Real Analytic Quasi-Periodic Solutions with More Diophantine Frequencies for Perturbed KdV Equations

Abstract

In this paper, we consider the perturbed KdV equation with Fourier multiplier

$$\begin{aligned} u_{t} =- u_{xxx} + \big (M_{\xi }u+u^3 \big )_{x},\quad u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$

with analytic data of size $\varepsilon$. We prove that the equation admits a Whitney smooth family of small amplitude, real analytic quasi-periodic solutions with $\tilde{J}$ Diophantine frequencies, where the order of $\tilde{J}$ is $O(\frac{1}{\varepsilon })$. The proof is based on a conserved quantity $\int _0^{2\pi } u^2 dx$, Töplitz–Lipschitz property 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, our normal form part is independent of angle variables in spite of the unbounded perturbation.

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 [13]. $\square$

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 [24]. $\square$