Local well-posedness of the Benjamin-Ono equation with spatially quasiperiodic data

2024/12/17 by Sultan Aitzhan, David M. Ambrose, Aitzhan, Sultan +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Algebraic structures and combinatorial models #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2412.12457

openalex publication_date 2024/12/17 · openalex created_date 2024/12/19 · openalex updated_date 2026/07/28

Abstract

We consider the Benjamin-Ono equation in the spatially quasiperiodic setting. We establish local well-posedness of the initial value problem with initial data in quasiperiodic Sobolev spaces. This requires developing some of the fundamental properties of Sobolev spaces and the energy method for quasiperiodic functions. We discuss prospects for global existence. We demonstrate that while conservation laws still hold, these quantities no longer control the associated Sobolev norms, thereby preventing the establishment of global results by usual arguments.

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