Diophantine Conditions in Well-Posedness Theory of Coupled KdV-Type Systems: Local Theory

Tadahiro Oh*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
Pages (from-to)3516-3556
Number of pages41
JournalInternational Mathematics Research Notices
Volume2009
Issue number18
DOIs
Publication statusPublished - 2009

Keywords

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

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