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

TL;DR

This paper investigates the local well-posedness of coupled KdV-type systems, revealing how Diophantine conditions influence resonance phenomena and establishing sharp results in both periodic and non-periodic settings.

Contribution

It introduces a Diophantine-based characterization of resonances and proves sharp local well-posedness results depending on the coupling parameter in various function spaces.

Findings

Resonances depend on the coupling parameter 

Sharp local well-posedness in H^s(\

Sharp global well-posedness in L^2(

Abstract

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.