We consider the local well-posedness (LWP) problem of a one-parameter family of coupled Korteweg-de Vries-type systems in both the periodic and nonperiodic settings. In particular, we show that certain resonances occur, closely depending on the value of a coupling parameter alpha when alpha not equal 1. In the periodic setting, we use the Diophantine conditions to characterize the resonances, and establish a sharp LWP of the system in H(s)(T(lambda)), s >= s*, where s* = s*(alpha) is an element of (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)(R). In the Appendix, we briefly discuss the LWP result in H(-1/2) (T(lambda)) for alpha = 1 without the mean zero assumption, by introducing the vector-valued X(s,b) spaces.
- bilinear estimates
- Diophantine condition