A Nekhoroshev-Type Result for a Generalized Boussinesq Equation

Abstract

In this paper, we consider a generalized Boussinesq equation with hinged boundary conditions. We first observe that the nonlinear terms possess a TAME structure and then prove a Nekhoroshev-type result, which implies that any solution with a small initial value remains small in a high-index Sobolev norm over a power long time.

1. Introduction

Background

Nekhoroshev theory is established by Nekhoroshev [1] to study the long-time stability of nearly integrable Hamiltonian systems:

$$H(I,\theta )=h\left(I\right)+ϵf(I,\theta ),$$

(1)

where $(I,\theta )\in V\times T^n,$ $V\subseteq {\mathbb{R}}^n$ is a bounded domain, h satisfies the so-called steepness condition, f is analytic, and $ϵ$ is a small scalar. The author obtained that $|I\left(t\right)-I\left(0\right)|⩽r_0{ϵ}^a$ for $t⩽T_{Nek}=:t_0{exp}^{{\left({\displaystyle \frac{{ϵ}_0}{ϵ}}\right)}^b}$, with suitable positive constants $r_0,t_0,{ϵ}_0,a,b.$ Here, a and b are usually called stability exponents, which depend on the steepness condition. This fact shows that a solution remains close to the initial value over an exponentially long time scale. The optimal result on a and b was obtained in [2] with a generic steepness condition. Later, similar results for (1) were proven to be subject to some special steepness conditions, such as convex conditions [3], quasi-convex ones [4], and rational convexity ones [5,6].

For the following degenerate Hamiltonian system:

$$H(I,\theta ,p,q)=h\left(I\right)+ϵf(I,\theta ,p,q),$$

(2)

where degenerate variables are $(p,q)\in U\subseteq {\mathbb{R}}^{2m},$ Nekhoroshev [1] assumed that $h\left(I\right)$ is convex, f is analytic, and the degenerate variables are bounded in principle only by $(\dot{p},\dot{q})\sim O\left(ϵ\right)$, then the author obtained that there exist positive constants ${ϵ}_0,r_0,t_0$ such that if $ϵ<{ϵ}_0$, then $|I\left(t\right)-I\left(0\right)|⩽r_0{ϵ}^{{\displaystyle \frac{1}{2n}}}$ for $t⩽min\{t_0{e}^{{\left({\displaystyle \frac{{ϵ}_0}{ϵ}}\right)}^{{\displaystyle \frac{1}{2n}}}},T_e\},$ where $T_e$ is the time of solutions escaping out of the boundary U. Since then, the Nekhoroshev theory for degenerate Hamiltonian systems was greatly developed and enriched; see [7,8,9] and the references therein.

In the infinite-dimensional Hamiltonian partial differential equations (HPDEs), additional challenges arise because of the resonance problem of infinitely many frequencies. The first result is obtained by Bourgain [10] for some typical nonlinear wave equations (NLWs) and nonlinear Schrödinger equations (NLSs) subject to Dirichlet boundary conditions. The author proved a power long stability time ${ϵ}^{-M}$ for solutions with smooth initial values. Comparing with the exponentially long time scales result in [1] for finite-dimensional Hamiltonian systems, infinite-dimensional systems can only attain power long time scales. For this reason, Bougain’s result is referred to as a Nekhoroshev-type result.

Later, Bambusi and Grébert [11] provided similar results for some general NLWs and NLSs under Dirichlet and periodic boundary conditions. It is worth emphasizing that their proof is based on the famous Tame structure, which is first observed in [11]. Tame structure means that most of the nonlinear terms are so small that can be neglected. Tame structure is usually related to the local property of eigenfunctions, which is a decay property of the Fourier coefficients in the Fourier series expansion of the eigenfunctions. Tame structure is very important and it is widely used in study of some HPDEs such as NLSs and NLWs [12,13]. In particular, based on Tame structure, some Nekhoroshev-type results are proved; for some related results we refer to [14,15,16,17,18] and the references therein.

In this paper, we are concerned with the Boussinesq equation, a shallow water wave equation that admits solitary wave solutions. Boussinesq [19,20] first consider the following equation

$$u_{tt}-u_{xx}-{\displaystyle \frac{3}{2}}ϵ{\left(u^2\right)}_{xx}-{\displaystyle \frac{ϵ}{3}}{\left(u\right)}_{xxxx}=0,$$

(3)

and obtained solitary wave solutions. Later, Bona and Sachs [21] proposed to study the following Boussinesq equation:

$$u_{tt}-m{u}_{xx}+n{u}_{xxxx}+{\left(u^2\right)}_{xx}=0.$$

(4)

The authors obtained some solutions corresponding to solitary waves and proved the global existence of smooth solutions. Subsequently, numerous variants of the Boussinesq equation have emerged. We refer to refs. [22,23,24] for generalized Boussinesq models, refs. [25,26] for the Oberbeck–Boussinesq Equation, and refs. [27,28] for a Boussinesq system in higher dimensions.

As is widely recognized, there are many perspectives to investigate dynamical systems. For example, the authors [29] consider a kind of fractional order system and provide the dynamical behavior by analyzing the characteristics of the explicit solutions around the equilibrium point. Recently, from the perspective of infinite-dimensional dynamical systems, Shi et al. [30] have systematically investigated the following generalized Boussinesq equation with hinged boundary conditions:

$$\left\{\begin{array}{c}{u}_{tt}-u_{xx}+u_{xxxx}+{\left(u^3\right)}_{xx}=0,\hfill \\ u(0,t)=u(\pi ,t)=u_{xx}(0,t)=u_{xx}(\pi ,t)=0.\hfill \end{array}\right.$$

(5)

Here, the hinged boundary conditions are derived from the nonlinear Beam equation (NLB) [31]. The authors observed that the Euler–Bernoulli (E-B) operator $Lu=-u_{xx}+b{u}_{xxxx}$ in (5) has a discrete spectrum. Then, they applied the KAM method to obtain KAM tori and the corresponding quasi-periodic solutions for (5). The result indicates that under a small perturbation, most of the invariant tori are preserved with only slight deformations. Correspondingly, there exist a large number of quasi-periodic solutions with small initial values on these invariant tori. The KAM-type result is a kind of stability that lasts forever. A natural question arises: For this equation, does a Nekhoroshev-type result hold? That is, due to the breakdown of some invariant tori, will certain small solutions move away from the original invariant tori? If so, what is the rate of this divergence?

In this paper, we aim to investigate the Nekhoroshev-type problem for the following generalized Boussinesq equation with hinged boundary conditions:

$$\left\{\begin{array}{c}{u}_{tt}-m{u}_{xx}+u_{xxxx}+{\left(u^3\right)}_{xx}=0,\hfill \\ u(0,t)=u(\pi ,t)=u_{xx}(0,t)=u_{xx}(\pi ,t)=0.\hfill \end{array}\right.$$

(6)

Here, a parameter m is introduced to satisfy some non-degeneracy conditions. By verifying the Tame structure of nonlinear terms and the strong non-resonance property of its frequencies, we prove a normal-form theorem and then derive a Nekhoroshev-type dynamical theorem, which demonstrates that small initial value solutions remain small in the high-index Sobolev norm over a power-law long time scale.

The proof of this paper strongly relies on the Tame structure of nonlinear terms, which determines whether a normal form exists for the Boussinesq equation. The Tame structure is usually related to the locality of eigenvectors. The locality means that the Fourier coefficients of the Fourier series expansion of the eigenvectors possess decay property. By the decay property of the eigenvectors of the Sturm–Liouville operator of NLSs and NLWs, the authors [11] prove that the nonlinear terms possess a Tame structure. In this paper, we will prove that the eigenvectors of the Euler–Bernoulli operator also possess locality, and consequently admit Tame structure for the Boussinesq Equation (6).

The proof also requires the strong non-resonance property of the eigenvalues (frequencies), which determines whether a normal form exhibits the Nekhoroshev-type dynamical behavior. The authors [11] have already proven that under Dirichlet boundary conditions, the frequencies of NLS and NLW possess the strong non-resonance property; however, under periodic boundary conditions, the result does not hold again. In this paper, we will prove that the strong non-resonance property still holds for (6).

In recent years, the research on the Nekhoroshev-type theory mainly focus on HPDEs with Sturm–Liouville operator, for example, NLS [16,17,18,32,33,34], NLW [35], nonlinear Beam equation (NLB) [36], and nonlinear Klein–Gordon equation (NLKG) [14]. To the best of our knowledge, our result is the first Nekhoroshev-type result for HPDEs with Euler–Bernoulli operator. Under periodic boundary conditions, there has been some progress made regarding the Nekhoroshev problem for NLS [35]. By this motivation, in the future, we will investigate the Nekhoroshev problem for the Boussinesq equation with periodic boundary conditions. The main challenge lies in the lack of strong non-resonance due to multiple eigenvalues (${\omega }_j={\omega }_{-j}$).

The remainder of this paper is organized as follows:

In Section 2, we summarize the main results of this work, including a normal form theorem and a Nekhoroshev-type dynamical behavior theorem for (6). In Section 3, we first verify Tame structure of (6) by exploiting the locality of the eigenvectors of the Boussinesq equation and then establish sharp estimates for the nonlinear terms. In Section 4, we confirm the strong non-resonance property of the frequencies. In Section 5, we first prove an iteration lemma, then use this iteration lemma to establish the normal form theorem, and finally apply this normal form theorem to prove the dynamical theorem. In Appendix A, we will give the Hamiltonian structure for (6) along with its associated eigenvalues, eigenvectors, and symplectic basis. In Appendix B, we give the detailed proof of Lemma 7 (iteration lemma), which concerns the small divisor problem and the entire iterative procedure.

2. Main Result

The Hamiltonian structure of (6) is very important for Nekhoroshev-type results. For convenience, the Hamiltonian structure of the Boussinesq equation is given in Appendix A, including its eigenvalues, eigenvectors, symplectic basis, and other relevant content.

By Lemma A2 in Appendix A, we consider the Hamiltonian function of Boussinesq (6):

$$H(q,\overline{q})=\Lambda +G$$

(7)

$$=\sum _{j⩾1}{\omega }_j{I}_j-{\displaystyle \frac{1}{4}}{\int }_0^{\pi }{u}^{4}dx$$

(8)

with actions $I_j=q_j{q}_{-j}$ and frequencies ${\omega }_j=\sqrt{j^4+m{j}^2}$ for $j⩾1.$

Let ${\mathbb{P}}_s:=l_s^2\left(\mathbb{C}\right)\times l_s^2\left(\mathbb{C}\right)$ be a phase space, where

$$l_s^2\left(\mathbb{C}\right)=\{{\left(q_j\right)}_{j⩾1}\in {\mathbb{C}}^{\mathbb{N}}|\sum _j{j}^{2s}|x_j{|}^2<\infty ,s\in \mathbb{R}\}$$

is a complex Hilbert space endowed with the standard norm ${\parallel z\parallel }_s^2={\sum }_j{j}^{2s}{\left|q_j\right|}^2$ with $z={\left(q_j\right)}_{j\ne 0}\in {\mathbb{P}}_s$. Here, ${\mathbb{P}}_s$ is constructed by symplectic basis ${\left\{r_{\pm j}{\varphi }_{\pm j}\right\}}_{j\in \mathbb{N}}$, where ${\varphi }_{\pm j}$ are eigenfunctions of frequencies ${\omega }_j=\sqrt{j^4+m{j}^2}$ and $r_{\pm j}$ are weights. Moreover, ${\mathbb{P}}_s$ is endowed with symplectic structure $\mathrm{i}{\sum }_{\mathrm{j}⩾1}{\mathrm{dq}}_{\mathrm{j}}\wedge {\mathrm{dq}}_{-\mathrm{j}}.$

Let ${\mathcal{B}}_s={\mathcal{B}}_{\mathbb{R},s}\left(R\right)$ be an open ball in ${\mathbb{P}}_s$ with origin as the center and $R>0$ as radius. Let C be a constant independent of R. Denote $z={(q_j,q_{-j})}_{j\in \mathbb{N}}.$ For $r⩾1,$$\iota >3$, and $\alpha \in \mathbb{R},$ define $s_{*}(r,\alpha )={\displaystyle \frac{r+\iota -3}{a}}+1$ with $a={\displaystyle \frac{2-\iota }{\alpha (r+2)}}.$

Theorem 1 (Dynamical Behavior Theorem).

Let $z\left(t\right)$ be the solution of (7) with initial value $z_0\in {\mathbb{P}}_s$ at $t=0.$ If $s⩾s_{*}(r,\alpha ),$ and $\parallel z_0{\parallel }_s<{\displaystyle \frac{R}{8}},$ then we have ${\parallel z\left(t\right)\parallel }_s⩽R\mathrm{for}\left|t\right|⩽C{R}^{1-r}$.

The proof of Theorem 1 relies on the following normal form theorem.

Theorem 2 (Normal Form Theorem).

For any $s⩾s_{*}(r,\alpha )$, there exists a $R_s>0$ such that for any $R<R_s,$ we can construct an analytic canonical transformation ${\Gamma }_R:{\mathcal{B}}_s\left({\displaystyle \frac{R}{3}}\right)\to {\mathcal{B}}_s\left(R\right),$ which takes the Hamiltonian (7) into the form

$$(H_0+P)\circ {\Gamma }_R=H_0+\mathcal{L}+\mathcal{R}.$$

(9)

Moreorve, we have the following:

$\left(i\right)$ $\mathcal{L}$ is a polynomial with degree at most $r+2$ and depends only on $I_j$;

$\left(ii\right)$ $\mathcal{R}\in C^{\infty }\left(B_s\left({\displaystyle \frac{R}{3}}\right)\right)$ is a small term with the estimate

$$su{p}_{\parallel z\parallel ⩽{\displaystyle \frac{R}{3}}}{\parallel X_{\mathcal{R}}\left(z\right)\parallel }_s⩽C{R}^{r+\iota };$$

(10)

$\left(iii\right)$ ${\Gamma }_R$ is close to the identity with the estimate

$$su{p}_{\parallel z\parallel ⩽{\displaystyle \frac{R}{3}}}{\parallel z-{\Gamma }_R\left(z\right)\parallel }_s⩽C{R}^3.$$

(11)

Theorems 1 and 2 will be proven in Section 5.

Remark 1.

We consider the Boussinesq Equation (6) from the perspective of infinite-dimensional dynamical systems. By the eigenvectors of the Euler–Bernoulli operator, we construct an invariant subspace ${\mathbb{P}}_s$ endowed with a symplectic structure such that the Hamiltonian vector field of Equation (6) is analytic and bounded, which implies that the Cauchy problem for (6) is well-posed on ${\mathbb{P}}_s$.

3. Tame Structure and Tame Norm

In this section, we will verify the Tame structure of the generalized Boussinesq equation and estimate the vector field of nonlinear terms.

3.1. Tame Structure

At first, we note that the Moser inequality in [37] shows

$${\parallel uv\parallel }_s⩽{\parallel u\parallel }_1{\parallel v\parallel }_s+{\parallel u\parallel }_s{\parallel v\parallel }_1,$$

(12)

with $u={\sum }_{j\in \mathbb{Z}}{u}_j{e}^{\mathrm{i}jx}$ and $v={\sum }_{j\in \mathbb{Z}}{v}_j{e}^{\mathrm{i}jx}.$ This inequality is important in dealing with nonlinear terms of HPDEs on exponentials basis ${\left\{e^{\mathrm{i}jx}\right\}}_{j\in \mathbb{Z}}$. It means that

$$\parallel u^2{\parallel }_1⩽{\displaystyle \frac{{\parallel u\parallel }_s^2}{N^{s-1}}}$$

(13)

for $u={\sum }_{j\ge N}{u}_j{e}^{\mathrm{i}jx}$ with large N and $s,$ that is, most parts of nonlinear terms in the vector field are not relevant since their norms are small.

To study HPDEs with general basis ${\left\{{\varphi }_j\right\}}_{j\in \mathbb{Z}}$ fulfilling some local property, Bambusi and Grébert [11] generalized Moser inequality from exponential basis ${\left\{e^{\mathrm{i}jx}\right\}}_{j\in \mathbb{Z}}$ to general basis ${\left\{{\varphi }_j\right\}}_{j\in \mathbb{Z}}$ and called it Tame (modulus) property. In this subsection, we will briefly summarize their works.

We first give some notations and definitions. Let

$$H_r=\left\{\mathrm{homogeneous}\mathrm{polynomials}\mathrm{with}r\mathrm{degree}\mathrm{such}\mathrm{that}{\mathbb{P}}_s\to {\mathbb{P}}_s\right\}.$$

For $f\in H_r,$ we denote its r-SF (r-linear symmetric form) by $\tilde{f}(\underset{r}{\underbrace{z,\cdots ,z}})=f\left(z\right).$

Definition 1.

$f\in H_r$ is a tame map on the general basis ${\left\{{\varphi }_j\right\}}_{j\in \mathbb{Z}}$ if

$$\parallel \tilde{f}(z^{\left(1\right)},\cdots ,z^{\left(r\right)}){\parallel }_s⩽C_s\sum _{l=1}^r\parallel z^{\left(1\right)}{\parallel }_1\cdots \parallel z^{(l-1)}{\parallel }_1\parallel z^{\left(l\right)}{\parallel }_s\parallel z^{(l+1)}{\parallel }_1\cdots {\parallel z^{\left(r\right)}\parallel }_1,$$

for some $C_s$ with $z^{\left(l\right)}=\sum z_j^{\left(l\right)}{\varphi }_j,$ $l⩽r.$

Remark 2.

For a vector field $X\in H_r:{\mathbb{P}}_s\to {\mathbb{P}}_s,$ rewrite $X\left(z\right)={\sum }_{l\in \mathbb{Z}}{X}_l\left(z\right){\mathbf{e}}_l$, with $X_l\in H_r$ being the l-th component of X and ${\mathbf{e}}_l=({\underbrace{0,\cdots ,0}}_{l-1},1,0,\cdots ,0).$ Denote the r-SF of vector field X by $\tilde{X}\left({\underbrace{z,\cdots ,z}}_{rtimes}\right)=X\left(z\right).$ Then, one has $\tilde{X}={\sum }_l{\tilde{X}}_l{\mathbf{e}}_l$, with ${\tilde{X}}_l$ being the r-SF of $X_l$.

Definition 2.

For $f\in H_r$, denote its modulus by $\left[f\right]\left(z\right)={\sum }_{\left|j\right|=r}\left|f_j\right|z^j$ with

$$j=(\cdots ,j_{-l},\cdots ,j_{-1},j_1,\cdots ,j_l,\cdots ),\left|j\right|=\sum _l\left|j_l\right|$$

and

$$z^j=\cdots z_{-l}^{j_{-l}}\cdots z_{-1}^{j_{-1}}{z}_1^{j_1}\cdots z_l^{j_l}\cdots .$$

For a vector field $X\in H_r$, its modulus is denoted by $\left[X\right]\left(z\right)={\sum }_{\left|j\right|=r}\left[X_l\right]\left(z\right){\mathbf{e}}_l.$

Now, we can introduce the famous Tame structure.

Definition 3.

A vector field $X\in H_r$ has a Tame structure if its modulus $\left[X\right]$ is a Tame map. For $f\in H_r$, we denote by

$$T_M^s=\{f:X_f\mathrm{has}\mathrm{Tame}\mathrm{structure}\},$$

with $X_f$ being its vector field.

Remark 3.

By Definitions 1–3, for $f\in H_{r+1}$ and $f\in T_M^s$, one can obtain

$$\parallel X_f(z^{\left(1\right)},\cdots ,z^{\left(r\right)}){\parallel }_s⩽C_s\sum _{l=1}^r\parallel z^{\left(1\right)}{\parallel }_1\cdots \parallel z^{(l-1)}{\parallel }_1\parallel z^{\left(l\right)}{\parallel }_s\parallel z^{(l+1)}{\parallel }_1\cdots {\parallel z^{\left(r\right)}\parallel }_1,$$

(14)

on the basis ${\left\{{\varphi }_j\right\}}_{j\in \mathbb{Z}}.$ Similar to the discussion for Moser inequality, (14) means that most parts of the vector fields $X_f$ are so small that can be ignored.

Then, Bambusi and Grébert provided an efficient way to verify the Tame structure.

Lemma 1

(Theorem 3.6 [11]). A vector field X has Tame structure on the basis ${\left\{{\varphi }_j\right\}}_{j\in \mathbb{Z}}$ if the following assumptions are true:

(i) ${\varphi }_j$ are well localized on exponentials, namely

$${\varphi }_j=\sum _{j,k\in \mathbb{Z}}{\varphi }_j^k{e}^{\mathrm{i}jx}\mathrm{with}\left|{\varphi }_j^k\right|⩽\underset{\pm }{max}{\displaystyle \frac{C_n}{{(1+|k\pm j\left|\right)}^n}},$$

(15)

for a constant $C_n.$

(ii) X has a Tame structure with respect to exponentials, namely ${\Phi }^{-1}X\Phi$, where

$$\Phi :(u_j,v_j)\in {\mathbb{P}}_s\mapsto (\sum _j{u}_j{e}^{\mathrm{i}jx},\sum _j{v}_j{e}^{\mathrm{i}jx})\in {\mathcal{B}}_s,$$

(16)

is an isomorphism with $(u_j,v_j)$ fulfilling $u={\sum }_j{u}_j{\varphi }_j$ and $v={\sum }_j{v}_j{\varphi }_j.$

Remark 4.

By Definitions 1–3 and Lemma 1, a bounded linear map $A:{\mathbb{P}}_s\to {\mathbb{P}}_s$ has a Tame structure; $X\circ Y$ has a Tame structure if X and Y have Tame structures.

3.2. The Verification of Tame Structure

According to [11], it is simpler to verify Tame structure in real variables, so we introduce the following transformation:

$$\left\{\begin{array}{c}{z}_j={\displaystyle \frac{1}{\sqrt{2}}}(q_j+q_{-j}),\hfill \\ z_{-j}={\displaystyle \frac{1}{\mathrm{i}\sqrt{2}}}(q_j-q_{-j}).\hfill \end{array}\right.$$

(17)

By (A6) and (17), the Poisson bracket shows

$$\{F,G\}(z_j,z_{-j})=\sum _{j⩾1}({\displaystyle \frac{\partial F}{\partial z_j}}{\displaystyle \frac{\partial G}{\partial z_{-j}}}+{\displaystyle \frac{\partial F}{\partial z_j}}{\displaystyle \frac{\partial G}{\partial z_{-j}}}),$$

and the symplectic structure takes the form

$$\sum _{j⩾1}d{z}_j\wedge d{z}_{-j}.$$

Thus, the Hamiltonian system (A7) is equivalent to

$$\left\{\begin{array}{c}{z}_j={\displaystyle \frac{\partial H}{\partial z_{-j}}},\hfill \\ z_{-j}=-{\displaystyle \frac{\partial H}{\partial z_j}},\hfill \end{array}\right.$$

(18)

where the Hamiltonian function is

$$H=H_0+P,$$

(19)

with

$$H_0=\sum _{j⩾1}{\omega }_j{\displaystyle \frac{(z_j^2+z_{-j}^2)}{2}},$$

(20)

and

$$P=-{\displaystyle \frac{1}{4}}{\int }_0^{\pi }{u}^{4}dx.$$

(21)

Obviously, the action takes the form $I_j={\displaystyle \frac{z_j^2+z_{-j}^2}{2}}$ and

$$u(t,x)=\sum _{j⩾1}\sqrt{2}{\nu }_j{z}_j\left(t\right)sinjx$$

(22)

is the solution of (6).

Below, we will verify the Tame structure. The idea is to decompose the original vector field map into several component maps. After verifying the Tame structure in each component map, we can obtain the Tame structure in the original map.

Proposition 1.

The vector field of P in (21) has Tame structure.

Proof. 

By (21) one obtains $X_P=\{-u^3,0\}$ with

$$z=(z_j,z_{-j})\mathrm{and}u\left(z\right)=\sum _{j⩾1}\sqrt{2}{\nu }_j{z}_j\left(t\right)sinjx.$$

The vector field $X_P$ decomposes into following mappings,

$$z\stackrel{{\Psi }_1}{\to }u\stackrel{{\Psi }_2}{\to }{u}^3.$$

Firstly, we verify Tame structure in

$${\Psi }_1:z\to u\left(z\right)=\sum _{j⩾1}\sqrt{2}{\nu }_j{z}_j\left(t\right)sinjx$$

$$(z_{-j},z_j)\mapsto (u_{-j},u_j),j⩾1,$$

with $u_{-j}=0$, $u_j=\sqrt{2}{\nu }_j{z}_j\left(t\right)sinjx$, and ${\nu }_j={\displaystyle \frac{1}{\pi }}\sqrt[4]{{\displaystyle \frac{j^2}{1+j^2}}}.$ Obviously, ${\Psi }_1$ is a linear bounded map, and hence ${\Psi }_1$ has Tame structure by Remark 4.

Secondly, we seek Tame structure in ${\Psi }_2$ by Lemma 1$.$ Recall that ${\{\sqrt{2}{\nu }_{j}sinjx\}}_{j⩾1}$ is a basis. By Euler’s formula, we have

$$\sqrt{2}{\nu }_{j}sinjx={\displaystyle \frac{\sqrt{2}{\nu }_j}{2\mathrm{i}}}{e}^{\mathrm{i}jx}+{\displaystyle \frac{\sqrt{2}{\nu }_j}{2\mathrm{i}}}{e}^{-\mathrm{i}jx}.$$

Since $|{\nu }_j|=|{\displaystyle \frac{1}{\sqrt{\pi }}}\sqrt[4]{{\displaystyle \frac{j^2}{1+j^2}}}|<1,$ it follows that $|{\displaystyle \frac{\sqrt{2}{\nu }_j}{2\mathrm{i}}}|\le {\displaystyle \frac{\sqrt{2}}{2}}.$ Thus, ${\{\sqrt{2}{\nu }_{j}sinjx\}}_{j⩾1}$ are well localized with respect to the exponentials according to (15). Thus, $\left(i\right)$ in Lemma 1 is satisfied.

Consider

$${\Psi }_2:H^s\left(\mathbb{T}\right)\to H^s\left(\mathbb{T}\right)$$

$$u\mapsto u^3$$

where $H^s\left(\mathbb{T}\right)$ is Sobolev space and $u={\sum }_{l\in \mathbb{Z}}{u}_l{e}^{\mathrm{i}lx}\in H^s\left(\mathbb{T}\right).$ Let ${\tilde{\Psi }}_2$ be the trilinear form of ${\Psi }_2$ such that

$${\tilde{\Psi }}_2(u,u,u)=u^3={\Psi }_2\left(u\right).$$

Then,

$${\tilde{\Psi }}_2(u,v,w)=uvw=\sum _{i+j+l=k\in \mathbb{Z}}{u}_i{v}_j{w}_l{e}^{\mathrm{i}(i+j+l)x}.$$

Using Moser inequality twice, we have

$$\parallel {\tilde{\Psi }}_2{(u,v,w)\parallel }_s⩽C_s{(\parallel u\parallel }_1{\parallel v\parallel }_1{\parallel w\parallel }_s+{\parallel u\parallel }_1{\parallel v\parallel }_s{\parallel w\parallel }_1+{\parallel u\parallel }_s{\parallel v\parallel }_1{\parallel w\parallel }_1).$$

It means that ${\Psi }_2$ has Tame structure with respect to the exponentials. This proves $\left(ii\right)$ in Lemma 1, and we have thus verified Lemma 1. Thus, ${\Psi }_2$ has Tame structure on the basis ${\{\sqrt{2}{\nu }_{j}sinjx\}}_{j⩾1}$.

At last, $X_P={\Psi }_2\circ {\Psi }_1$ has Tame structure by Remark 3. Thus, we prove Proposition 1. □

Remark 5.

Proposition 1 shows that the nonlinear terms has Tame structure with respect to real variables ${(z_j,z_{-j})}_{j⩾1}.$ Similarly, it is true for complex variables ${(q_j,q_{-j})}_{j⩾1}$.

3.3. Tame Norm

In this subsection, we will estimate the small part of nonlinear terms in the vector field by the Tame norm.

Firstly, we decompose the nonlinear terms. Fixing $N>0,$ one can rewrite z as $z=\overline{z}+\widehat{z}$ with $\overline{z}={(q_{-j},q_j)}_{j⩽N}$ and $\widehat{z}={(q_{-j},q_j)}_{j>N}.$ We decompose the nonlinear term P in (22) into $P=P_0+P_N,$ where

$$P_N\left(z\right)=\sum _{l=3,4}{P}_{N_l}\left(z\right)=\sum _{l=3,4}{\tilde{P}}_{N_l}(\underset{(4-l)-times}{\underbrace{\overline{z},\cdots ,\overline{z}}},\underset{\left(l\right)-times}{\underbrace{\widehat{z},\cdots ,\widehat{z}}}),$$

is the small part and $P_0=P-P_N$ is the main one. Below, we will estimate $P_N,$ while $P_0$ will be dealt with in the homological equation in Section 5.

We first introduce the Tame norm. We then rewrite $z=(\cdots ,q_{-l},\cdots ,q_{-1},q_1,\cdots ,q_l,\cdots )$ with $q_{-l}={\overline{q}}_l.$ Denoted by

$${\parallel w\parallel }_{s,1}={\displaystyle \frac{1}{r}}\sum _{l=1}^r\parallel z^{\left(1\right)}{\parallel }_1\cdots \parallel z^{(l-1)}{\parallel }_1\parallel z^{\left(l\right)}{\parallel }_s\parallel z^{(l+1)}{\parallel }_1\cdots {\parallel z^{\left(r\right)}\parallel }_1,$$

with $w=(z^{\left(1\right)},\cdots ,z^{\left(r\right)}).$

Definition 4.

(i) For $f\in H_{r+1}$ and $f\in T_M^s$, the Tame norm of $X_f$ is defined as the infimum of constants $C_s$ such that

$$\parallel \tilde{X}{\left(w\right)\parallel }_s⩽C_s{\parallel w\parallel }_{s,1}$$

and it is denoted by ${\left|X\right|}_s^T.$ In particular, the Tame norm of $X_{\left[f\right]}$ is written as ${\left|f\right|}_s.$

(ii) For a nonhomogeneous function $f=\sum f_r\in T_M^s$ with $f_r\in H_r$, denote by

$${\langle \left|f\right|\rangle }_{s,R}=\sum _{r⩾2}{\left|f_r\right|}_s{R}^{r-1}.$$

Those with finite norm, in particular, are written as

$$T_{s,R}=\left\{f:{\langle \left|f\right|\rangle }_{s,R}<+\infty ,f\in T_M^s\right\}.$$

Moreover, $T_{s,R}$ is a Banach space with norm ${\langle \left|f\right|\rangle }_{s,R}.$

At last, similar to Lemma 4.11 in [11], we can give the estimate of the homogeneous polynomial $P_N\in H_4$ in the vector field.

Proposition 2.

For $R>0$ and a fixed N, the vector field $X_{P_N}$ has the following estimate:

$$\parallel X_{P_N}{\parallel }_{s,R}⩽{\displaystyle \frac{\langle |P_N{|\rangle }_{s,R}}{N^{s-1}}}.$$

Remark 6.

For a nonhomogeneous polynomial $P_N,$ one can rewrite $P_N={\sum }_{j⩾4}{P}_{Nj}\left(z\right)$ with homogeneous polynomial $P_{Nj}\left(z\right)\in H_j$ and $P_{Nj}\in T_M^s.$ We observe that the homogeneous polynomial $P_{Nj}\left(z\right)$ has a zero of at least order 3 in $\widehat{z}.$ Applying Proposition 2 to each $P_{Nj}$ with $j⩾4,$ one still obtains

$$\parallel X_{P_N}{\parallel }_{s,R}⩽{\displaystyle \frac{\langle |P_N{|\rangle }_{s,R}}{N^{s-1}}}.$$

4. Non-Resonant Condition

In this section, we aim to verify that the frequencies of the Boussinesq equation possess the strong non-resonance property:

Definition 5

(r-NR [11]). One says that the frequencies are r-NR non-resonant, if there exist constants $\gamma >0$ and $\alpha \in \mathbb{R}$ such that for any N large enough, we have

$$|\sum _{j⩾1}{\omega }_j{k}_j|⩾{\displaystyle \frac{\gamma }{N^{\alpha }}}$$

for all $k\in {\mathbb{Z}}^{\infty }$ with $\left|k\right|={\sum }_j\left|k_j\right|⩽r+2$ and ${\sum }_{j>N}\left|k_j\right|⩽2.$

The verification about the frequency non-resonance requires the non-degeneracy property and an appropriate separation property of frequencies.

Lemma 2.

Denote by ${\omega }_j^{\left(i\right)}={\displaystyle \frac{d^i{\omega }_j}{d{m}^i}}.$ One has

$$D=\left|\begin{array}{cccc}{\omega }_{j_1}& {\omega }_{j_2}& \cdots & {\omega }_{j_K}\\ {\omega }_{j_1}^{\left(1\right)}& {\omega }_{j_2}^{\left(1\right)}& \cdots & {\omega }_{j_K}^{\left(1\right)}\\ ⋮& ⋮& ⋮& ⋮\\ {\omega }_{j_1}^{(K-1)}& {\omega }_{j_2}^{(K-1)}& \cdots & {\omega }_{j_K}^{(K-1)}\end{array}\right|⩾{\displaystyle \frac{C}{N^{(r+2)(r+1)}}}>0$$

(23)

with $0<|j_i|⩽N$ and $0<K⩽r+2.$

Proof. 

It is obvious that D is a Vandermonde determinant. Similar to [11,35], one has $\left|D\right|⩾{\displaystyle \frac{C}{N^{(r+2)(r+1)}}}>0$ by direct calculations. Thus, Lemma 2 is proven. □

Remark 7.

In fact, this Lemma provides the Rüssmann non-degeneracy condition.

Lemma 3

(Proposition of Appendix B in [35]). For K-independent vectors $u^{\left(1\right)},u^{\left(2\right)},\cdots ,u^{\left(K\right)}$, with $\parallel u^{\left(i\right)}{\parallel }_{l^1}⩽1$, and an arbitrary vector $\omega \in {\mathbb{R}}^K$, there is an $i\in [1,2,\cdots ,K]$ such that

$$|u^{\left(i\right)}·\omega |⩾{\displaystyle \frac{{\parallel \omega \parallel }_{l^1}det(u^{\left(1\right)},u^{\left(2\right)},\cdots ,u^{\left(K\right)})}{K^{\frac{3}{2}}}}.$$

By Lemmas 2 and 3, it follows that

Corollary 1.

For any m and vector $k\in {\mathbb{R}}^{+\infty }$ with $0\ne \left|k\right|⩽r+2,$ one obtains $|\langle k,{\displaystyle \frac{d^i\omega }{d{m}^i}}\rangle |⩾C{\displaystyle \frac{1}{N^{(r+2)(r+1)+2}}}>0$.

When the frequencies are degenerate, Xu et al. [38,39] gave an effective way to estimate the Lebesgue measure.

Lemma 4

(Lemma 1.1 [38]). Suppose that $g\left(x\right)$ is an m-th differentiable function on the closure $\overline{I}$ of I, where $I\in {\mathbb{R}}^1$ is an interval. Let $I_h=\{x:|g\left(x\right)|<h,h>0\}.$ If there exists a constant $d>0$ such that $|g^{\left(m\right)}|⩾d$ for $\forall x\in I,$ we have the Lebesgue measure $|I_h|⩽c{h}^{{\displaystyle \frac{1}{m}}}$ with $c=2(2+3+\cdots +m+d^{-1}).$

The following inequalities are summarized in [38].

Lemma 5

(Lemma 1.2, 1.3 [38]). Let $d⩾1,\delta <d-1,$$\kappa =min(d,d-1-\delta )$ and ${\left[l\right]}_d=i^d-j^d.$ Then, we have

$${|\left[l\right]}_d|⩾{\displaystyle \frac{1}{2}}|i-j|(i^{d-1}+j^{d-1})$$

and

$$(i^{\delta }+j^{\delta })(i^{\kappa }+j^{\kappa })⩽5{\left[l\right]}_d.$$

Remark 8.

The frequencies ${\omega }_j$ defined in (8) can be rewritten as

$${\omega }_j=\sqrt{j^4+m{j}^2}=j^2+{\displaystyle \frac{m}{2}}+O\left(j^{-2}\right).$$

Thus, we have

$${\displaystyle \frac{\langle l,\Omega \left(m\right)\rangle }{{\left[l\right]}_2}}={\displaystyle \frac{{\omega }_i-{\omega }_j}{i^2-j^2}}\to 1$$

uniformly in m.

By Corollary 1, Lemma 4, and Remark 8, we have the following estimate.

Lemma 6.

For any $△>m_0$, there is a full-measure subset $\mathcal{T}\subset [m_0,△]$ such that the nonresonance condition (r-NR) holds if $m\in \mathcal{T}$.

5. Proofs of the Main Results

In this section, we will prove Theorems 1 and 2. It is based on an iteration lemma. We first introduce the so-called $(\gamma ,\alpha ,N)$-normal form, which is used in the iteration lemma.

Definition 6

(Definition 2.12 [11]). For parameters $\gamma ,\alpha >0$ and a fixed $N>0,$ one says that

$$f(q,\overline{q})=\sum _{k,l\in {\mathbb{N}}^{\mathbb{N}}}{f}_{k,l}{q}^k{\overline{q}}^l$$

is in $(\gamma ,\alpha ,N)$-normal form with respect to ω, if for the subscript $(k,l)$ such that $f_{k,l}\ne 0,$ we have

$$|\omega ·(k-l)|<{\displaystyle \frac{\gamma }{N^{\alpha }}}\mathrm{and}\sum _{j⩾N+1}{k}_j+l_j⩽2.$$

Remark 9.

By Definitions 5 and 6, a polynomial f in $(\gamma ,\alpha ,N)$-normal form with respect to ω depends only on actions if ω is r-NR nonresonant.

We also need some iterative parameters.

$r_{*}⩾2$ means the step length of the iteration. In general, a bigger $r_{*}$ means a more accurate estimate.

$r⩽r_{*}$ means the r-step of the iteration.

${\mathcal{B}}_s\left(R\right)$ means a complex ball with the radius $R>0.$

$\delta ={\displaystyle \frac{R}{2r_{*}}}$ means the loss of radius in each iteration.

$R_r=R-r\delta$ means the radius of a complex ball ${\mathcal{B}}_s\left(R\right)$ after r step.

Lemma 7 (Iteration Lemma).

For all $N>1$ and a fixed $s⩾1$, there exist a positive number $R_{\ast r}\ll 1$ and a canonical transformation ${\mathcal{T}}^{\left(r\right)},$ such that the Hamiltonian system (19) can be transformed into

$$H^{\left(r\right)}:=H^{\left(r\right)}\circ {\mathcal{T}}^{\left(r\right)}=H_0+{\mathcal{L}}^{\left(r\right)}+f^{\left(r\right)}+{\mathcal{R}}_N^{\left(r\right)}+{\mathcal{R}}_T^{\left(r\right)}.$$

(24)

Moreover, for any $R<{\displaystyle \frac{R_{\ast r}}{N^{\alpha }}},$ one has the following results.

$\left(i\right)$ The transformation ${\mathcal{T}}^{\left(r\right)}$ satisfies

$$\underset{z\in B_s\left(R_r\right)}{sup}{\parallel z-{\mathcal{T}}^{\left(r\right)}\left(z\right)\parallel }_s\le C{N}^{\alpha }{R}^3.$$

(25)

$\left(ii\right)$ The $(r+2)$-degree polynomial ${\mathcal{L}}^{\left(r\right)}$ has a zero of order 4 at origin. Moreover, ${\mathcal{L}}^{\left(r\right)}\in T_{s,R}$ is in $(\gamma ,\alpha ,N)$-normal form with

$$|\langle |{\mathcal{L}}^{\left(r\right)}{|\rangle }_{s,R_r}\le C{R}^3,\forall r⩾1.$$

(26)

$\left(iii\right)$ The polynomial $f^{\left(r\right)}$ has a degree no more than $r_{*}+2$ and an order of zero of $r+4$ at origin. Moreover, it has

$$|\langle |f^{\left(r\right)}{|\rangle }_{s,R_r}\le C{R}^{r+3}{\left(R{N}^{\alpha }\right)}^r,\forall r⩾1.$$

(27)

$\left(iv\right)$${\mathcal{R}}_T^r\in T_{s,R}$ and ${\mathcal{R}}_N^r\in T_{s,R}$ are small terms with

$$|\langle |{\mathcal{R}}_T^r{|\rangle }_{s,R_r}\le C{\left(R{N}^{\alpha }\right)}^{r_{*}+2},$$

(28)

and

$$|\langle |{\mathcal{R}}_N^r{|\rangle }_{s,R_r}\le C{\displaystyle \frac{R^3}{N^{s-1}}}.$$

(29)

Remark 10.

Denote by ${\Gamma }_R={\mathcal{T}}^{\left(r\right)}\cdots {\mathcal{T}}^{\left(0\right)}.$ Let R be small enough such that $R<{\displaystyle \frac{R_{*r}}{N^{\alpha }}}.$ By (25), the canonical transformation ${\Gamma }_R$ satisfies

$$\underset{z\in B_s({\displaystyle \frac{R}{3}})}{sup}{\parallel z-{\Gamma }_R\left(z\right)\parallel }_s⩽N^{\alpha }{R}^3⩽{\displaystyle \frac{R}{6}},$$

(30)

which means

$${\Gamma }_R:B_s\left({\displaystyle \frac{R}{3}}\right)\to B_s\left({\displaystyle \frac{R}{2}}\right).$$

By (30), the inverse ${\left({\Gamma }_R\right)}^{-1}$ exists with

$$\parallel z-{\left({\Gamma }_R\right)}^{-1}\left(z\right){\parallel }_{s,{\displaystyle \frac{R}{3}}}⩽\sum _{n=1}^{\infty }{\parallel z-{\Gamma }_R\left(z\right)\parallel }_{s,{\displaystyle \frac{R}{3}}}^n⩽2C_{(s,r)}{N}^{\alpha }{R}^3<{\displaystyle \frac{R}{3}}.$$

(31)

This means

$${\left({\Gamma }_R\right)}^{-1}:B_s\left({\displaystyle \frac{R}{3}}\right)\to B_s\left({\displaystyle \frac{2R}{3}}\right)\subset B_s\left(R\right).$$

Proof of Lemma 7 being similar to that in [11] and only the exponent of R is different. For convenience, we provide the proof in Appendix B.

We now use Lemma 7 to prove normal-form Theorem 2.

Proof of Theorem 2. 

(9) and (11) are the direct conclusions of (24) and (25) in Lemma 7, respectively. Denote $\mathcal{R}\left(z\right)={\mathcal{R}}_T^r+{\mathcal{R}}_N^r$. By choosing $N=R^{-a}$ and $s>{\displaystyle \frac{r_{*}+\iota -3}{a}}+1$ with $a={\displaystyle \frac{2-\iota }{\alpha (r_{*}+2)}},$ we have

$$su{p}_{\parallel z\parallel ⩽{\displaystyle \frac{R}{3}}}{\parallel X_{\mathcal{R}}\left(z\right)\parallel }_s⩽C{R}^{r+\iota },\iota >3.$$

Hence, (10) is proven. At last, by Lemma 6 and Remark 9, $\mathcal{L}$ depends only on actions. Thus Theorem 2 is proved. □

At last, we use Theorem 2 to prove Theorem 1.

Proof of Theorem 1. 

By Remark 10, one has

$${\Gamma }_R:z\in B_s\left({\displaystyle \frac{R}{8}}\right)\to z^{\prime }\in B_s\left({\displaystyle \frac{R}{4}}\right)$$

for R small enough. Denote by $I_{j\in \mathbb{Z}}^{\prime }$ the actions at $z^{\prime }$.

Since $H_0\left(z^{\prime }\right)$ only depends on $I_{j\in \mathbb{Z}}^{\prime }$ and $\parallel z^{\prime }{\parallel }_s^2={\sum }_{j\in \mathbb{Z}}{j}^{2s}{I}_j^{\prime },$ one has

$$\{H_0\left(z^{\prime }\right),\parallel z^{\prime }{\parallel }_s^2\}=0.$$

(32)

By Remark 9, one obtains

$$\{\mathcal{L}\left(z^{\prime }\right),\parallel z^{\prime }{\parallel }_s^2\}=0.$$

(33)

According to (10), (32), and (33), it follows that

$$\underset{z^{\prime }\in B_s({\displaystyle \frac{R}{3}})}{sup}\{H_0+\mathcal{L}+\mathcal{R},\parallel z^{\prime }{\parallel }_s^2\}=\underset{z^{\prime }\in B_s({\displaystyle \frac{R}{3}})}{sup}\{\mathcal{R},\parallel z^{\prime }{\parallel }_s^2\}⩽C{R}^{r+3}.$$

Denote by $T_P$ the escape time of the flow $z^{\prime }\left(t\right)$ from $B_s({\displaystyle \frac{R}{3}}),$ i.e., $\parallel z^{\prime }\left(T_P\right)\parallel ⩾{\displaystyle \frac{R}{3}}.$ Then, the flow $z^{\prime }\left(t\right)$ with initial value $z^{\prime }\left(t\right)\in B_s({\displaystyle \frac{R}{4}})$ satisfies

$$-C_s{R}^{r+3}⩽{\displaystyle \frac{d(\parallel z^{\prime }{\parallel }_s^2\parallel )}{dt}}=\{H_0+\mathcal{L}+\mathcal{R},\parallel z^{\prime }{\parallel }_s^2\}⩽C{R}^{r+3},$$

(34)

for $t<T_P.$ Integrating (34) from $t=0$ to $t=T_P,$ one has

$$\parallel z^{\prime }\left(T_P\right){\parallel }_s^2⩽{\parallel z^{\prime }\left(0\right)\parallel }_s^2+C{R}^{r+3}{T}_P.$$

(35)

Since $\parallel z^{\prime }\left(T_P\right){\parallel }_s⩾{\displaystyle \frac{R}{3}}$ and $\parallel z^{\prime }{\left(0\right)\parallel }_s⩽{\displaystyle \frac{R}{4}},$ one obtains

$$T_P⩾{\displaystyle \frac{\parallel z^{\prime }\left(T_P\right){\parallel }_s^2-{\parallel z^{\prime }\left(0\right)\parallel }_s^2}{C{R}^{r+3}}}$$

$$⩾{\displaystyle \frac{({\left(\frac{1}{3}\right)}^2-{\left(\frac{1}{4}\right)}^2)R^2}{C{R}^{r+3}}}=C{R}^{-r+1}.$$

It follows that

$$\parallel z^{\prime }\left(T_P\right){\parallel }_s⩽{\displaystyle \frac{R}{3}}\mathrm{for}t<T_P.$$

(36)

By Remark 10, one acquires that

$$\parallel {\left({\Gamma }_R\right)}^{-1}(z^{\prime }\left(t\right)-z\left(t\right)){\parallel }_s<{\displaystyle \frac{2R}{3}}$$

(37)

for $t<T_P$ and $z^{\prime }\left(t\right)\in B_s({\displaystyle \frac{R}{3}}).$

By (36) and (37), it is concluded that

$${\parallel z\left(t\right)\parallel }_s={\parallel {\left({\Gamma }_R\right)}^{-1}\left(z^{\prime }\left(t\right)\right)\parallel }_s$$

$$⩽\parallel {\left({\Gamma }_R\right)}^{-1}\left(z^{\prime }\left(t\right)\right)-z^{\prime }\left(t\right){\parallel }_s+{\parallel z^{\prime }\left(t\right)\parallel }_s$$

$$⩽{\displaystyle \frac{2}{3}}R+{\displaystyle \frac{1}{3}}R=R.$$

This means ${\parallel z\left(t\right)\parallel }_s⩽R$ for $\left|t\right|⩽C{R}^{-r+1}.$ Thus, Theorem 1 is proven. □

6. Conclusions

We investigate a class of Boussinesq equation associated with a symplectic structure under hinged boundary conditions and derive a Nekhoroshev-type dynamical theorem. The main result demonstrates that small initial-value solutions remain small in the high-index Sobolev norm over a power-law long time scale.

Our result is a complement to the KAM-type results. The KAM theory reveals the preservation of most invariant tori under small perturbations and the quasi-periodic solutions on these tori are always stability. The Nekhoroshev-type result further quantifies the long-term behavior of small solutions, clarifying their “stable duration” after the partial breakdown of invariant tori.

Notably, our result should be contextualized within the broader framework of long-term dynamics in Hamiltonian systems, particularly in relation to Arnold diffusion. As a fundamental global instability phenomenon in non-integrable Hamiltonian systems, Arnold diffusion describes the slow global drift of certain solutions across different action-level regions over extremely long time scales. The Nekhoroshev-type theorem and Arnold diffusion represent two complementary aspects of the systems dynamical behavior: the former characterizes the local boundedness of small solutions before the onset of global drift (i.e., the power-law estimate of the “pre-escape time”), while the latter predicts the potential long-term global instability of the system. Thus, our result directly describes a “temporal boundary” for Arnold diffusion: the global drift of Arnold diffusion can only occur after the “stable period” predicted by Nekhoroshev theory.

Ultimately, our result enables the construction of a complete dynamical understanding of the Boussinesq equation: local stability (KAM-type), transitional boundedness (Nekhoroshev-type), and global drift (Arnold diffusion).