Long time bounds for coupled KdV equations

Abstract

In this paper, we consider the coupled KdV equationηt+wx+(wη)x+16wxxx=0,wt+ηx+wwx+16ηxxx=0on T=R/2πZ with initial data of small amplitudes ɛ in Sobolev spaces. If the first three Fourier modes of initial data are of size ɛ1+μ for any 0≤μ≤12, we prove that the solutions remain smaller than 2ɛ for a time scale of order ɛ−(1+μ) via a normal form transformation. Further, we show this order of time scale is sharp.