Rough solutions for the periodic Schrödinger - Kortweg-deVries system

2005/11/19 by Alexander Arbieto, Arbieto, Alexander, Adán J. Corcho +3
Mathematics · Physics and Astronomy · #35Q99 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.math/0511491

openalex publication_date 2005/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove two new mixed sharp bilinear estimates of Schrödinger-Airy type. In particular, we obtain the local well-posedness of the Cauchy problem of the Schrödinger - Kortweg-deVries (NLS-KdV) system in the periodic setting. Our lowest regularity is H^1/4× L², which is somewhat far from the naturally expected endpoint L²× H^-1/2. This is a novel phenomena related to the periodicity condition. Indeed, in the continuous case, Corcho and Linares proved local well-posedness for the natural endpoint L²× H^-3/4+. Nevertheless, we conclude the global well-posedness of the NLS-KdV system in the energy space H¹× H¹ using our local well-posedness result and three conservation laws discovered by M. Tsutsumi.

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