Soliton resolution conjecture for the Benjamin-Ono equation in the sharp well-posedness space

Abstract

We prove the soliton resolution conjecture for the Benjamin-Ono equation in the sharp well-posedness space $H^s\left(\mathbb{R}\right)$ for $s>-\frac{1}{2}$. This extends soliton resolution to the sharp regularity, whereas previous results required weighted $H^{1,1}\left(\mathbb{R}\right)$ initial data. We first prove the soliton resolution conjecture in the sharp well-posedness space for the integrable equation. In particular, our work solves three important open problems, including the problem of embedded singular continuous spectrum, the extension of Gérard's explicit formula, and the inverse spectral problem for the Lax operator. A central ingredient of our work is a new modular operator-theoretic framework combining the closed quadratic-form theory of rough Lax operators, weighted limiting absorption, and distorted Fourier analysis, which yields a unified treatment of distributional spectral theory, continuous-spectrum scattering, and radiative asymptotics at the sharp well-posedness threshold. Another important ingredient is a Fourier-side regularity and stability result for the eigenfunctions of the rough Lax operator, established through a zero-frequency identity, resolvent commutator estimates, and an isolating-contour argument. This yields well-defined soliton centers that are stable under rough approximation. Our framework provides a broadly applicable framework for proving soliton resolution conjecture for integrable dispersive equations in low-regularity spaces.

arXiv comment72 pages. Comments are welcome! Major revision. Title and abstract changed. Soliton resolution extended to the sharp well-posedness space $H^s(\mathbb{R}),s>-1/2$. New results on Gérard's formula, bound states, the positive spectrum, and sharpness of the spatial weight. The inverse spectral result is retained with a revised proof. Previous explicit $L^{\infty }$ error estimates are omitted