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

2009/04/18 by Tadahiro Oh, Oh, Tadahiro · 2 citations
Mathematics · Physics and Astronomy · #35Q53 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.0904.2813

openalex publication_date 2009/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 αwhen α≠ 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(\mathbbT_λ), s ≥ s^∗, where s^∗ = s^∗(α) ∈ (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 L²(ℝ). In the appendix, we briefly discuss the local well-posedness result in H^-1/2(\mathbbT_λ) for α= 1 without the mean 0 assumption, by introducing the vector-valued X^{s, b} spaces.

Cited by

• A Coupled Generalized Korteweg-de Vries System Driven by White Noise
• Global well-posedness of the Majda-Biello system in the resonant case on the real line

Related

• Invariant Gibbs Measures and a.s. Global Well-Posedness for Coupled KdV Systems
• Local well-posedness for the Schrödinger-KdV system in H^s₁× H^s₂
• Local well-posedness for the Schrödinger-KdV system in H^s₁× H^s₂, II
• Global Well-posedness of a System of Nonlinearly Coupled KdV equations of Majda and Biello
• Sharp well-posedness for a coupled system of mKdV type equations