Diophantine conditions in global well-posedness for coupled KdV-type systems

We consider the global well-posedness problem of a one-parameter family of coupled KdV-type systems both in the periodic and non-periodic setting. When the coupling parameter $alpha = 1$, we prove the global well-posedness in $H^s(mathbb{R}) $ for $s > 3/4$ and $H^s(mathbb{T}) $ for $s geq -1/2$ via the I-method developed by Colliander-Keel-Staffilani-Takaoka-Tao [5]. When $alpha e 1$, as in the local theory [14], certain resonances occur, closely depending on the value of $alpha$. We use the Diophantine conditions to characterize the resonances. Then, via the second iteration of the I-method, we establish a global well-posedness result in $H^s(mathbb{T})$, $s geq widetilde{s}$, where $widetilde{s}= widetilde{s}(alpha) in (5/7, 1]$ is determined by the Diophantine characterization of certain constants derived from the coupling parameter $alpha$. We also show that the third iteration of the I-method fails in this case.