Quasi-periodic Solutions for Two Dimensional Modified Boussinesq Equation

Abstract

In this paper, two dimensional modified Boussinesq equation

$$\begin{aligned} u_{tt} +\Delta ^2u+ \Delta ( u^3 ) =0, \quad x\in {\mathbb {T}}^2,~t\in {\mathbb {R}} \end{aligned}$$

under periodic boundary conditions is considered. It is proved that the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form.

Appendix

For $w\in \Sigma ,$ we have

$$\begin{aligned} w= \sum \limits _{ n\in {\mathbb {Z}}_{odd}^2 } r_nq_n\varphi _n+r_n{\bar{q}}_n{\bar{\varphi }}_n, \end{aligned}$$

where $\{ q_n,{\bar{q}}_n \}_{ n\in {\mathbb {Z}}_{odd}^2 }$ are the coordinates of w on the bases $\{\varphi _n,{\bar{\varphi }}_n \}_{ n\in {\mathbb {Z}}_{odd}^2 } ,$ and $\{r_n\}_{ n\in {\mathbb {Z}}_{odd}^2 }$ are weights with $r_n=r_{-n}.$ In this appendix, we want to compute the Poisson product $\{F,G\}$ in the coordinates $\{ q_n,{\bar{q}}_n \}_{ n\in {\mathbb {Z}}_{odd}^2 }.$ Especially, we choose suitable weights $\{r_n\}( { n\in {\mathbb {Z}}_{odd}^2 })$ such that the Poisson product is a standard form. One has

$$\begin{aligned} \int _{{\mathbb {T}}^2} \langle w, \varphi _n \rangle \ dx= & {} \int _{ {\mathbb {T}}^2 } \left\langle \sum \limits _{ n\in {\mathbb {Z}}_{odd}^2 } r_nq_n\varphi _n+r_n{\bar{q}}_n{\bar{\varphi }}_n, \varphi _n \right\rangle \ dx \\= & {} r_n q_{-n} \int _{ {\mathbb {T}}^2 } \langle \varphi _{-n}, \varphi _n \rangle \ dx + r_n{\bar{q}}_n \int _{ {\mathbb {T}}^2 } \langle {\bar{\varphi }}_n , \varphi _n \rangle \ dx \\= & {} r_n q_{-n} \frac{ 1-|n|^2}{ 1+|n|^2 }+ r_n{\bar{q}}_n,\\ \int _{{\mathbb {T}}^2} \langle w, {\bar{\varphi }}_n \rangle \ dx= & {} \int _{ {\mathbb {T}}^2 } \left\langle \sum \limits _{ n\in {\mathbb {Z}}_{odd}^2 } r_nq_n\varphi _n+r_n{\bar{q}}_n{\bar{\varphi }}_n, {\bar{\varphi }}_n \right\rangle \ dx \\= & {} r_n q_n \int _{ {\mathbb {T}}^2 } \langle \varphi _n, {\bar{\varphi }}_n \rangle \ dx + r_n{\bar{q}}_{-n} \int _{ {\mathbb {T}}^2 } \langle {\bar{\varphi }}_{-n} , {\bar{\varphi }} _n \rangle \ dx \\= & {} r_n q_n + r_n{\bar{q}}_{-n} \frac{ 1-|n|^2}{ 1+|n|^2 }. \end{aligned}$$

Let $A_n= \frac{ 1-|n|^2}{ 1+|n|^2 },$ then

$$\begin{aligned}&\int _{{\mathbb {T}}^2} \langle w, \varphi _n \rangle \ dx = r_n q_{-n} A_n+ r_n{\bar{q}}_n, \nonumber \\&\int _{{\mathbb {T}}^2} \langle w, \varphi _{-n} \rangle \ dx = r_n q_n A_n+ r_n{\bar{q}}_{-n}, \nonumber \\&\int _{{\mathbb {T}}^2} \langle w, {\bar{\varphi }}_n \rangle \ dx = r_n q_n + r_n{\bar{q}}_{-n} A_n,\nonumber \\&\int _{{\mathbb {T}}^2} \langle w, {\bar{\varphi }}_{-n} \rangle \ dx = r_n q_{-n} + r_n{\bar{q}}_n A_n. \end{aligned}$$

(8.1)

By (8.1), one has

$$\begin{aligned} q_n= & {} \frac{1}{ r_n( A_n^2-1 ) } \left[ \ A_n \int _{{\mathbb {T}}^2} \langle w, \varphi _{-n} \rangle \ dx - \int _{{\mathbb {T}}^2} \langle w, {\bar{\varphi }}_n \rangle \ dx \ \right] ,\\ {\bar{q}}_n= & {} \frac{1}{ r_n( A_n^2-1 ) } \left[ \ A_n \int _{{\mathbb {T}}^2} \langle w, {\bar{\varphi }}_{-n} \rangle \ dx -\int _{{\mathbb {T}}^2} \langle w, \varphi _n \rangle \ dx \ \right] . \end{aligned}$$

Then the Frechet derivatives of $q_n,{\bar{q}}_n$ with respect to w are

$$\begin{aligned} \frac{ dq_n}{ dw }= & {} \frac{1}{ r_n( A_n^2-1 ) } [ \ A_n \varphi _{-n}- {\bar{\varphi }}_n \ ] ,\\ \frac{ d {\bar{q}}_n }{ dw }= & {} \frac{1}{ r_n( A_n^2-1 ) } [ \ A_n {\bar{\varphi }}_{-n} - \varphi _n \ ]. \end{aligned}$$

Denote by

$$\begin{aligned} {\mathbb {J}}= \begin{pmatrix} 0 &{} (-\Delta )^{\frac{1}{2}}\\ -(-\Delta )^{\frac{1}{2}}&{} 0 \end{pmatrix}. \end{aligned}$$

For $n,m \in {\mathbb {Z}}_{odd}^2$ we have

$$\begin{aligned} \int _{{\mathbb {T}}^2} \langle \varphi _ n , {\mathbb {J}} {\bar{\varphi }}_m \rangle \ dx= & {} \left\{ \begin{array}{ll} -\frac{ \mathrm{i} 2 |n|^2 }{ 1+|n|^2 } , &{} \quad m=n,\\ &{}\\ 0, &{} \quad m\ne n. \end{array} \right. \\ \int _{{\mathbb {T}}^2} \langle \varphi _ n , {\mathbb {J}} \varphi _m \rangle \ dx= & {} 0, \ \ \ \int _{{\mathbb {T}}^2} \langle {\bar{\varphi }}_ n , {\mathbb {J}} {\bar{\varphi }} _m \rangle \ dx = 0, \end{aligned}$$

then

$$\begin{aligned} \int _{{\mathbb {T}}^2} \left\langle \frac{ dq_n}{ dw } , {\mathbb {J}} \frac{ dq_m}{ dw } \right\rangle \ dx= & {} 0, \quad \int _{{\mathbb {T}}^2} \left\langle \frac{ d {\bar{q}}_n}{ dw } , {\mathbb {J}} \frac{ d {\bar{q}}_m}{ dw } \right\rangle \ dx = 0, \nonumber \\ \int _{{\mathbb {T}}^2} \left\langle \frac{ dq_n}{ dw } , {\mathbb {J}} \frac{ d {\bar{q}}_m}{ dw } \right\rangle \ dx= & {} \left\{ \begin{array}{ll} \frac{ \mathrm{i} 2 |n|^2 }{ r_n^2 \cdot ( 1-A_n^2)( 1+|n|^2 ) } , &{} \quad m=n,\\ 0, &{} \quad m\ne n. \end{array} \right. \end{aligned}$$

(8.2)

We choose $r_n=r_{-n} =\sqrt{ \frac{ 1+|n|^2 }{2} }$ and then

$$\begin{aligned} \int _{{\mathbb {T}}^2} \left\langle \frac{ dq_n}{ dw } , {\mathbb {J}} \frac{ d {\bar{q}}_m}{ dw } \right\rangle \ dx = \left\{ \begin{array}{ll} \mathrm{i } , &{} \quad m=n,\\ &{}\\ 0, &{} \quad m\ne n. \end{array} \right. \end{aligned}$$

(8.3)

Since

$$\begin{aligned} \nabla F= & {} \frac{ d F}{ dw } = \sum \limits _{ n\in {\mathbb {Z}}_{odd}^2 } \left( \frac{ \partial F}{ \partial q_n } \frac{ d q_n}{ dw } + \frac{ \partial F}{ \partial {\bar{q}}_n } \frac{ d {\bar{q}}_n}{ dw } \right) ,\\ \nabla G= & {} \frac{ d G}{ dw } = \sum \limits _{ n\in {\mathbb {Z}}_{odd}^2 } \left( \frac{ \partial G}{ \partial q_n } \frac{ d q_n}{ dw } + \frac{ \partial G}{ \partial {\bar{q}}_n } \frac{ d {\bar{q}}_n}{ dw } \right) , \end{aligned}$$

and ${\mathbb {J}}$ is anti-self-adjoint, we have

$$\begin{aligned} \{F,G\}= & {} \int _{{\mathbb {T}}^2} \ \nabla F^T {\mathbb {J}} \nabla G \ dx \\= & {} \int _{{\mathbb {T}}^2} \ \left\langle \sum \limits _{ n\in {\mathbb {Z}}_{odd}^2 } \left( \frac{ \partial F}{ \partial q_n } \frac{ d q_n}{ dw } + \frac{ \partial F}{ \partial {\bar{q}}_n } \frac{ d {\bar{q}}_n}{ dw } \right) , {\mathbb {J}} \nabla G \right\rangle \ dx \\= & {} - \sum \limits _{ n\in {\mathbb {Z}}_{odd}^2 } \int _{{\mathbb {T}}^2} \frac{ \partial F}{ \partial q_n } \left\langle \nabla G, {\mathbb {J}} \frac{ d q_n}{ dw } \right\rangle + \frac{ \partial F}{ \partial {\bar{q}}_n } \left\langle \nabla G, {\mathbb {J}} \frac{ d {\bar{q}}_n}{ dw } \right\rangle \ dx. \end{aligned}$$

Inserting the gradient of G,  then

$$\begin{aligned} \{F,G\}= & {} - \sum \limits _{ n,m\in {\mathbb {Z}}_{odd}^2 } \int _{{\mathbb {T}}^2} \frac{ \partial F}{ \partial q_n } \frac{ \partial G}{ \partial q_m } \left\langle \frac{ d q_m}{ dw } ,{\mathbb {J}} \frac{ d q_n}{ dw } \right\rangle \ dx + \frac{ \partial F}{ \partial q_n } \frac{ \partial G}{ \partial {\bar{q}}_m } \left\langle \frac{ d {\bar{q}}_m}{ dw } ,{\mathbb {J}} \frac{ d q_n}{ dw } \right\rangle \ dx \\&+ \frac{ \partial F}{ \partial {\bar{q}}_n } \frac{ \partial G}{ \partial q_m } \left\langle \frac{ d q_m}{ dw } ,{\mathbb {J}} \frac{ d {\bar{q}}_n}{ dw } \right\rangle \ dx + \frac{ \partial F}{ \partial {\bar{q}}_n } \frac{ \partial G}{ \partial {\bar{q}}_m } \left\langle \frac{ d {\bar{q}}_m}{ dw } ,{\mathbb {J}} \frac{ d {\bar{q}}_n}{ dw } \right\rangle \ dx. \end{aligned}$$

By (8.2) and (8.3), we have

$$\begin{aligned} \{F,G\}= \mathrm{i}\sum \limits _{ n \in {\mathbb {Z}}_{odd}^2 } \left( \frac{ \partial F}{ \partial q_n } \frac{ \partial G}{ \partial {\bar{q}}_n} - \frac{ \partial F}{ \partial {\bar{q}}_n } \frac{ \partial G}{ \partial q_n} \right) . \end{aligned}$$