Quasi-periodic Solutions for a Generalized Higher-Order Boussinesq Equation

Abstract

In this paper one-dimensional generalized eighth-order Boussinesq equation

$$\begin{aligned} u_{tt} - \partial _x^2 u + \beta \partial _x^4 u - \partial _x^6 u +\partial _x^8 u + ( u^3 )_{xx } =0,~~~ \beta =\pm 1 \end{aligned}$$

with the boundary conditions $u(0,t)= u(\pi ,t)=u_{xx}(0,t)= u_{xx}(\pi ,t)=u_{xxxx}(0,t)= u_{xxxx}(\pi ,t)=0$ is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form.

Appendix

For $w\in \Sigma$, we have

$$\begin{aligned} w= \sum \limits _{n\ge 1} r_n q_n \varphi _n + r_nq_{-n}\varphi _{-n}, \end{aligned}$$

where $\{r_n \}\ ( n\ne 0)$ with $r_n =r_{-n}$ are weights and on the bases $\{\varphi _n, \varphi _{-n} \}_{n\ge 1},$ $\{q_n,q_{-n}\}_{ n\ge 1}$ are the coordinates of w. In the coordinates $\{q_n,q_{-n}\}_{n\ge 1},$ we want to compute the Poisson product $\{ F,G\}.$ Especially, we choose suitable weights $\{r_n \}\ ( n\ge 1)$ such that the Poisson product is a standard form.

Let

$$\begin{aligned} A_n{} & {} = \int _0^ \pi \langle \varphi _n, \varphi _n \rangle \, dx = \int _0^ \pi \langle \varphi _{-n}, \varphi _{-n} \rangle \, dx =\frac{\pi }{ 2 }(\alpha _n ^2-\beta _n^2) \\{} & {} = -\frac{ n^6+ n^4+ \beta n^2 }{ n^6+ n^4+ \beta n^2 +2}, ~~~\beta =\pm 1, \end{aligned}$$

where $\langle \cdot , \cdot \rangle$ denotes the Euclidean inner product in $R^2.$

It follows easily that

$$\begin{aligned} \int _0^ \pi \langle \varphi _n, \varphi _{-n} \rangle \, dx=1,~~~ \int _0^ \pi \langle \varphi _n, \varphi _m \rangle \, dx= 0,~~~ n \pm m \ne 0. \end{aligned}$$

Then one has

$$\begin{aligned} \left\{ \begin{array}{lll} \int _0 ^\pi \langle w,\varphi _n \rangle \, dx &{}=&{} A_n r_n q_n+ r_n q_{-n}, \\ \int _0 ^\pi \langle w, \varphi _{-n} \rangle \, dx &{}=&{} r_n q_n + A_n r_n q_{-n}. \end{array}\right. \end{aligned}$$

(7.1)

By simple computation, one obtains

$$\begin{aligned} q_n = \frac{1}{r_n} \cdot \frac{1}{ 1-A_n^2 } \left( -A_n \int _0 ^\pi \langle w,\varphi _n \rangle \, dx + \int _0 ^\pi \langle w, \varphi _{-n} \rangle \, dx \right) ,\\ q_{-n} = \frac{1}{r_n} \cdot \frac{1}{ 1-A_n^2 } \left( -A_n \int _0 ^\pi \langle w, \varphi _{-n} \rangle \, dx + \int _0 ^\pi \langle w, \varphi _n \rangle \, dx \right) , \end{aligned}$$

where $A_n=\ -\frac{ n^6+ n^4+ \beta n^2 }{ n^6+ n^4+ \beta n^2 +2},~\beta =\pm 1.$

Then the Frechet derivatives of $q_n, q_{-n}$ with respect to w are

$$\begin{aligned} \frac{d q_n}{ d w} = \frac{1}{r_n} \cdot \frac{1}{ 1-A_n^2 } \left( -A_n \varphi _n + \varphi _{-n} \right) ,\\ \frac{d q_{-n}}{ d w} = \frac{1}{r_n} \cdot \frac{1}{ 1-A_n^2 } \left( -A_n \varphi _{-n} + \varphi _n \right) . \end{aligned}$$

For $n \ge 1, m\ne 0,$ we have $\int _0 ^\pi \langle \varphi _n, \mathbb {J} \varphi _m \rangle \, dx =\textrm{i}n\alpha _n \beta _n \pi ,$ if $n+m=0;$ $\int _0 ^\pi \langle \varphi _n, \mathbb {J} \varphi _m \rangle \, dx =0,$ otherwise. Then, one obtains

$$\begin{aligned} {} \int _0 ^\pi \left\langle \frac{ d q_n }{d w }, \mathbb {J} \frac{ d q_m}{ d w } \right\rangle \, dx = \left\{ \begin{array}{lll} -\textrm{i} \frac{1}{r_n^2} \frac{n}{ \pi \alpha _n \beta _n }, &{}&{} n+m=0 \\ 0,&{}&{} n+m\ne 0 \end{array}.\right. \end{aligned}$$

(7.2)

By choosing $r_n=\sqrt{ \frac{n}{ \pi \alpha _n \beta _n } },$ then

$$\begin{aligned} {} \int _0 ^\pi \left\langle \frac{ d q_n }{d w }, \mathbb {J} \frac{ d q_m}{ d w } \right\rangle \, dx = \left\{ \begin{array}{lll} -\textrm{i}, &{}&{} n+m=0 \\ 0, &{}&{} n+m\ne 0 \end{array}.\right. \end{aligned}$$

(7.3)

Since $\mathbb {J}$ is anti-self-adjoint, and the gradient of F and G are $\nabla F = \frac{dF}{ dw} = \sum \limits _{n\ge 1} \frac{\partial F }{\partial q_n} \frac{ d q_n }{ d w} +\frac{\partial F }{\partial q_{-n}} \frac{ d q_{-n}}{ d w},$ and $\ \nabla G = \frac{dG}{ dw}= \sum \limits _{n\ge 1} \frac{\partial G }{\partial q_n} \frac{ d q_n }{ d w} +\frac{\partial G }{\partial q_{-n}} \frac{ d q_{-n}}{ d w},$ then we have

$$\begin{aligned} \begin{array}{lll} \{ F,G\}&{}=&{} \int _0 ^\pi \nabla F^T \mathbb {J} \nabla Gdx= \int _0 ^\pi \left\langle \sum \limits _{n\ge 1} \frac{\partial F }{\partial q_n} \frac{ d q_n }{ d w} +\frac{\partial F }{\partial q_{-n}} \frac{ d q_{-n}}{ d w},\mathbb {J} \nabla G \right\rangle dx \\ &{}=&{} - \sum \limits _{n\ge 1} \int _0 ^\pi \frac{\partial F}{ \partial q_n} \left\langle \nabla G, \mathbb {J}\frac{ d q_n }{ d w} \rangle + \frac{\partial F}{ \partial q_{-n}} \langle \nabla G, \mathbb {J}\frac{ d q_{-n} }{ d w} \right\rangle dx \\ &{}=&{} - \sum \limits _{n,m\ge 1} \int _0 ^\pi \frac{\partial F}{ \partial q_n} \frac{\partial G }{\partial q_m} \left\langle \frac{ d q_m }{ d w}, \mathbb {J}\frac{ d q_n }{ d w} \rangle + \frac{\partial F}{ \partial q_n} \frac{\partial G }{\partial q_{-m}} \langle \frac{ d q_{-m}}{ d w}, \mathbb {J}\frac{ d q_n }{ d w} \right\rangle \\ &{} &{} + \frac{\partial F}{ \partial q_{-n}} \frac{\partial G }{\partial q_m} \left\langle \frac{ d q_m }{ d w}, \mathbb {J}\frac{ d q_{-n} }{ d w} \right\rangle + \frac{\partial F}{ \partial q_{-n}} \frac{\partial G }{\partial q_{-m}}\left\langle \frac{ d q_{-m}}{ d w}, \mathbb {J}\frac{ d q_{-n} }{ d w} \right\rangle dx \\ &{}=&{} \textrm{i}\sum \limits _{n\ge 1} \left( \frac{\partial F }{\partial q_{-n}} \frac{ \partial G}{ \partial q_n } - \frac{\partial F }{ \partial q_n } \frac{ \partial G}{\partial q_{-n}}\right) \end{array} \end{aligned}$$