABLE]

the last second equation follows from

[TABLE]

and $(k-k^{0},l-l^{0},{i^{0}}-2i)\in{\cal A}_{f^{-i}_{t,kl}};$

when $\theta=1$ for any $l,k\in\mathbb{N}^{\mathbb{Z}^{*}}$ and any $i\in\mathbb{Z}$,

[TABLE]

Hence, the coefficients of $f^{w_{\theta}}(u,\bar{u})$ are symmetric.

Let $({\mathcal{H}^{p}}(\mathbb{Z}^{*},\mathbb{C}),w_{\theta})$ ($\theta\in\{0,1\}$) be a symplectic space endowed with symplectic form

[TABLE]

When $\theta=0$, $\mathbb{Z}^{*}$ can be either $\mathbb{Z}$ or $\mathbb{Z}\backslash\{0\}$. When $\theta=1$, $\mathbb{Z}^{*}$ means $\mathbb{Z}\backslash\{0\}$ only.

The possion bracket of differential functions $f_{1}$ and $f_{2}$ defined in the domain of ${\mathcal{H}^{p}}(\mathbb{Z}^{*},\mathbb{C})$ under the symplectic form $w_{\theta}\ (\theta\in\{0,1\})$ has the following form

[TABLE]

Given a differential function $f$, its corresponding Hamiltonian vector field under the symplectic form $w_{\theta}$ is defined as

[TABLE]

where $I_{0}$ is an identity operator on space $\ell^{2}_{p}(\mathbb{Z}^{*},\mathbb{C})$, and for any $u=(u_{j})_{j\in\mathbb{Z}^{*}}\in\ell_{p}^{2}(\mathbb{Z}^{*},\mathbb{C})$,

[TABLE]

3.2 Result of Hamiltonian system with ($\beta,\theta$)-Type symmetric coefficients

In order to use an uniformly formula to describe two kinds of Hamiltonian equations, in this paper, denote $0^{0}=1$.

Let $\theta\in\{0,1\}$. Consider Hamiltonian systems defined in $({\mathcal{H}}^{p}(\mathbb{Z}^{*},\mathbb{C}),w_{\theta})$, for any $j\in\mathbb{Z}^{*}$

[TABLE]

with a Hamiltonian function

[TABLE]

where $H_{0}^{w_{\theta}}:=\sum_{j\in\mathbb{Z}^{*}}\omega^{w_{\theta}}_{j}|u_{j}|^{2}$.

Theorem 3**.**

Suppose that equation (3.10) satisfies the following assumptions:

${\cal A}_{\theta}$

: $\omega^{w_{\theta}}:=(\omega^{w_{\theta}}_{j})_{j\in\mathbb{Z}^{*}}$, $\ \omega^{w_{\theta}}_{j}\in\mathbb{R}$. $\omega^{w_{\theta}}$ satisfies strong non resonant condition.†††See Definition 5.1 in section 5.**

${\cal B}_{\theta}$

: $P^{w_{\theta}}(u,\bar{u})$ is a power series beginning with at least at order 2 in $(u,\bar{u})$ and has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{\theta}>0$ ($\beta$* is big enough positive number).*

Given integer $r_{*}>1$, there exists an integer $p_{r_{*}}>0$, for any $p$ fulfilling $(\beta-4)/2>p>p_{r_{*}}$ there exists $\varepsilon_{r_{*},p}>0$ such that the solution to (3.10) satisfies

[TABLE]

if the initial data fulfills

[TABLE]

Let us give the basic procedure of proving Theorem 3 which consists of the following steps.

The first step is to construct a bounded symplectic transformation around the origin under which the nonlinearity of Hamiltonian function (3.11) becomes into the sum of the following three parts: one is a high order $(\theta,\gamma,\alpha,N)$-normal form $Z^{(r_{*},w_{\theta})}(u,\bar{u})$‡‡‡See Definition 5.1., one of the others is ${\cal R}^{T(r_{*},w_{\theta})}(u,\bar{u})$ which vanishes at $r_{*}+3$ order of $(u,\bar{u})$ at origin and the last one denoted as ${\cal R}^{N(r_{*},w_{\theta})}(u,\bar{u})$ has zero at least order 3 about high index variable $(u_{j},\bar{u}_{j})_{|j|>N}$($N$ is big enough). Moreover, when $P^{w_{\theta}}(u,\bar{u})$ in (3.11) has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{\theta}>0$, the new Hamiltonian function is still of $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C(\theta,r)>0$ ($C(\theta,r)$ is defined in Theorem 4). In order to guarantee the boundedness of the symplectic transformation, a strong non-resonant condition is presented in Definition 5.1. See Theorem 4 in section 5 for details.

Since (3.10) is a Hamiltonian system, the following equation holds true

[TABLE]

Researching the Possion bracket of Hamiltonian function $H^{w_{\theta}}(u,\bar{u})$ and $\|u\|_{p}^{2}$ under the corresponding symplectic form $w_{\theta}$ is important. The second step is to estimate the Possion bracket of function $f^{w_{\theta}}(u,\bar{u})$ and $\|u\|_{p}^{2}$. Suppose that $f^{w_{\theta}}(u,\bar{u})$ has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{\theta}>0$. If the momentum of $f^{w_{\theta}}(u,\bar{u})$ are bounded, partial result can be found in [37] and [38]. When the set of the momentum of $f^{w_{\theta}}(u,\bar{u})$ is unbounded, the corresponding Hamiltonian vector field of $f^{w_{\theta}}(u,\bar{u})$ is small under $H^{p-1}$ norm but not $H^{p}$ norm (see Proposition 4.1 and Remark 4.1). In order to deal with it, I will make use of the Hamiltonian structure and $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{\theta}>0$ to get the estimate of Possion bracket of Hamiltonian function $H^{w_{\theta}}(u,\bar{u})$ and $\|u\|_{p}^{2}$ under the corresponding symplectic form $w_{\theta}$. See Proposition 4.2 and Corollary 1 for details. By Proposition 4.2 and Corollary 1, $|\{{\cal R}^{T(r_{*},w_{\theta})}(u,\bar{u})+{\cal R}^{N(r_{*},w_{\theta})}(u,\bar{u}),\|u\|_{p}^{2}\}_{w_{\theta}}|\prec R^{r_{*}+1}$ holds true for $\|(u,\bar{u})\|_{p}\leq R$. From Remark 5.2, $(\theta,\gamma,\alpha,N)$-normal form $Z^{(r_{*},w_{\theta})}(u,\bar{u})$ is not a standard Birkhoff normal form. By Lemma 5.1 when $Z^{(r_{*},w_{\theta})}(u,\bar{u})$ has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C(\theta,r_{*})$, for any $\|(u,\bar{u})\|_{p}<R\ll 1$ and $N$ satisfying (5.33), $|\{Z^{(r_{*},w_{\theta})}(u,\bar{u}),\|u\|_{p}^{2}\}|\prec R^{r_{*}+1}$.

4 Estimate \{f_{r}^{w_{\theta}}(u,\bar{u}),\|u\|_{p}^{2}\}_{w_{\theta}}\ and Hamiltonian vector field $X_{f_{r}^{w_{\theta}}}$ ($f_{r}^{w_{\theta}}(u,\bar{u})$ has $(\beta,\theta)$-type symmetric coefficients $\theta\in\{0,1\}$)

Suppose that an $r$-degree homogeneous power series $f_{r}^{w_{\theta}}(u,\bar{u})$ defined on ${\cal H}^{p}(\mathbb{Z}^{*},\mathbb{C})$ is of $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{\theta}>0$.

First of all I present that the Hamiltonian vector field of $f_{r}^{w_{\theta}}(u,\bar{u})$ under symplectic form $w_{\theta}$ is unbounded with order 1. See Proposition 4.1.

Next, the estimate of the possion bracket of the power series $f_{r}^{w_{\theta}}(u,\bar{u})$ and $\|u\|_{p}^{2}$ is given in Proposition 4.2.

Last but not least, I introduce truncated operators $\Gamma^{N}_{\leq 2}$ and $\Gamma^{N}_{>2}$, and estimate the Hamiltonian vector fields of the functions $\Gamma^{N}_{\leq 2}f_{r}^{w_{\theta}}(u,\bar{u})$, $\Gamma^{N}_{>2}f_{r}^{w_{\theta}}(u,\bar{u})$ in ${{\mathcal{H}}^{p-1}}(\mathbb{Z}^{*},\mathbb{C})$-norm, $|\{\Gamma^{N}_{\leq 2}f_{r}^{w_{\theta}}(u,\bar{u}),\|u\|_{p}^{2}\}|$ and $|\{\Gamma^{N}_{>2}f_{r}^{w_{\theta}}(u,\bar{u}),\|u\|_{p}^{2}\}|$ for any $(u,\bar{u})\in{{\mathcal{H}}^{p}}(\mathbb{Z}^{*},\mathbb{C})$ in Corollary 1.

These results will be used in proving Theorem 3.

Proposition 4.1**.**

Suppose that an $r$-degree ($r\geq 3$) homogeneous polynomial $f^{w_{\theta}}_{r}(u,\bar{u})$ ($\theta\in\{0,1\}$) defined on ${{\mathcal{H}}^{p}}(\mathbb{Z}^{*},\mathbb{C})$ has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{f^{w_{\theta}}}>0$, and $\beta-p\geq 2$. Then for any $(u,\bar{u})\in{{\mathcal{H}}^{p}}(\mathbb{Z}^{*},\mathbb{C})$ and any $\theta\in\{0,1\}$,

[TABLE]

If an $r$-degree homogeneous polynomial $f_{r}(u,\bar{u})$ has $\beta$-bounded symmetric coefficients bounded by $C_{f}>0$, then the following inequality holds true for any $(u,\bar{u})\in{{\mathcal{H}}^{p}}(\mathbb{Z}^{*},\mathbb{C})$ and any $\theta\in\{0,1\}$

[TABLE]

Remark 4.1**.**

If an $r$-degree homogeneous polynomial $f^{w_{\theta}}_{r}(u,\bar{u}):B_{p}(R_{*})\rightarrow\mathbb{C}$ ($\theta\in\{0,1\},\ R_{*}>0$) has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{f^{w_{\theta}}}>0$ ($\theta\in\{0,1\}$), from Proposition 4.1 it holds that

• •

The Hamiltonian vector field $X^{w_{\theta}}_{f^{w_{\theta}}_{r}}$ is from $B_{p}(R_{*})$ to ${{\cal H}}^{p-1}(\mathbb{Z}^{*},\mathbb{C})$, but not to ${{\cal H}^{p}}(\mathbb{Z}^{*},\mathbb{C})$. It means that $X_{f_{r}^{w_{\theta}}}^{w_{\theta}}$ is unbounded with order 1.

• •

There exists a constant $\delta\in(0,1)$ such that

[TABLE]

Together with Cauchy estimate and (4.1), one has that the function $f_{r}^{w_{\theta}}(u,\bar{u})$ $(\theta\in\{0,1\})$ is analytic about $(u,\bar{u})$ on some $B_{p}(R)\subset{{\cal H}^{p}}(\mathbb{Z}^{*},\mathbb{C})$ ($p>1$). (In this paper, when I mention the “analyticity” of functions or vector fields, I take $u$ and $\bar{u}$ as independent variables).

Proposition 4.2**.**

Suppose that an $r$-degree ($r\geq 3$) homogeneous polynomial $f^{w_{\theta}}_{r}(u,\bar{u})$ $(\theta\in\{0,1\})$ has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{f_{r}^{w_{\theta}}}>0$. Then the following inequality holds true for any $(u,\bar{u})\in{\mathcal{H}}^{p}(\mathbb{Z}^{*},\mathbb{C})$ ($p\geq 2,\ \beta-p\geq 2$)

[TABLE]

Given an integer $N>0$, two projection operators $\Gamma_{>N}$ and $\Gamma_{\leq N}$ on $\mathbb{N}^{\mathbb{Z}^{*}}$ and $\ell^{2}_{p}(\mathbb{Z}^{*},\mathbb{C})$ are defined as follows. For any $l=(l_{j})_{j\in\mathbb{Z}^{*}}\in\mathbb{N}^{\mathbb{Z}^{*}}$, $\Gamma_{>N}l$ and $\Gamma_{\leq N}l\in\mathbb{N}^{\mathbb{Z}^{*}}$ with

[TABLE]

For any $u=(u_{j})_{j\in\mathbb{Z}^{*}}\in\ell^{2}_{p}(\mathbb{Z}^{*},\mathbb{C})$, $\Gamma_{>N}u$, $\Gamma_{\leq N}u\in\ell^{2}_{p}(\mathbb{Z}^{*},\mathbb{C})$ with

[TABLE]

Now I will introduce two truncated operators $\Gamma^{N}_{\leq 2}$ and $\Gamma^{N}_{>2}$ defined as follows. For any power series

[TABLE]

denote

[TABLE]

[TABLE]

Remark 4.2**.**

Fix a positive integer $N$. Suppose that a power series $f^{w_{\theta}}(u,\bar{u})$ has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{f^{w_{\theta}}}>0$ $(\theta\in\{0,1\})$. Then $\Gamma^{N}_{\leq 2}f^{w_{\theta}}(u,\bar{u})$, $\Gamma^{N}_{>2}f^{w_{\theta}}(u,\bar{u})$ also have $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{f^{w_{\theta}}}>0.$

Corollary 1**.**

Suppose that an $r$-degree ($r\geq 3$) homogeneous polynomials $f^{w_{\theta}}_{r}(u,\bar{u})$ ($\theta\in\{0,1\}$) defined on ${{\mathcal{H}}^{p}}(\mathbb{Z}^{*},\mathbb{C})$ has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{f^{w_{\theta}}}>0$, and $\beta-p\geq 2$. Given an integer $N>0$, then

[TABLE]

The proof of Corollary 4.5 is similar with Proposition 4.1-4.2 and I omit it. In order to give the proof of Proposition 4.1-4.2, the following Lemmas are needed.

Lemma 4.1**.**

If power series $f^{w_{\theta}}(u,\bar{u})$ and $g^{w_{\theta}}(u,\bar{u})$ have $(\beta,\theta)$-type symmetric coefficients ($\theta\in\{0,1\}$) semi-bounded by $C_{f^{w_{\theta}}}>0$ and $C_{g^{w_{\theta}}}>0$, respectively, then for any $a,b\in\mathbb{R}$, $(af^{w_{\theta}}+bg^{w_{\theta}})(u,\bar{u})$ also has $(\beta,\theta)$-type symmetric coefficients semi-bounded by

[TABLE]

Proof.

I only give the proof in the case $\theta=0$, while in the case $\theta=1$ the proof is similar to the case $\theta=0$. For any $a$, $b\in\mathbb{R}$,

[TABLE]

where $M_{(af^{w_{0}}+bg^{w_{0}})_{t}}:=M_{f^{w_{0}}_{t}}\cup M_{g^{w_{0}}_{t}}.$ For any $l,k\in\mathbb{N}^{\mathbb{Z}^{*}}$ with ${\cal M}(l,k)=i\in M_{(af^{w_{0}}+bg^{w_{0}})_{t}}$, the corresponding coefficient of $af^{w_{0}}+bg^{w_{0}}$ has the following form

[TABLE]

where

[TABLE]

and

[TABLE]

It is easy to check that

[TABLE]

and

[TABLE]

where $C_{af^{w_{0}}+bg^{w_{0}}}$ is defined in (4.7). ∎

Lemma 4.2**.**

Given real numbers $q_{i}\leq p$ ($1\leq i\leq r$), suppose that ${F}=(F_{j})_{j\in\mathbb{Z}^{*}}$ is an $r$-multiple linear vector field defined as following

[TABLE]

where u^{(1)}:=\big{(}u^{(1)}_{j_{(1)}}\big{)}_{j_{(1)}\in\mathbb{Z}^{*}}, $\cdots$ u^{(r)}:=\big{(}u^{(r)}_{j_{(r)}}\big{)}_{j_{(r)}\in\mathbb{Z}^{*}}\in\ell^{2}_{p}(\mathbb{Z}^{*},\mathbb{C}). If there exist a positive constant $C$ and an integer $n\in\{1,\cdots,r\}$ such that

[TABLE]

then

[TABLE]

where $c:=\sqrt{\sum_{j\in\mathbb{Z}}\langle j\rangle^{-2}}.$

Proof.

Using Young’s inequality§§§ Suppose $a\in\ell^{p}(\mathbb{Z}^{*},\mathbb{C})$, $b\in\ell^{q}(\mathbb{Z}^{*},\mathbb{C})$ and $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}+1,$ with $1\leq p,q,r\leq\infty$. Then $\|f\ast g\|_{\ell^{r}(\mathbb{Z}^{*},\mathbb{C})}\leq\|f\|_{\ell^{p}(\mathbb{Z}^{*},\mathbb{C})}\cdot\|g\|_{\ell^{q}(\mathbb{Z}^{*},\mathbb{C})}.$ the following inequality holds true for any $a\in\ell^{2}(\mathbb{Z}^{*},\mathbb{C})$ and $b\in\ell^{1}(\mathbb{Z}^{*},\mathbb{C})$

[TABLE]

Together with (4.9), using (4.10) repeatedly, one has

[TABLE]

Since $q_{i}\leq p$, the following inequality holds true for any $u=(u_{j})_{j\in\mathbb{Z}^{*}}\in\ell^{2}_{p}(\mathbb{Z}^{*},\mathbb{C})$,

[TABLE]

In view of (4.12) and (4.11), one has

[TABLE]

∎

Corollary 2**.**

Given integers $p>q\geq 1$ and real number $\rho\geq 2$, suppose that there exists a positive number $C_{f}>0$ such that the coefficients of an $r$-degree homogeneous polynomial

[TABLE]

satisfy that for any $l,k\in\mathbb{N}^{\mathbb{Z}^{*}}$ with ${\cal M}(l,k)=i\in M_{f_{r}}$

[TABLE]

Then for any $(u,\bar{u})\in{{\mathcal{H}}^{p}}(\mathbb{Z}^{*},\mathbb{C})$, it satisfies that

[TABLE]

Remark 4.3**.**

The result of Corollary 2 still holds true for $\rho>1$. To simplify the process of proof, assume $\rho\geq 2$.

Proof.

By Cauchy estimate,

[TABLE]

where $G:=(\langle j\rangle^{p}|u_{j}|)_{j\in\mathbb{Z}^{*}}$ and $F(u,\bar{u})=(F_{j}(u,\bar{u}))_{j\in\mathbb{Z}^{*}}$

[TABLE]

For any $n\in\{0,\ 1,\cdots,r-1\}$, there exist an $(r-1)$-multiple linear vector field

[TABLE]

such that

[TABLE]

By condition (4.13), the coefficients of each $(r-1)$-multiple linear vector fields $\widetilde{F}^{n}$ satisfy the condition (4.9) of Lemma 4.2 with $q_{n}=p$ and $q_{i}=q\leq p\ (i\neq n)$. Hence, by Lemma 4.2, for any $(u,\bar{u})\in{\mathcal{H}}^{p}(\mathbb{Z}^{*},\mathbb{C})$, one has

[TABLE]

In view of (4.14) and (4.15), the following inequality holds true

[TABLE]

∎

Now give the proof of Proposition 4.1.

Proof.

In the case $\theta=0$,

[TABLE]

Note that $\|\nabla_{\bar{u}}f^{w_{0}}_{r}(u,\bar{u})\|_{p-1}$ equals to equals to $\ell^{2}(\mathbb{Z}^{*},\mathbb{C})$ norm of the following vector field

[TABLE]

Accordingly, $\|\nabla_{{u}}f^{w_{0}}_{r}(u,\bar{u})\|_{p-1}$ equals to equals to $\ell^{2}(\mathbb{Z}^{*},\mathbb{C})$ norm of the following vector field

[TABLE]

In order to use Lemma 4.2 to give the $\ell^{2}$-norm of (4.17) and (4.18), it is required to estimate the coefficients of vector fields (4.17) and (4.18). Note that for any fixed $l,k\in\mathbb{N}^{\mathbb{Z^{*}}}$ and $j\in\mathbb{Z}^{*}$ with $(f^{w_{0}})^{i}_{r,lk}\neq 0$ and $k_{j}\neq 0$, the indices satisfy

[TABLE]

Since $p>2$, it follows

[TABLE]

the last inequality holds true by the fact that for any integer $a,\ b\geq 1$,

[TABLE]

Furthermore, for any $(l^{0},k^{0},i^{0})\in{\cal A}_{(f^{w_{0}})_{r,lk}^{i}}$, it holds that

[TABLE]

and the momentum of $(l^{0},k^{0})$ equals to

[TABLE]

Together with (4.19), (4.21), (4.22) and (4.23), one has

[TABLE]

In view of (4.19), (4.20), (4.24) and $f^{w_{0}}(u,\bar{u})$ having the ($\beta,0$)-type symmetric coefficients semi-bounded by $C_{f^{w_{0}}}>0$, one has

[TABLE]

Therefor, using (4.25) and Lemma 4.2 by taking $q_{n}=p$ and $q_{i}=2\ (i\neq n)$, the following inequality holds true

[TABLE]

Similarly, one has

[TABLE]

Thus,

[TABLE]

In the case $\theta=1$,

[TABLE]

Similarly, $\|\nabla_{\bar{u}}f^{w_{1}}_{r}(u,\bar{u})\|_{p-1}$ and $\|\nabla_{u}f^{w_{1}}_{r}(u,\bar{u})\|_{p-1}$ equal to $\ell^{2}(\mathbb{Z}^{*},\mathbb{C})$-norm of the following vector fields respectively

[TABLE]

and

[TABLE]

For any fixed $l,k\in\mathbb{N}^{\mathbb{Z}^{*}}$ and $j\in\mathbb{Z}^{*}$ with $(f^{w_{1}})^{i}_{r,lk}\neq 0$. ${\cal M}(l,k)=i$ and $k_{j}\neq 0$, (4.19) still holds true. By (4.19) and (4.20), the coefficients of the vector field in (4.27) are bounded by

[TABLE]

Using Lemma 4.2 and (4), one has

[TABLE]

Similarly,

[TABLE]

Thus,

[TABLE]

By the same approach, when $f$ has $\beta$-bounded symmetric coefficients bounded by $C_{f}>0$, one has

[TABLE]

∎

Next the proof of Proposition 4.2 is given.

Proof.

Step 1: (delete unbounded part)

In the case $\theta=0$, since $f^{w_{0}}_{r}(u,\bar{u})$ has $(\beta,0)$-type symmetric coefficients, assume that $f^{w_{0}}_{r}(u,\bar{u})$ has the following form

[TABLE]

Under the definition of Possion bracket, it holds that

[TABLE]

For the sake of convenience, rewrite $\{f^{w_{0}}_{r}(u,\bar{u}),\|u\|_{p}^{2}\}_{w_{0}}$ as the sum of the following two parts

[TABLE]

[TABLE]

Since the coefficients of $f^{w_{0}}_{r}(u,\bar{u})$ are $(\beta,0)$-type symmetric, take complex conjugation to $O^{-}(u,\bar{u})$ and obtain

[TABLE]

Together with (4.30), rewrite $\{f^{w_{0}}_{r}(u,\bar{u}),\|u\|_{p}^{2}\}_{w_{0}}$ as the following

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

The estimate of $\{f_{r}^{w_{0}}(u,\bar{u}),\|u\|_{p}^{2}\}_{w_{0}}$ follows the estimates of $A^{0}(u,\bar{u})$, $A^{1}(u,\bar{u})$ and $A^{2}(u,\bar{u})$. In fact, I can not estimate $A^{2}(u,\bar{u})$ by Corollary 2 directly, because the coefficients of $A^{2}(u,\bar{u})$ are not satisfy condition (4.13). Fortunately the bad unbounded part (not satisfy the condition (4.13)) in $A^{2}(u,\bar{u})$ can be handled by $(\beta,0)$-type symmetric property of $f^{w_{0}}_{r}(u,\bar{u})$. Then $A^{2}(u,\bar{u})$ is transformed into a new form, the coefficients of which satisfy (4.13). Thus, the estimate of $A^{2}(u,\bar{u})$ can be obtained by Corollary 2.

Now the details of deleting the unbounded terms in $A^{2}(u,\bar{u})$ are given in the follows. For any $(l^{0},k^{0},i^{0})\in{\cal A}_{(f^{w_{0}})^{i}_{r,lk}}$, it holds that

[TABLE]

Using (4.34)

[TABLE]

Since the two parts in the right side of (4.35) are real value functions, they are invariant under complex conjugation. Taking complex conjugation to the second part of the right side of (4.35) and using that fact that $f_{r}^{w_{0}}(u,\bar{u})$ has $(\beta,0)$-type symmetric coefficients, it leads to

[TABLE]

Together with (4.35) and (4.36), it follows

[TABLE]

In the case $\theta=1$,

[TABLE]

where

[TABLE]

Since the coefficients of $f^{w_{1}}_{r}(u,\bar{u})$ are $(\beta,1)$-type symmetric, it holds that

[TABLE]

From (4.38) and (4.39), one has

[TABLE]

the last equation is obtained by $\mbox{sgn}(j)|j|=j$. For any $l,k\in\mathbb{N}^{\mathbb{Z}^{*}}$ and any $j\in\mathbb{Z}^{*}$ with ${\cal M}(l,k)=i\in M_{f_{r}^{w_{1}}}$ and $l_{j}\neq 0$, one has that

[TABLE]

Together with (4.40) and (4.41),

[TABLE]

where

[TABLE]

In order to estimate of $E(u,\bar{u})$ and $B(u,\bar{u})$, $E(u,\bar{u})$ is rewritten as the sum of $E^{1}(u,\bar{u})$ and $E^{2}(u,\bar{u})$, where

[TABLE]

and $B(u,\bar{u})$ is rewritten as the sum of $B^{1}(u,\bar{u})$ and $B^{2}(u,\bar{u})$, where

[TABLE]

Using the $(\beta,1)$-type symmetric property of $f^{w_{1}}_{r}(u,\bar{u})$, delete the bad unbounded parts (not satisfy the condition (4.13)) in $B^{2}(u,\bar{u})$ and $E^{2}(u,\bar{u})$ in the followings. For any $l,k\in\mathbb{N}^{\mathbb{Z}^{*}}$ and any $j,n\in\mathbb{Z^{*}}$ with $k_{n}\neq 0$, $l_{j}\neq 0$ and ${\cal M}(l,k)=i$, the following equation holds true

[TABLE]

Take (4.50) into $E^{2}(u,\bar{u})$, one obtains that

[TABLE]

Take complex conjugation to the second part of the right side of (4.51) and get

[TABLE]

the last equation is obtained by the coefficients of $f^{w_{1}}_{r}(u,\bar{u})$ being $(\beta,1)$-type symmetric. Equation (4.52) leads to

[TABLE]

Similarly, by the fact

[TABLE]

the following equation holds true

[TABLE]

Summarize this step, it satisfies that

[TABLE]

and

[TABLE]

Step 2:Estimate $A^{0}(u,\bar{u})$-$A^{2}(u,\bar{u})$, $B^{1}(u,\bar{u})$, $B^{2}(u,\bar{u})$, $E^{1}(u,\bar{u})$, $E^{2}(u,\bar{u})$ and $D(u,\bar{u})$.

It is clear that $A^{0}(u,\bar{u})$ can be written as an inner product of the vector fields $G:=(\langle j\rangle^{p}\bar{u}_{j})_{j\in\mathbb{Z}}$ and $F:=(F_{j})_{j\in\mathbb{Z}}$, where

[TABLE]

and

[TABLE]

Noting the fact the momentum of $(f^{w_{0}})_{r,lk}^{i}u^{l}\bar{u}^{k}$ being $i$ and $0\leq l_{j}^{0}\leq l_{j}$, $0\leq k_{j}^{0}\leq k_{j}$, the following equation holds true

[TABLE]

Using the fact that $|x|^{p}$ ($p\geq 2$) is convex function and (4.21), it follows that

[TABLE]

In view of (4.57) and (4.58), it holds that

[TABLE]

Since $f_{r}^{w_{0}}$ has $(\beta,0)$-type symmetric coefficients semi-bounded by $C_{f^{w_{0}}}$, together with (4.59), the coefficients of vector field $F$ in (4.55) are bounded by the following

[TABLE]

Then by Corollary 2, the following inequality holds true

[TABLE]

Using the same method, one has

[TABLE]

In order to estimate $A^{1}(u,\bar{u})$, the following inequality is given for any $j,\ m\in\mathbb{Z}^{*}$

[TABLE]

with $a\geq 2$.

Take $a=p$ into (4.62). Given $l,k\in\mathbb{N}^{\mathbb{Z}^{*}}$ fulfilling ${\cal M}(l,k)=i$ with $k_{j}\neq 0$ and ${l}^{0}_{m}\neq 0$, together with (4.21), it holds that

[TABLE]

The similar inequality holds in the case $k_{j}\neq 0$, $k^{0}_{m}\neq 0$ ($m\neq j$).

Since

[TABLE]

take the right side of (4.64) as an inner product of vectors $F$ and $G$, where

[TABLE]

and

[TABLE]

By (4.63) and $f^{w_{0}}_{r}(u,\bar{u})$ having $(\beta,0)$-type symmetric coefficients semi-bounded by $C_{f^{w_{0}}}$, the coefficients of $G_{j}$ are bounded by

[TABLE]

Using Corollary 2, one has

[TABLE]

By the same method, $E^{1}(u,\bar{u})$ and $B^{1}(u,\bar{u})$ satisfy the following inequalities

[TABLE]

and

[TABLE]

Since $A^{2}(u,\bar{u})$, $B^{2}(u,\bar{u})$ and $E^{2}(u,\bar{u})$ can be estimated by the same method, I only give the details of estimate of $A^{2}(u,\bar{u})$ in (4.37).

[TABLE]

where $F=(F_{j})_{j\in\mathbb{Z}^{*}}$ and $G=(G_{j})_{j\in\mathbb{Z}^{*}}$ with $F_{j}=\langle j\rangle^{p}u_{j}$ and

[TABLE]

From (4.21) and (4.50), one has

[TABLE]

Then the coefficients of $G$ are bounded by

[TABLE]

Using Corollary 2, it holds that

[TABLE]

Similarly,

[TABLE]

Thus, the following inequalities hold true

[TABLE]

and

[TABLE]

∎

5 Birkhoff Normal form and non resonant condition

5.1 $(\theta,\gamma,\alpha,N)$-normal form

In order to guarantee the boundedness of the symplectic transformation, it is required a strong non resonant condition. Given integers $N>0$ and $r\geq 3$, let

[TABLE]

Definition 5.1**.**

Given $\gamma>0$, $\alpha>0$, $\theta\in\{0,1\}$ and $N,r\in\mathbb{N}$, frequencies $\omega=(\omega_{j})_{j\in\mathbb{Z}^{*}}$ is said to be $r$-degree $(\theta,\gamma,\alpha,N)$-non resonant*, if for any $(l,k)$ belongs to*

[TABLE]

it satisfies

[TABLE]

where

[TABLE]

With the $r$-degree $(\theta,\gamma,\alpha,N)$-non resonant condition, a symplectic transformation will be obtained. Under this transformation the Hamiltonian function $H(u,\bar{u})$ are transformed into the sum of an $r$-degree normal form and a remainder term. However this $r$-degree normal form is not a standard Hamiltonian Birkhoff normal form (A standard $2r$-degree Hamiltonian Birkhoff normal form in variables $(u,\bar{u})$ is an $r$-degree polynomial which only depends on variables $(|u_{j}|^{2})_{j\in\mathbb{Z}^{*}}$). Now I introduce a definition to describe this normal form.

Definition 5.2**.**

Given $\gamma>0$, $\alpha>0$ and an integer $N>0$, call an $r$-degree polynomial

[TABLE]

$\bm{(\theta,\gamma,\alpha,N)}$-normal form with respect to $\omega\in\mathbb{R}^{\mathbb{Z}^{*}}$, if for any $(l,k)\in E_{r,N}$ ($E_{r,N}$ is defined in (5.1)) with ${\cal M}(l,k)=i\in M_{f_{t}}$ and $f^{i}_{t,lk}\neq 0$, it satisfies that

[TABLE]

where $\langle\ ,\ \rangle$ is the inner product of space $\ell^{2}(\mathbb{Z}^{*},\mathbb{C})$, $I_{\theta}$ and $M_{l,k}$ are defined in (3.9) and (5.8).

Remark 5.1**.**

Let $f(u,\bar{u})$ be an $r$-degree polynomial. For any given $\gamma>0$, $\alpha>0$, $\theta\in\{0,1\}$, integers $N>0$ and $\omega\in\mathbb{R}^{\mathbb{Z}^{*}}$, denote

[TABLE]

with

[TABLE]

as $\bm{(\theta,\gamma,\alpha,N)}$-normal form of $f(u,\bar{u})$ with respect to $\omega$. Moreover, suppose that $f^{w_{\theta}}(u,\bar{u})$ has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{\theta}>0$ ($\theta\in\{0,1\}$). So does $\Gamma^{\omega}_{(\theta,\gamma,\alpha,N)}f^{w_{\theta}}(u,\bar{u})$.

Remark 5.2**.**

Assume that $\omega=(\omega_{j})_{j\in\mathbb{Z}^{*}}$ is an $r_{*}$-degree $(\theta,\gamma,\alpha,N)$-non resonant frequencies and $f^{w_{\theta}}(u,\bar{u})$ is an $r_{*}$-degree $(\theta,\gamma,\alpha,N)$-normal form with respect to $\omega$. Then $f^{w_{\theta}}(u,\bar{u})$ has the following form

[TABLE]

where

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

Lemma 5.1**.**

Let $\omega=(\omega_{j})_{j\in\mathbb{Z}^{*}}$ be an $r$-degree $(\theta,\gamma,\alpha,N)$ non-resonant frequency. Suppose that $f_{r}^{w_{\theta}}(u,\bar{u})$ is an $r$-degree ($r\geq 3$) homogeneous $(\theta,\gamma,\alpha,N)$-normal form with respect to $\omega$ and has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{f_{r}^{w_{\theta}}}>0$ ($\beta-p\geq 2$). Then for any $(u,\bar{u})\in{{\mathcal{H}}^{p}}(\mathbb{Z}^{*},\mathbb{C})$ it has

[TABLE]

Remark 5.3**.**

Although $f_{r}^{w_{\theta}}(u,\bar{u})$ in Lemma 5.1 is at most 2 degree about $(\Gamma_{>N}u,\Gamma_{>N}\bar{u})$, it still satisfies an inequality similar to (1) in Corollary 1. But for the general polynomials being at most 2 degree about $(\Gamma_{>N}u,\Gamma_{>N}\bar{u})$, this inequality dose not always hold.

Proof of Lemma 5.1.

From Remark 5.2,

[TABLE]

where ${\cal A}^{w_{\theta}}_{r}(u,\bar{u})$, ${\cal B}_{r}^{w_{\theta}}(u,\bar{u})$ and ${\cal C}_{r}^{w_{\theta}}(u,\bar{u})$ are defined in (5.9)-(5.11) in Remark 5.2.

Step 1: Calculate $\{{\cal A}_{r}^{w_{\theta}}(u,\bar{u}),\|u\|_{p}^{2}\}_{w_{\theta}}$.

It is easy to verify that

[TABLE]

and

[TABLE]

Step 2: Estimate $\{{\cal B}_{r}^{w_{\theta}}(u,\bar{u}),\|u\|_{p}^{2}\}_{w_{\theta}}$.

Since the function ${\cal B}_{r}^{w_{\theta}}(u,\bar{u})$ depends on $(u_{j},\bar{u}_{j})_{|j|\leq N}$ and $(|u_{j}|^{2})_{|j|>N}$, the following equation holds true

[TABLE]

Thus,

[TABLE]

From the definition of $\{,\}_{w_{\theta}}$ and the structure of ${\cal B}^{w_{\theta}}_{r}(u,\bar{u})$, $\{{\cal B}^{w_{\theta}}_{r}(u,\bar{u}),\sum_{|j|\leq N}\langle j\rangle^{2p}|u_{j}|^{2}\}_{w_{\theta}}$ still depends on $(|u_{j}|^{2})_{|j|>N}$ and $(\Gamma_{\leq N}u,\Gamma_{\leq N}\bar{u})$. To be more specific,

[TABLE]

For any no zero term $u^{l}\bar{u}^{k}$ of the right side of (5.17), its index $(l,k)$ satisfies

[TABLE]

From (5.18), for any $|j|\leq N$ with $l_{j}\neq 0$, it satisfies that

[TABLE]

Moreover, given $p\geq 2$, by (4.21) and (5.19), the following inequalities hold true

[TABLE]

and

[TABLE]

I will estimate the coefficients of $\{{\cal B}_{r}^{w_{\theta}}(u,\bar{u}),\|u\|_{p}^{2}\}_{w_{\theta}}$. When $\theta=0$, for any $(l^{0},k^{0},i^{0})\in{\cal A}_{(f^{w_{0}})^{i}_{r,lk}}$ with $(l,k)\in\Omega^{\theta}_{{\cal B}_{r}^{N}}$, it holds that

[TABLE]

the last inequality hold by (4.21). From (5.20) and (5.22), the coefficients of $\{{\cal B}_{r}^{w_{0}}(u,\bar{u}),\|u\|_{p}^{2}\}_{w_{0}}$ are bounded by

[TABLE]

Similarly, in the case $\theta=1$, using (5.21), the coefficients of $\{{\cal B}_{r}^{w_{1}}(u,\bar{u}),\|u\|_{p}^{2}\}_{w_{1}}$ are bounded by

[TABLE]

By Corollary 2, it holds that

[TABLE]

Step 3: Estimate $\{{\cal C}_{r}^{w_{\theta}}(u,\bar{u}),\|u\|_{p}^{2}\}_{w_{\theta}}$.

When $\theta=0$,

[TABLE]

the last equation holds by the fact that

[TABLE]

It is easy to verify that $\{{\cal C}_{r}^{w_{0}}(u,\bar{u}),\sum_{|t|\leq N}|u_{t}|^{2}\langle t\rangle^{2p}\}_{w_{0}}$ is still dependent on $(u_{j_{0}}\bar{u}_{-j_{0}})_{|j_{0}|>N}$ and $(\Gamma_{\leq N}u,\Gamma_{\leq N}\bar{u})$. Using the method of estimate $\{{\cal B}_{r}^{(w_{\theta})}(u,\bar{u}),\sum_{|t|\leq N}|u_{t}|^{2}\langle t\rangle^{2p}\}_{w_{\theta}}$, the estimate of $\{{\cal C}_{r}^{w_{0}}(u,\bar{u}),\sum_{|t|\leq N}|u_{t}|^{2}\langle t\rangle^{2p}\}_{w_{0}}$ is obtained.

When $\theta=1$,

[TABLE]

Using the method of estimate $\{{\cal B}_{r}^{w_{\theta}}(u,\bar{u}),\sum_{|t|\leq N}|u_{t}|^{2}\langle t\rangle^{2p}\}_{w_{\theta}}$ in step 2, the estimate of

$\{{\cal C}_{r}^{w_{1}}(u,\bar{u}),\sum_{|t|\leq N}|u_{t}|^{2}\langle t\rangle^{2p}\}_{w_{1}}$ can be obtained. The estimate of $\{{\cal C}_{r}^{w_{1}}(u,\bar{u}),\sum_{|t|>N}|u_{t}|^{2}\langle t\rangle^{2p}\}_{w_{1}}$ will be obtained by the following. For any nonzero term of ${\cal C}_{r}^{w_{1}}(u,\bar{u})$ with index $(l,k)$, there exists $|j_{0}|>N$ with $l_{j_{0}}=1$, $k_{-j_{0}}=1$ (or $l_{-j_{0}}=1$, $k_{j_{0}}=1$) such that

[TABLE]

which follows from ${\cal M}(l,k)=i$. From the relation (5.27), using (4.21) it holds that

[TABLE]

and

[TABLE]

By (5.28) and (5.29), the coefficients of

[TABLE]

are smaller than

[TABLE]

Using Corollary 2, it holds that

[TABLE]

Summing (5.14), (5.15), (5.25) and (5.32), inequality (5.12) is obtained. ∎

5.2 Birkhoff normal form theorem

In this subsection, construct a coordinate transformation under which the Hamiltonian system (3.10) will have an $r_{*}+3$ degree $(\theta,\gamma,\alpha,N)$-normal form, for any given positive $r_{*}$.

Theorem 4** (Birkhoff normal form theorem).**

Suppose that system (3.10) satisfies assumptions ${\cal A}_{\theta}$-${\cal B}_{\theta}$ and $P^{w_{\theta}}(u,\bar{u})$ in $H^{w_{\theta}}(u,\bar{u})$ defined in (3.11) has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{\theta}>0$ ($\beta$ is big enough). Given $\alpha>1$, $0<\gamma\ll 1$ and integer $r_{*}>0$, take $p$ satisfying $(\beta-4)/2>p>2+4\alpha(r_{*}+1)r_{*}^{2}$. There exist a positive real number $\widetilde{R}>0$ and a Lie-transformation ${\cal T}_{w_{\theta}}^{(r_{*})}$: $B_{p}(\widetilde{R}/3)\rightarrow B_{p}(\widetilde{R})$ such that:

For any $R<\widetilde{R}$ and any integer $N$ fulfilling

[TABLE]

the transformation ${\cal T}_{w_{\theta}}^{(r_{*})}$ puts Hamiltonian $H^{w_{\theta}}$ into

[TABLE]

which satisfies that

Both $Z^{(r_{*},w_{\theta})}$ and $\mathcal{R}^{N(r_{*},w_{\theta})}$ are $(r_{*}+3)$-degree polynomials and $\mathcal{R}^{T(r_{*},w_{\theta})}$ is a power series which starts with $r_{*}+4$ degree polynomial. All of them have $(\beta,\theta)$-type symmetric coefficients semi-bounded by C({\theta},r_{*}):=C_{\theta}\big{(}\frac{2^{\beta}N^{2\alpha}}{\gamma}\big{)}^{(r_{*}+1)}.

• 2)

The polynomial $Z^{(r_{*},w_{\theta})}(u,\bar{u})$ is $r_{*}+3$-degree $(\theta,\gamma,\alpha,N)$-normal form with respect to $\omega^{w_{\theta}}$.

• 3)

The polynomial ${\mathcal{R}}^{N(r_{*},w_{\theta})}(u,\bar{u}):=\sum_{r=3}^{r_{*}+3}\Gamma_{>2}^{N}g_{r+4}^{(r,w_{\theta})}$, where $g_{r+4}^{(r,w_{\theta})}$ is an $(r+4)$-degree homogeneous polynomial with $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C({\theta},r)$;

• 4)

The canonical Lie-transformation ${\cal T}^{(r_{*})}_{w_{\theta}}$ satisfies

[TABLE]

where $C(\theta,p,r_{*})$ is a constant dependent on $\theta,p$ and $r_{*}$.

5.3 Important Lemmas in the Proof of Theorem 4

In order to prove Theorem 4, it need not only to construct a bounded canonical transformation under which the Hamiltonian $H^{w_{\theta}}$ in (3.11) has an $r_{*}+3$-degree $(\theta,\gamma,\alpha,N)$-normal form, but also to show that the new Hamiltonian function has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C(\theta,r_{*})$. First, let us review the definition of canonical transformation.

Definition 5.3**.**

Call a map $\phi:{\cal H}^{p}(\mathbb{Z}^{*},\mathbb{C})\ni(u,\bar{u})\rightarrow(\xi,\bar{\xi})\in{\cal H}^{p}(\mathbb{Z}^{*},\mathbb{C})$ canonical transformation under a symplectic form $w_{\theta}$ (or a symplectic change of coordiantes), if $\phi$ is a diffeomorphism and preserves the Poisson bracket, i.e. $\{f,g\}_{w_{\theta}}\circ\phi=\{f\circ\phi,g\circ\phi\}_{w_{\theta}}.$

A convenient way of constructing canonical transformations is as followings. Let $\Phi_{S^{w_{\theta}}}^{t}$ be the flow generated by a regular function $S^{w_{\theta}}(u,\bar{u})$ defined in ${\cal H}^{p}(\mathbb{Z}^{*},\mathbb{C})$ with respect to the symplectic structure $w_{\theta}$. $\Phi_{S^{w_{\theta}}}^{t}|_{t=0}=id$. If $\Phi_{S^{w_{\theta}}}^{t}$ is well defined up to $t=1$, then the map $\Phi_{S^{w_{\theta}}}^{t}|_{t=1}$ is called a Lie transformation associated to $S^{w_{\theta}}(u,\bar{u})$ under symplectic form $w_{\theta}$. $\Phi_{S^{w_{\theta}}}^{1}$ is canonical.

Given a regular function $g$, the new function $g\circ{\Phi}_{S^{w_{\theta}}}^{t}$ satisfies

[TABLE]

Thus the Taylor expansion of $g\circ\Phi_{S^{w_{\theta}}}^{t}$ in the variable $t$ is

[TABLE]

where

[TABLE]

Take $t=1$ and it follows that

[TABLE]

In this paper, denote $\prec$ as $\leq\tilde{C}\cdot$, where $\tilde{C}>0$ is independent of $R$. To improve the order of the $(\theta,\gamma,\alpha,N)$-normal form of $H^{w_{\theta}}$, it needs to solve a linear equation to find a suitable generated function $S^{w_{\theta}}$ under symplectic form $w_{\theta}$. The following lemma is to do this with respect to $w_{\theta}$-Possion bracket.

Lemma 5.2**.**

(Homological Equation)* Given an integer $N>0$, real numbers $\gamma>0$ and $\alpha>0$, suppose that an $r$-degree homogeneous polynomial $g^{w_{\theta}}(u,\bar{u})$ has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{g^{w_{\theta}}}>0$ ($\theta\in\{0,1\}$). Then there exists an unique $S^{w_{\theta}}(u,\bar{u})$ such that*

[TABLE]

where $H_{0}^{w_{\theta}}:=\sum_{j\in\mathbb{Z}^{*}}\omega^{w_{\theta}}_{j}|u_{j}|^{2}$ with $\omega^{w_{\theta}}_{j}\in\mathbb{R}$. Moreover for any $(u,\bar{u})\in{\cal H}^{p}(\mathbb{Z}^{*},\mathbb{C})$ the Hamiltonian vector of $S^{w_{\theta}}(u,\bar{u})$ holds

[TABLE]

Proof.

By the definition of Possion bracket $\{\ ,\ \}_{w_{\theta}}$, the solution $S^{w_{\theta}}(u,\bar{u})$ of (5.36) is still an $r$-degree homogeneous polynomial and has the following form

[TABLE]

with undetermined coefficients. Since $g^{w_{\theta}}(u,\bar{u})$ has $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{g^{w_{\theta}}}$, by Remark 4.2 and Remark 5.1, $\Gamma^{\omega_{\theta}}_{(\theta,\gamma,\alpha,N)}\Gamma_{\leq 2}^{N}g^{w_{\theta}}(u,\bar{u})$ is an $r$-degree $(\theta,\gamma,\alpha,N)$-normal form of $\Gamma_{\leq 2}^{N}g^{w_{\theta}}(u,\bar{u})$ with $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{g^{w_{\theta}}}$, and its coefficients have the following form

[TABLE]

where $M_{l,k}$ is defined in Definition 5.1. Take (5.37) into equation (5.36) and get that for any $i\in M_{g^{w_{\theta}}}$ and any $l,k\in\mathbb{N}^{\mathbb{Z}^{*}}$ with $|l+k|=r$ and ${\cal M}(l,k)=i$,

[TABLE]

which means that the coefficients of $S^{w_{\theta}}(u,\bar{u})$ has the following form

[TABLE]

and satisfy that

[TABLE]

the second equality holds by $\Gamma_{\leq 2}^{N}g^{w_{\theta}}(u,\bar{u})$ having symmetric coefficients from Remark 4.2 and $\omega^{w_{\theta}}_{j}\in\mathbb{R}$ ($j\in\mathbb{Z}^{*}$).

The norm of Hamiltonian vector field $X_{S^{w_{\theta}}}^{w_{\theta}}$

[TABLE]

equals to the $\ell^{2}$ norm of the vector fields

[TABLE]

and

[TABLE]

When $\theta=0$, for any $l,k\in\mathbb{N}^{\mathbb{Z}^{*}}$ with ${\cal M}(l,k)=i$ and $k_{j}\neq 0$, by (4.21), it holds

[TABLE]

For any $(l^{0},k^{0},i^{0})\subset{\cal A}_{f_{r,lk}^{i}}$, by (4.21) the following inequality holds

[TABLE]

By (5.40) and (5.43), the coefficients of $S^{w_{0}}$ satisfy that

[TABLE]

From (5.42) and (5.44), the coefficients of vector fields $Q^{w_{0}}_{1}(u,\bar{u})$ and $Q^{w_{0}}_{2}(u,\bar{u})$ fulfill

[TABLE]

By (5.45), using Corollary 2, it holds that

[TABLE]

When $\theta=1$, in order to estimate the $\ell^{2}$-norm of $Q^{w_{1}}_{1}(u,\bar{u})$ and $Q^{w_{1}}_{2}(u,\bar{u})$, let us consider the coefficients of $Q_{1}^{w_{1}}(u,\bar{u})$ and $Q_{2}^{w_{1}}(u,\bar{u})$ firstly. For any $i\in M_{S^{w_{1}}}$ and any $l,k\in\mathbb{N}^{{\mathbb{Z}}^{*}}$ satisfying $|l+k|=r,\ {\cal M}(l,k)=i$ and $k_{j}\neq 0$ (or $l_{j}\neq 0$), using (5.40), the coefficients of $u^{l}\bar{u}^{k-e_{j}}$ in $Q^{w_{1}}_{1}$ are bounded by the following

[TABLE]

and the coefficients of $u^{l-e_{j}}\bar{u}^{k}$ in $Q_{2}^{w_{1}}(u,\bar{u})$ are bounded by

[TABLE]

By Corollary 2 and (5.47)-(5.48), the following estimate is obtained

[TABLE]

∎

The following Lemma shows that the Possion bracket of an $\tilde{r}$-degree homogeneous polynomial $f^{w_{\theta}}(u,\bar{u})$ with $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{f^{w_{\theta}}_{\tilde{r}}}>0$ and the solution $S^{\omega_{\theta}}(u,\bar{u})$ to equation (5.36) is still of $(\beta,\theta)$-type symmetric coefficients. Moreover, its coefficients satisfy some inequalities.

Lemma 5.3**.**

Let an $\tilde{r}$-degree homogeneous polynomial $f^{w_{\theta}}(u,\bar{u})$ ($\theta\in\{0,1\}$) have $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C_{f^{w_{\theta}}_{\tilde{r}}}>0$. Then the possion bracket of $f^{w_{\theta}}(u,\bar{u})$ and the solution $S^{w_{\theta}}(u,\bar{u})$ to equation (5.36) under the symplectic form $w_{\theta}$ is an $(\tilde{r}+r-2)$-degree homogeneous polynomial with $(\beta,\theta)$-type symmetric coefficients and it holds that

• •

when $\theta=0$, for any $l^{\prime},\ k^{\prime},\ i$ fulfilling $|l^{\prime}+k^{\prime}|=\tilde{r}+r-2$ and ${\cal M}(l^{\prime},k^{\prime})=i$, it holds that

[TABLE]

• •

when $\theta=1$ the following inequality holds true

[TABLE]

Remark 5.4**.**

Under the same assumptions of Lemma 5.3, for any integer $\nu\geq 1$, $f^{w_{\theta}}_{(\nu,S^{w_{\theta}})}$ is an \big{(}\tilde{r}+\nu(r-2)\big{)}-degree homogeneous polynomial with $(\beta,\theta)$-type symmetric coefficients.

• •

When $\theta=0$, the following inequality holds

[TABLE]

• •

When $\theta=1$, it holds that

[TABLE]

Before proving Lemma 5.3, I denote a set of indexes and give a Lemma to count the number of this set. This Lemma is used to prove Lemma 5.3.

For any $(l^{\prime},k^{\prime})\in\mathbb{N}^{\mathbb{Z}^{*}}\times\mathbb{N}^{\mathbb{Z}^{*}}$ and any $i^{\prime}\in\mathbb{Z}$, let

[TABLE]

where

A:

$|l+k|=\tilde{r},\ |L+K|={r},\ |\Gamma_{>N}(L+K)|\leq 2;$

B:

${\cal M}(l,k)=i_{1},\ {\cal M}(L,K)=i_{2},\ i^{\prime}=i_{1}+i_{2};$

D1:

$(l-e_{j})+L=l^{\prime},\ k+(K-e_{j})=k^{\prime},\ \mbox{with}\ l_{j}>0\ \mbox{and}\ K_{j}>0;$

D2:

$l+(L-e_{j})=l^{\prime},\ (k-e_{j})+K=k^{\prime},\ \mbox{with}\ L_{j}>0\ \mbox{and}\ k_{j}>0.$

From the definition of set $\Omega(l^{\prime},k^{\prime},i^{\prime})$, if element $((l,k,i_{1}),\ (L,K,i_{2}),j)\in\Omega(l^{\prime},k^{\prime},i^{\prime})$, then $((k,l,-i_{1}),\ (K,L,-i_{2}),j)\in\Omega(k^{\prime},l^{\prime},-i^{\prime})$.

Lemma 5.4**.**

Fix $\beta\geq 2$. For any given $l^{\prime},k^{\prime}\in\mathbb{N}^{\mathbb{Z}^{*}}$ with $|l^{\prime}+k^{\prime}|=r+\tilde{r}-2$ and ${\cal M}(l^{\prime},k^{\prime})=i^{\prime}$, it holds

[TABLE]

Proof.

Consider the non-zero components of vectors $k^{\prime}$ and $l^{\prime}$. For example, $e_{j}$ has only one non-zero component with index $j$, being 1; Taking multiplicity into account, regard that $ke_{j}$($k$ is a positive integer) has $k$ non-zero components whose values are 1 and their indexes are $j$. So $(l^{\prime},k^{\prime})$ with $|l^{\prime}+k^{\prime}|=r+\tilde{r}-2$ have $r+\tilde{r}-2$ non-zero components whose values are $1$.

Denote

[TABLE]

It follows

[TABLE]

The element in $\Omega_{j,i_{2}}(l^{\prime},k^{\prime},i^{\prime})$ is unique determined, if $(l-e_{j},k)$ is fixed. The estimate of $\tilde{\sharp}\Omega_{j,i_{2}}(l^{\prime},k^{\prime},i^{\prime})$ is obtained as follows.

In the case $|j|\leq N$, since $|\Gamma_{>N}({L}+{K})|\leq 2$, there are at least $r-3$ non-zero components of $(l^{\prime},k^{\prime})$ coming from $(L,K)$ with the indexes being bounded by $N$ and the choices of that is smaller than $(2N+1)^{r-3}$. As for the remaining three components of $(L,K)$ whose values are 1, one of their positions is $j$ with $|j|\leq N$; One position among the other two can be selected from the rest non-zero components of $(l^{\prime},k^{\prime})$ and the choices is $\tilde{r}+1$; The last one may be determined by the fact that ${\cal M}(L,K)=i_{2}$. It holds

[TABLE]

In the case $|j|>N$, there are at least $r-2$ value-1 components of $(l^{\prime},k^{\prime})$ coming from $(L,K)$ whose indexes are bounded by $N$, and there are at most $(2N+1)^{r-2}$ choices; One position of the last two value-1 components of $(L,K)$ is chosen from the rest $\tilde{r}$ non-zero components of $(l^{\prime},k^{\prime})$ and the choices is $\tilde{r}$; The position of the last component of $(L,K)$ is determined by the momentum of $(L,K)$ being $i_{2}$. It holds

[TABLE]

The result is obtained from (5.50) and (5.51). ∎

In the following, the proof of the Lemma 5.3 is given.

Proof.

By the definition of $\{\cdot,\cdot\}_{w_{\theta}}$, the following equation holds

[TABLE]

with $M_{f^{w_{\theta}}_{(1,S^{w_{\theta}})}}:=\{i=i_{1}+i_{2}\ |\ i_{1}\in M_{f^{w_{\theta}}}\subset\mathbb{Z},\ i_{2}\in M_{S^{w_{\theta}}}\subset\mathbb{Z}\}$ and

[TABLE]

In the case $\theta=0$, I will give the exact definition of ${\cal A}_{(f^{w_{0}}_{(1,S^{w_{0}})})^{i^{\prime}}_{\tilde{r}+r-\!2,l^{\prime}k^{\prime}}}\!$ and $(\!f^{w_{0}}_{(1,S^{w_{0}})}\!)_{\tilde{r}+r-2,l^{\prime}k^{\prime}}^{i^{\prime}(l^{\prime 0},k^{\prime 0},i^{\prime 0})}$ and prove that the coefficients $(\!f^{w_{0}}_{(1,S^{w_{0}})})^{i^{\prime}}_{\tilde{r}+r-2,l^{\prime}k^{\prime}}$ can be rewritten as the following form

[TABLE]

and satisfy

[TABLE]

In order to describe the set ${\cal A}_{(f^{w_{0}}_{(1,S^{w_{0}})})^{i^{\prime}}_{\tilde{r}+r-2,l^{\prime}k^{\prime}}}$ clearly, for any fixed \big{(}(l,k,i_{1}),\ (L,K,i_{2}),\ j\big{)}\in\Omega^{w_{0}}(l^{\prime},k^{\prime},i^{\prime}), define a mapping $D$ on set ${\cal A}_{(f^{w_{0}})_{\tilde{r},lk}^{i_{1}}}$, for any $(l^{0},k^{0},i_{1}^{0})\in{\cal A}_{(f^{w_{0}})_{\tilde{r},lk}^{i_{1}}}$,

[TABLE]

Base on the set $\Omega(l^{\prime},k^{\prime},i^{\prime})$ and the map $D$, denote

[TABLE]

and

[TABLE]

where

[TABLE]

It is easy to check that $D$ is not an inverse mapping from ${\cal A}_{(f^{w_{0}})_{\tilde{r},lk}^{i_{1}}}$ to $D{\cal A}_{(f^{w_{0}})_{\tilde{r},lk}^{i_{1}}}$. Denote

[TABLE]

where $D^{-1}(l^{\prime 0},k^{\prime 0},i^{\prime 0}):=\{(l^{0},k^{0},i^{0})\in{\cal A}_{(f^{w_{0}})^{i^{0}}_{\tilde{r},lk}}\ |\ D(l^{0},k^{0},i^{0})=(l^{\prime 0},k^{\prime 0},i^{\prime 0})\}.$ For any $(l^{\prime 0},k^{\prime 0},i^{\prime 0})\in{\cal A}_{(f^{w_{0}}_{(1,S^{w_{0}})})^{i^{\prime}}_{\tilde{r}+r-2,l^{\prime}k^{\prime}}}$, it is easy to verify that $(k^{\prime}-k^{\prime 0},l^{\prime}-l^{\prime 0},i^{\prime 0}-i^{\prime})\in{\cal A}_{(f^{w_{0}}_{(1,S^{w_{0}})})^{-i^{\prime}}_{\tilde{r}+r-2,k^{\prime}l^{\prime}}}$. Moreover, by (5.65) and the facts that $f^{w_{0}}$ having $(\beta,0)$-type symmetric coefficients and $S^{w_{0}}$ having symmetric coefficients, it holds

[TABLE]

the last second equation is holding by the definition of $D$ in (5.64) and

[TABLE]

So $f^{w_{0}}_{(1,S^{w_{0}})}$ has $(\beta,0)$-type symmetric coefficients semi-bounded by $C_{g^{w_{0}}_{r}}$. By (5.65), (5.44) in Lemma 5.2, it holds

[TABLE]

By (5.66) and Lemma 5.4, it follows that

[TABLE]

When $\theta=1$, the coefficients of $f^{w_{1}}_{(1,S^{w_{1}})}$ have the following form

[TABLE]

where

[TABLE]

Since $f^{w_{1}}(u,\bar{u})$ and $g^{w_{1}}(u,\bar{u})$ have $(\beta,1)$-type symmetric coefficients ($g^{w_{1}}(u,\bar{u})$ is given in equation (5.36)), from (5.69) the coefficients of ${f}^{w_{1}}_{(1,S^{w_{1}})}$ satisfy that

[TABLE]

From (5.40), for any $(l,k)$ with nonzero $(S^{w_{1}})_{r,lk}^{i}$,

[TABLE]

which implies that

[TABLE]

the last inequality holds by Lemma 5.4. ∎

The proof of Theorem 4 is a purely technical matter and is relegated to Appendix.

6 Proof of Theorem 3

For any given integer $r_{*}\geq 0$, using Theorem 4, there exists a transformation ${\cal T}^{(r_{*})}_{w_{\theta}}$ changing the system (3.10) into

[TABLE]

with Hamiltonian

[TABLE]

The solution $(\tilde{u},\bar{\tilde{u}})$ to (6.4) satisfies

[TABLE]

It is easy to get that

[TABLE]

Using Theorem 4, Proposition 4.2 and Corollary 1, when $N$ satisfies (5.33), it holds that

[TABLE]

the inequality is holding by the fact that for any $(\tilde{u},\bar{\tilde{u}})\in B_{p}(R)$

[TABLE]

By Lemma 5.1 and (6.9), when $N$ satisfies (5.33), it follows that

[TABLE]

Suppose that the initial value to (3.10) satisfies $(u(0),\bar{u}(0))\in B_{p}(R/6)$. If $R$ is small enough, the initial value $(u(0),\bar{u}(0))\in B_{p}(R/6)$ is transformed into

[TABLE]

Together with (6.6)-(6.8) and (6.10)-(6.11), the following inequality holds true

[TABLE]

where T:=\min\{\ |t|\ |\ \|\big{(}\tilde{u}(t),\bar{\tilde{u}}(t)\big{)}\|_{p}=R/2\}, which means that for any $|t|\leq T:=\frac{1}{144C(\theta,p,r_{*})R^{r_{*}-1}},$

[TABLE]

From Theorem 4, when $R\ll 1$, ${\cal T}_{w_{\theta}}^{(r_{*})}$ is an inverse transformation from $B_{p}(R/2)$ to $B_{p}(R)$. Then by (6.13), the solution $(u(t),\bar{u}(t))$ to systems (3.10) with $(u(0),\bar{u}(0))\in B_{p}(R/6)$ satisfies

[TABLE]

7 Proof of Theorem 1 and Theorem 2

7.1 Proof of Theorem 1

It is common knowledge that $(j^{2})_{j\in\mathbb{Z}}$ are the eigenvalues of $-\partial_{xx}$ under periodic boundary condition $\psi(x,t)=\psi(x+2\pi,t)$ with the corresponding eigenfunctions $\{\phi_{j}(x):=\frac{e^{{\bf i}jx}}{\sqrt{2\pi}}\}_{j\in\mathbb{Z}}$ . Take

[TABLE]

into equation (2.2) and obtain a Hamiltonian system,

[TABLE]

with respect to 2-form $w_{0}$ in (3.5), and the Hamiltonian function has the form

[TABLE]

where

[TABLE]

Under assumptions ${\bf A_{1}}$ and ${\bf A_{2}}$ in section 2.1, the power series $P^{w_{0}}(u,\bar{u})$ has the following form

[TABLE]

where ${\cal A}_{(P^{w_{0}})^{i}_{r,lk}}:=\{(l,0,-i)\ |\ i\in M_{P^{w_{1}}_{r}}\subset\mathbb{Z}\}$, $M_{P^{w_{1}}_{r}}$ is a symmetric set and

[TABLE]

Moreover, the following equation holds true for any $l,k$ with $|l+k|=r$ and ${\cal M}(l,k)=i$

[TABLE]

and there exists a constant $C_{1}>0$ such that

[TABLE]

which means that $P^{w_{0}}(u,\bar{u})$ has $(\beta,0)$-type symmetric coefficients semi-bounded by $C_{1}>0$.

Lemma 7.1**.**

For any given integers $r_{*},N>0$ and real numbers $\alpha>4m+r_{*}+8$, $1\gg\gamma>0$, there exists an open subset $\widetilde{\Theta}^{\theta}_{m}\subset\Theta^{\theta}_{m}$ ($\Theta^{\theta}_{m}$ defined in (2.4) and (2.14), respectively) such that for any $V\in\widetilde{\Theta}^{\theta}_{m}$ and any $(l,k)$ belongs to ${O}^{w_{\theta}}_{r_{*}+3,N}$ defined in (5.7), it satisfies

[TABLE]

where

[TABLE]

Moreover,

[TABLE]

Remark 7.1**.**

If $\gamma>0$ is small enough, the set $\widetilde{\Theta}_{m}^{\theta}$ will have a positive measure. In particular, if $\gamma$ approaches to 0, then the measure of $\widetilde{\Theta}_{m}^{\theta}$ will approach to the measure of ${\Theta}_{m}^{\theta}$.

Now give the proof of Lemma 7.1.

Proof.

Denote

[TABLE]

where

[TABLE]

and

[TABLE]

for $n\in\{0,1,2\}$ and $3\leq r\leq r_{*}+3$.

I only give the estimate of the measure of $\Theta^{\theta}_{r,lk}$ in the case $(l,k)\in O^{w_{\theta},r}_{r_{*}+3,N,2}$, which is more complex than the case $(l,k)\in O^{w_{\theta},r}_{r_{*}+3,N,0}\cup O^{w_{\theta},r}_{r_{*}+3,N,0}$.

When the multi-index $(l,k)\in O^{w_{\theta},r}_{r_{*}+3,N,2}$, estimate the measure of $\Theta^{\theta}_{r,lk}$ in two cases.

(1)The first case

[TABLE]

In this case, there exists $|j_{1}|,\ |j_{0}|>N$ such that $l_{j_{1}}=l_{j_{0}}=1$ or $k_{j_{1}}=k_{j_{0}}=1$. Without loss of generality, assume $|j_{0}|\geq|j_{1}|>N$ with $l_{j_{0}}=l_{j_{1}}=1$. So $M_{l,k}=|j_{0}|$ and $|\omega^{w_{\theta}}_{j_{0}}(V)|>j_{0}^{2}-\frac{1}{2}$, $|\omega^{w_{\theta}}_{j_{1}}(V)|>j_{1}^{2}-\frac{1}{2}$. The other frequencies $\omega^{w_{\theta}}_{j}(V)(|j|\leq N)$ are bounded by $|\omega^{w_{\theta}}_{j}|\leq N^{2}+1$.

If $|j_{0}|>4\sqrt{r}N$, it follows that

[TABLE]

That means when $|j_{0}|>4\sqrt{r}N$, the set ${\Theta}^{\theta}_{r,lk}$ is empty. So it is only need to calculate the measure of $\Theta^{\theta}_{r,lk}$ whose multi-index $(l,k)$ being in the following set,

[TABLE]

the number of which are bounded by

[TABLE]

For any fixed $(l,k)\in\widetilde{O^{w_{\theta},r}_{r_{*}+3,N,2a}}$, there exists $4\sqrt{r}N\geq|j_{0}|>N$ fulfilling

[TABLE]

such that

[TABLE]

The measure of $\Theta^{\theta}_{r,lk}$ has the following estimate by (7.8)

[TABLE]

From (7.7) and (7.9), it holds that

[TABLE]

(2)The second case

[TABLE]

Without loss of generality, assume $|j_{1}|>|j_{0}|$ and $M_{l,k}=|j_{1}|$. By (7.5), it holds $|\omega^{w_{\theta}}_{j_{1}}|\geq j_{1}^{2}-\frac{1}{2}$, $|\omega^{w_{\theta}}_{j_{0}}|\leq j_{0}^{2}+\frac{1}{2}$ and $\omega^{w_{\theta}}_{j}\leq N^{2}+1$, ($|j|\leq N$). If $|j_{1}|>4rN^{2}$, the following inequality holds

[TABLE]

That means when $M_{l,k}>4rN^{2}$, the set $\Theta^{\theta}_{r,lk}$ is empty. It only needs to calculate the sum of the set $\Theta^{\theta}_{r,lk}$ with $(l,k)$ being in the following set

[TABLE]

which is bounded by

[TABLE]

There exists $j$ with $4rN^{2}\geq|j|>N$ such that $l_{j}+l_{-j}-k_{j}-k_{-j}\neq 0$ and

[TABLE]

Denote a set

[TABLE]

When $(l,k)\in\widetilde{O^{w_{\theta},r}_{r_{*}+3,N,2b}},$ using the fact $\Theta^{\theta}_{r,lk}\subset\widehat{\Theta}^{\theta}_{r,lk}$ and (7.12), it implies that

[TABLE]

From (7.11) and (7.13), it holds that

[TABLE]

Using the same method, the following inequality holds true

[TABLE]

In view of (7.15) and (7.14), one has

[TABLE]

∎

Now Theorem 1 is obtained by Theorem 3 and Lemma 7.1. The transformation $\psi(x,t)=\sum_{j\in\mathbb{Z}}u_{j}(t)\phi_{j}(x)$ is from $\ell_{p}^{2}$ to $H^{p}([0,2\pi],\mathbb{C})$ and satisfies

[TABLE]

Take $\widetilde{{\varepsilon}}\ll 1$ small enough. For any $V(x)\in\widetilde{\Theta}_{m}^{0}$, if the initial value of $\psi(x,t)$ to (2.2) fulfilling $\|\psi(x,0)\|_{H^{p}([0,2\pi],\mathbb{C})}\leq\varepsilon/8<\widetilde{\varepsilon}/8,$ then it holds that $\|\psi(x,t)\|_{H^{p}([0,2\pi],\mathbb{C})}\leq\varepsilon,$ for any $|t|\prec\varepsilon^{-(r_{*}-1)}.$

7.2 Proof of Theorem 2

The following statements deal with the solution to equation (2.11). It is obvious that $j^{2}$ $(j\in\mathbb{Z}^{*})$ is the eigenvalue of $(-\partial_{xx})$ under periodic boundary condition and $\phi_{j}(x):=\frac{1}{\sqrt{2\pi}}e^{{\bf i}jx}$ is the corresponding eigenfunction. Precisely,

[TABLE]

For any $\psi\in H_{0}^{p+1/2}([0,2\pi],\mathbb{C})$,

[TABLE]

where $\widehat{\psi}_{j}(t):={\int_{0}^{2\pi}}\psi(x,t)\phi_{-j}(x)\mathrm{d}x$. In order to transform equation (2.11) into an infinite dimensional Hamiltonian system under a standard symplectic form, I will use a tool given in [30]

[TABLE]

It is easy to get that if $\psi\in H_{0}^{p+1/2}([0,2\pi],\mathbb{C})$ then the corresponding Fourier coefficients vector $(\widehat{\psi}_{j})_{j\in\mathbb{Z}^{*}}\in\ell^{2}_{p+1/2}({\mathbb{Z}}^{*},\mathbb{C})$ and $u\in\ell^{2}_{p}({\mathbb{Z}}^{*},\mathbb{C})$. Moreover, there exist constants $\tilde{C}_{1},\ \tilde{C}_{2}>0$ such that

[TABLE]

Under transformation (7.16), equation (2.11) therefore can be written into the following Hamiltonian system with respect to symplectic from $w_{1}$ defined in (3.5), for any $j\in\mathbb{Z}^{*}$,

[TABLE]

with the Hamiltonian

[TABLE]

where

[TABLE]

By assumptions ${\bf B_{1}}$-${\bf B_{2}}$ in Theorem 2, $P^{w_{1}}(u,\bar{u})$ has a zero at origin at last order 3 with the following form

[TABLE]

where $M_{P^{w_{1}}_{r}}\subset\mathbb{Z}$ is a symmetric set

[TABLE]

And there exists a constant $C_{2}>0$ such that for any $r\geq 3$ and any $i\in M_{P^{w_{1}}_{r}}\subseteq\mathbb{Z}$,

[TABLE]

and

[TABLE]

which means that the power series $P^{w_{1}}(u,\bar{u})$ has $(\beta,1)$-type symmetric coefficients semi-bounded by $C_{2}$.

From (7.20), the origin is the elliptic equilibrium point of the equation (7.18). Using Theorem 3, Lemma 7.1 and (7.17), for any $V\in\tilde{\Theta}^{1}_{m}$, there exists $\tilde{\varepsilon}\ll 1$ such that for any $0<\varepsilon<\tilde{\varepsilon}$, if

[TABLE]

then it satisfies

[TABLE]

8 Appendix

Now the proof of Theorem 4 is given in this section.

Proof.

For any $\theta\in\{0,1\},$ denote

[TABLE]

where $P^{w_{\theta}}_{r}(u,\bar{u})$ is an $r$-degree homogeneous polynomial of $P^{w_{\theta}}(u,\bar{u})$. Thus (3.11) can be rewritten as

[TABLE]

To start with, the results hold at rank $r=0$. For any $R<R_{*}$ and any $N$ satisfying (5.33), I will look for a bounded Lie-transformation ${\cal T}_{0}^{w_{\theta}}$ to eliminate the non-normalized monomials of $\Gamma^{N}_{\leq 2}g^{(-1,{w_{\theta}})}_{3}$. The Lie-transformation ${\cal T}_{0}^{w_{\theta}}$ is constructed from 1-time flow $\Phi_{S^{(0)}_{w_{\theta}}}^{t}$ of the following equations,

[TABLE]

where $S^{(0)}_{w_{\theta}}$ is undetermined. Under transformation ${\cal T}_{0}^{w_{\theta}}$ the new Hamiltonian $H^{(0,{w_{\theta}})}$ has the following form,

[TABLE]

where $(\ .\ )_{(\nu,S_{w_{\theta}}^{(0)})}$ is defined in (5.35). The auxiliary Hamiltonian function $S_{{w_{\theta}}}^{(0)}$ are obtained by solving the following homological equation

[TABLE]

Using Remark 4.2, $\Gamma^{N}_{>2}P_{3}^{w_{\theta}}$ and $\Gamma^{N}_{\leq 2}P_{3}^{w_{\theta}}$ are still having $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C(\theta,-1)=C_{\theta}>0.$ From Lemma 5.2, $Z^{{w_{\theta}}}_{3}$ is $(\theta,\gamma,\alpha,N)$-normal form of $\Gamma^{N}_{\leq 2}P^{w_{\theta}}_{3}$ and the Hamiltonian vector field of $S_{{w_{\theta}}}^{(0)}$ satisfies

[TABLE]

From (8.4), the following holds true

[TABLE]

The Lie-transformation ${\cal T}_{0}^{w_{\theta}}$ satisfies

[TABLE]

Use the bootstrap method to estimate ${\cal T}_{0}^{w_{\theta}}$. First, assume that

[TABLE]

By (8.5)-(8.7), the following inequality holds true

[TABLE]

Since $R$ is small enough, from (5.33) and (8.8), the transformation ${\cal T}_{0}^{w_{\theta}}$ satisfies

[TABLE]

which means

[TABLE]

Denote ${\cal T}^{w_{\theta}(0)}:={\cal T}_{0}^{w_{\theta}}$. By (5.33), (8.6) and (8.8), it is verified that (5.34) holds for rank $r=0$:

[TABLE]

Set

[TABLE]

Since $Z^{w_{\theta}}_{3}$ and ${g}^{(-1,{w_{\theta}})}_{3}$ having $(\beta,\theta)$-type symmetric coefficients, then $Z^{(0,w_{\theta})}$ and ${\cal R}^{N(0,{w_{\theta}})}$ are still having $(\beta,\theta)$-type symmetric coefficients. Denote the $r_{*}+3$-degree polynomial of power series (8.3) as $g^{(0,{w_{\theta}})}$ and the remainder as ${\cal R}^{T(0,{w_{\theta}})}$, i.e.,

[TABLE]

where for any $4\leq t\leq r_{*}+3$,

[TABLE]

and for any $t>r_{*}+3$

[TABLE]

From Remark 5.4 and Lemma 4.1, $g^{(0,w_{\theta})}$ and ${\cal R}^{T(0,w_{\theta})}$ have $(\beta,\theta)$-type symmetric coefficients. In order to estimate them, one needs to estimate the coefficients of functions $(H_{0}^{w_{\theta}})_{(t-2,S_{w_{\theta}}^{(0)})}$, $\sum_{n^{\prime}=1}^{t-3}(g^{(-1,w_{\theta})}_{t-n^{\prime}})_{(n^{\prime},S_{w_{\theta}}^{(0)})}$ and $\sum_{n^{\prime}=1}^{t-3}({\cal R}^{T(-1,w_{\theta})}_{t-n^{\prime}})_{(n^{\prime},S_{w_{\theta}}^{(0)})}$. By Remark 5.4, when $\theta=0$, for any $|l+k|=t\geq 4$, any $1\leq n^{\prime}\leq t-3$ and any $i\in M_{(g_{t-n^{\prime}}^{(-1,w_{0})})_{(n^{\prime},S_{w_{0}}^{(0)})}}$, it holds

[TABLE]

and when $\theta=1$, it holds

[TABLE]

By equation (8.4), when $\theta=0$ it follows

[TABLE]

when $\theta=1$ it follows

[TABLE]

When $N$ satisfies (5.33), using (8.12)-(8.14), in the case $\theta=0$ it holds that

[TABLE]

When $\theta=1$ it follows

[TABLE]

Similarly, ${\cal R}^{T(0,w_{\theta})}(u,\bar{u})$ has still $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C(\theta,0)$.

Now assume that the results hold for rank $r<r_{*}$. By these assumptions, there exist a real number $\tilde{R}<R_{*}$ and a Lie-transformation which changes Hamiltonian (8.1) into the following form

[TABLE]

which is defined in $B_{p}(R_{r})$ ($R<\tilde{R}<R_{*}$), where $R_{r}:=\frac{2r_{*}-r}{2r_{*}}R$. One should construct a bounded Lie-transformation ${\cal T}_{r}^{w_{\theta}}$ to eliminate the non-normalized monomials of $\Gamma^{N}_{\leq 2}g^{(r,w_{\theta})}_{r+4}$. Because $g_{r+4}^{(r,w_{\theta})}$ have $(\beta,\theta)$-type symmetric coefficients, by Remark 4.2, the coefficients of $\Gamma^{N}_{\leq 2}g^{(r,w_{\theta})}_{r+4}$ and $\Gamma^{N}_{>2}g^{(r,w_{\theta})}_{r+4}$ are $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C(\theta,r)$. Make use of the 1-time flow of the following equation, for any $j\in\mathbb{Z}^{*}$

[TABLE]

to define a Lie-transformation ${\cal T}_{r}^{w_{\theta}}$, under which the new Hamiltonian has the following form formally,

[TABLE]

The auxiliary Hamiltonian $S_{w_{\theta}}^{(r)}$ can be obtained by solving the following homological equation

[TABLE]

From Lemma 5.2, $Z_{r+4}$ is $(\theta,\gamma,\alpha,N)$-normal form of $\Gamma^{N}_{\leq 2}g^{(r,w_{\theta})}_{r+4}$ and

[TABLE]

The Hamiltonian vector field $X^{w_{\theta}}_{S^{(r)}_{w_{\theta}}}$ satisfies

[TABLE]

Using (8.18) and bootstrap method, suppose that

[TABLE]

for any $t\in[0,1]$.

[TABLE]

By (5.33) and (8.20), the transformation ${\cal T}_{r}$ satisfies

[TABLE]

which verifies (8.19). Denote ${\cal T}^{{}^{w_{\theta}}(r+1)}:={\cal T}^{{}^{w_{\theta}}(r)}\circ{\cal T}_{r}^{w_{\theta}}$. By (8.20) and (8.19), noting that $R<\tilde{R}<R_{*}<1$, it holds

[TABLE]

Because $C(\theta,t)\leq C(\theta,t+1)$ for any positive integer $t$, from (8.21), one has that

[TABLE]

Denote

[TABLE]

By Remark 5.4 and Lemma 4.1, $Z^{(r+1,w_{\theta})}$ and ${\cal R}^{N(r+1,w_{\theta})}$ have $(\beta,\theta)$-type symmetric coefficients. Denote

[TABLE]

where

[TABLE]

and

[TABLE]

where $[a]$ denotes the integer part of the real number $a$. Using Lemma 4.1 and Remark 5.4, from the fact that $g^{(r,w_{\theta})},\ {\cal R}^{T(r,w_{\theta})},\ {\cal R}^{N(r,w_{\theta})}$ and $Z^{(r,w_{\theta})}$ have $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C(\theta,r)$, then $g^{(r+1,w_{\theta})}$ and ${\cal R}^{T(r+1,w_{\theta})}$ also have $(\beta,\theta)$-type symmetric coefficients.

When $\theta=0$, using Remark 5.4, the followings estimates hold: for any $|l+k|=t$ with ${\cal M}(l,k)=i\in M_{(g^{(r,w_{0})}_{r+4})_{(\frac{t-r-4}{r+2},S_{w_{0}}^{(r)})}}$,

[TABLE]

for any $|l+k|=t$ with ${\cal M}(l,k)=i\in M_{(g^{(r,w_{0})}_{t-n^{\prime}(r+2)})_{(n^{\prime},S^{(r)}_{w_{0}})}}$,

[TABLE]

for any $|l+k|=t$ with ${\cal M}(l,k)=i\in M_{(Z^{(r,w_{0})}_{t-n^{\prime}(r+2)})_{(n^{\prime},S^{(r)}_{w_{0}})}},$

[TABLE]

for any $|l+k|=t$ with ${\cal M}(l,k)=i\in M_{({\cal R}^{N(r,w_{0})}_{t-n^{\prime}(r+2)})_{(n^{\prime},S^{(r)}_{w_{0}})}}$

[TABLE]

and for any $|l+k|=t$ with ${\cal M}(l,k)=i\in M_{({\cal R}^{T(r,w_{0})}_{t-n^{\prime}(r+2)})_{(n^{\prime},S^{(r)}_{w_{0}})}}$

[TABLE]

By (8)-(8) and assumption (5.33), for any $r+5\leq t\leq r_{*}+3$, $|l+k|=t$ and $i\in M_{g^{(r+1,w_{\theta})}_{t}}$, the following estimate holds

[TABLE]

which means that $g^{(r+1,w_{0})}(u,\bar{u})$ has $(\beta,0)$-type symmetric coefficients semi-bounded by $C(\theta,r+1)>0$.

Similarly, ${\cal R}^{T(r+1,w_{\theta})}(u,\bar{u})$ and ${\cal R}^{N(r+1,w_{1})}(u,\bar{u})$ are also of $(\beta,\theta)$-type symmetric coefficients semi-bounded by $C(\theta,r+1)>0$.

∎

Acknowledgement

This paper is supported in part by Science and Technology Commission of Shanghai Municipality (No. 18dz2271000).