Talbot effect for the periodic Benjamin--Ono equation
Abstract
We study the dependence on time of the spatial regularity of solutions to the periodic Benjamin--Ono equation with real, mean-zero initial data of bounded variation. For every datum with a nonzero jump, we prove that continuity and essential boundedness at an irrational time t=2πα are equivalent to the arithmetic condition ∑jlog(qj+1)/√(qj)<∞, where qj are the continued-fraction denominators of α. We also determine the exact Hölder regularity and the Besov scale for spatial exponents 2<p≤∞. We derive a high-frequency energy formula and prove the same sharp local criteria on every spatial interval at every irrational time for an open dense set of BV data. At rational times, piecewise smooth data give finite sums of logarithmic and jump kernels modulo a continuous remainder. For arbitrary BV data with a jump, the nonlinear and linear solutions have different logarithmic Poisson coefficients at every sufficiently large reduced denominator, uniformly in the numerator. The proofs combine the smoothing of Tao's gauge transform with estimates for quadratic Fourier series. A finite averaging argument isolates an atom of the initial gauge measure while retaining the real nonlinear reconstruction.
Related
- THE ZERO-DISPERSION LIMIT OF THE PERIODIC BENJAMIN-ONO EQUATION FOR SQUARE-INTEGRABLE DATA
- Long Time Well-posedness for the Benjamin-Ono Equation with Quasi-Periodic Initial Data
- Unique continuation for the periodic Benjamin-Bona-Mahony equation
- Infinite-order multisoliton solutions to the Benjamin--Ono equation and soliton resolution
- The virial--Kato method and the asymptotic stability of solitons of the generalized Benjamin, Bona and Mahony equation
BibTeX
@misc{chen2026talbot,
title = {{Talbot effect for the periodic Benjamin--Ono equation}},
author = {Xi Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.02567},
eprint = {2608.02567},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02567},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Talbot effect for the periodic Benjamin--Ono equation
AU - Xi Chen
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2608.02567
UR - https://arxiv.org/abs/2608.02567
AB - We study the dependence on time of the spatial regularity of solutions to the periodic Benjamin--Ono equation with real, mean-zero initial data of bounded variation. For every datum with a nonzero jump, we prove that continuity and essential boundedness at an irrational time t=2πα are equivalent to the arithmetic condition ∑_jlog(q_j+1)/√(q_j)<∞, where q_j are the continued-fraction denominators of α. We also determine the exact Hölder regularity and the Besov scale for spatial exponents 2<p≤∞. We derive a high-frequency energy formula and prove the same sharp local criteria on every spatial interval at every irrational time for an open dense set of BV data. At rational times, piecewise smooth data give finite sums of logarithmic and jump kernels modulo a continuous remainder. For arbitrary BV data with a jump, the nonlinear and linear solutions have different logarithmic Poisson coefficients at every sufficiently large reduced denominator, uniformly in the numerator. The proofs combine the smoothing of Tao's gauge transform with estimates for quadratic Fourier series. A finite averaging argument isolates an atom of the initial gauge measure while retaining the real nonlinear reconstruction.
ER -