On spatial decay for coherent states of the Benjamin-Ono equation

2025/05/21 by Gavin Stewart, Stewart, Gavin
Mathematics · Physics and Astronomy · #76B15 (Primary) 70K45 35C07 (Secondary) #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Cold Atom Physics and Bose-Einstein Condensates #FOS: Mathematics #Nonlinear Photonic Systems

paper · pdf · doi:10.48550/arxiv.2505.15915

openalex publication_date 2025/05/21 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

Abstract

We consider solutions to the Benjamin-Ono equation ∂_t u - H ∂_x² u = -∂_x(u²) that are localized in a reference frame moving to the right with constant speed. We show that any such solution that decays at least like ⟨ x⟩^-1-ε for some ε> 0 in a comoving coordinate frame must in fact decay like ⟨ x⟩^-2. In view of the explicit soliton solutions, this decay rate is sharp. Our proof has two main ingredients. The first is microlocal dispersive estimates for the Benjamin-Ono equation in a moving frame, which allow us to prove spatial decay of the solution provided the nonlinearity has sufficient decay. The second is a careful normal form analysis, which allows us to obtain rapid decay of the nonlinearity for a transformed equation assuming only modest decay of the solution. Our arguments are entirely time dependent, and do not require the solution to be an exact traveling wave.

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