Unique continuation for the periodic Benjamin-Bona-Mahony equation
Abstract
We prove unique continuation for the real periodic Benjamin--Bona--Mahony equation in the energy space: a solution that equals a constant on a nonempty space--time strip is that constant everywhere. In particular, Rosier's unique continuation conjecture [23] holds without any smallness or mean assumption. It even suffices to observe, at one point x0 and for t in an interval, the solution and the potential ψ=(1−∂x2)−1(u+u2/2) of its flux: u(x0,t)=C and ψ(x0,t)=C+C2/2 imply u≡ C, and the second condition cannot be dropped when C=−1. The proof uses an exponentially weighted Hamiltonian of indefinite sign with a coercive dissipation identity, and it applies to every nonconstant real polynomial flux. Consequently, Rosier's stabilization results for two localized feedback laws and for a boundary feedback hold unconditionally, for arbitrary data. Stationary step profiles show that the theorem fails in Hs for 0≤ s<1/2. We also give a generating-series proof of single-point unique continuation for the linearized equation and discuss related control results.
Related
- The virial--Kato method and the asymptotic stability of solitons of the generalized Benjamin, Bona and Mahony equation
- Almost sure global well-posedness for the Benjamin-Bona-Mahony equation
- 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
- Talbot effect for the periodic Benjamin--Ono equation
BibTeX
@misc{filho2026unique,
title = {{Unique continuation for the periodic Benjamin-Bona-Mahony equation}},
author = {Roberto de A. Capistrano Filho and Victor Hugo Gonzalez Martinez and Ailton Campos do Nascimento},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36204},
eprint = {2609.36204},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36204},
note = {CC0},
}
RIS
TY - UNPB
TI - Unique continuation for the periodic Benjamin-Bona-Mahony equation
AU - Roberto de A. Capistrano Filho
AU - Victor Hugo Gonzalez Martinez
AU - Ailton Campos do Nascimento
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.36204
UR - https://arxiv.org/abs/2609.36204
AB - We prove unique continuation for the real periodic Benjamin--Bona--Mahony equation in the energy space: a solution that equals a constant on a nonempty space--time strip is that constant everywhere. In particular, Rosier's unique continuation conjecture [23] holds without any smallness or mean assumption. It even suffices to observe, at one point x_0 and for t in an interval, the solution and the potential ψ=(1−∂_x^2)^−1(u+u^2/2) of its flux: u(x_0,t)=C and ψ(x_0,t)=C+C^2/2 imply u≡ C, and the second condition cannot be dropped when C=−1. The proof uses an exponentially weighted Hamiltonian of indefinite sign with a coercive dissipation identity, and it applies to every nonconstant real polynomial flux. Consequently, Rosier's stabilization results for two localized feedback laws and for a boundary feedback hold unconditionally, for arbitrary data. Stationary step profiles show that the theorem fails in H^s for 0≤ s<1/2. We also give a generating-series proof of single-point unique continuation for the linearized equation and discuss related control results.
ER -