Invariant Tori of Full Dimension for Second KdV Equations with the External Parameters

Abstract

In this paper, we consider the second KdV equation with the external parameters

$$\begin{aligned} u_{t} =\partial _x^5 u +(M_{\sigma }u+u^3)_{x}, \end{aligned}$$

under zero mean-value periodic boundary conditions

$$\begin{aligned} u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$

where $M_\sigma$ is a real Fourier multiplier. It is proved that the equations admit a Whitney smooth family of small amplitude, real analytic almost periodic solutions with all frequencies. The proof is based on a conserved quantity $\int _0^{2\pi } u^2 dx$, Töplitz–Lipschitz property of the perturbation and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity $\int _0^{2\pi } u^2 dx$ and Töplitz–Lipschitz property of the perturbation, our normal form part is independent of angle variables in spite of the unbounded perturbation. This is the first attempt to prove the almost periodic solutions for the unbounded perturbation case.

Appendix

Lemma 6.5

If

$$\begin{aligned} \Vert X_F\Vert _{ D_\rho (r,s),\mathcal O }< \varepsilon ',\ \Vert X_G\Vert _{ D_\rho (r,s),\mathcal O }< \varepsilon '', \end{aligned}$$

then

$$\begin{aligned} \Vert X_{\{F,G\}}\Vert _{D_\rho (r-\sigma ,\eta s),\mathcal O}<c\sigma ^{-1}\eta ^{-2}\varepsilon '\varepsilon '',\ \eta \ll 1. \end{aligned}$$

In particular, if $\eta \sim \varepsilon ^{\frac{1}{3}}$, $\varepsilon ', \varepsilon ''\sim \varepsilon$, we have $\Vert X_{\{F,G\}}\Vert _{D_\rho (r-\sigma ,\eta s),\mathcal O}\sim \varepsilon ^{\frac{4}{3}}$.

For the proof, see [18]. $\square$

Let V be an open domain in a real Banach space E with norm$\Vert \cdot \Vert$, B a subset of another real Banach space, and $X:V\times B\rightarrow E$ a parameter dependent vector field on V,  which is $C^{1}$ on V and Lipschitz on B. Let $\phi ^{t}$ be its flow. Suppose there is a subdomain $U\subset V$ such that $\phi ^{t} :U\times B\rightarrow V$ for $-1\le t\le 1$.

Lemma 6.6

Under the preceding assumptions,

$$\begin{aligned} \Vert \phi ^{t}-id\Vert _{U}\le & {} \Vert X\Vert _{V}\\ \Vert \phi ^{t}-id\Vert _{U}^{\mathcal L }\le & {} \mathrm{exp}(\Vert DX_{V}\Vert )\Vert X\Vert _{V}^{\mathcal L} \end{aligned}$$

For the proof, see [35]. $\square$