Diophantine Conditions in Well-Posedness Theory of Coupled KdV-Type Systems: Local Theory
Abstract
Description
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 αwhen α\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}_λ), s \geq s^\ast, where s^\ast = s^\ast(α) \in ({1/2}, 1] is determined by the Diophantine characterization of certain constants derived from the coupling parameter α. 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}_λ) for α= 1 without the mean 0 assumption, by introducing the vector-valued X^{s, b} spaces.
31 pages. To appear in Internat. Math. Res. Not. "Global Theory" is available in Electron. J. Diff. Eqns
31 pages. To appear in Internat. Math. Res. Not. "Global Theory" is available in Electron. J. Diff. Eqns