Long time bounds for coupled KdV equations

Abstract

In this paper, we consider the coupled KdV equation

$\left(\begin{array}{cc}{\eta }_t+w_x+{\left(w\eta \right)}_x+\frac{1}{6}{w}_{xxx}& =0,\\ w_t+{\eta }_x+w{w}_x+\frac{1}{6}{\eta }_{xxx}& =0\end{array}\right)$

on $T=R/2\pi Z$ with initial data of small amplitudes $\varepsilon$ in Sobolev spaces. If the first three Fourier modes of initial data are of size ${\varepsilon }^{1+\mu }$ for any $0\le \mu \le \frac{1}{2}$, we prove that the solutions remain smaller than $2\varepsilon$ for a time scale of order ${\varepsilon }^{-(1+\mu )}$ via a normal form transformation. Further, we show this order of time scale is sharp.