We consider the local well-posedness problem of a one-parameter family of coupled KdV-type systems both in the periodic and non-periodic setting. In particular, we show that certain resonances occur, closely depending on the value of a coupling parameter \alpha when \alpha \ne 1. In the periodic setting, we use the Diophantine conditions to characterize the resonances, and establish sharp local well-posedness of the system in H^s(\mathbb{ T}_\lambda), s \geq s^\ast, where s^\ast = s^\ast(\alpha) \in ({ 1/2}, 1] is determined by the Diophantine characterization of certain constants derived from the coupling parameter \alpha. We also present a sharp local (and global) result in L^2(\mathbb{ R}). In the appendix, we briefly discuss the local well-posedness result in H^{ -{ 1/2}}(\mathbb{ T}_\lambda) for \alpha= 1 without the mean 0 assumption, by introducing the vector-valued X^{ s, b} spaces.
Categories: math.AP