## Abstract

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.

Original language | English |
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Pages (from-to) | 3516-3556 |

Number of pages | 41 |

Journal | International Mathematics Research Notices |

Volume | 2009 |

Issue number | 18 |

DOIs | |

Publication status | Published - 2009 |

## Keywords

- KdV
- Well-posedness
- Ill-posedness
- bilinear estimates
- Diophantine condition