On the periodic Korteweg-de Vries equation: a normal form approach

2011/08/10 by Seungly Oh, Oh, Seungly
Mathematics · Physics and Astronomy · #35B10 #35B30 #35Q53 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #math.AP #msc:35B10 #msc:35B30 #msc:35Q53

paper · pdf · doi:10.48550/arxiv.1108.2249

Minor revisions in the abstract and the introduction. Added one reference item

openalex publication_date 2011/08/10 · arxiv created 2011/08/17 · arxiv updated 2011/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper discusses an improved smoothing phenomena for low-regularity solutions of the Korteweg-de Vries (KdV) equation in the periodic settings by means of normal form transformation. As a result, the solution map from a ball on H^-1/2+ to C₀^t ([0,T], H^-1/2+) can be shown to be Lipschitz in a H⁰⁺_x topology, where the Lipschitz constant only depends on the rough norm ‖u₀‖_H^-1/2+ of the initial data. A similar episode has been observed in a recent paper on 1D quadratic Schrödinger equation in low-regularity setting.

Citations

• Normal forms and quadratic nonlinear Klein‐Gordon equations

Related

• On the regularization mechanism for the periodic Korteweg-de Vries equation
• Resonant phase-shift and global smoothing of the periodic Korteweg-de Vries in low regularity settings
• Global Smoothing for the Periodic KdV Evolution
• Smoothing and growth bound of periodic generalized Korteweg-de Vries equation
• On Local Well-posedness of the Periodic Korteweg-de Vries Equation Below H^-(1)/(2)(\mathbbT)