Hong-Yu Pan | Searcharxiv

arXiv2026

Soliton resolution conjecture and soliton gases for the Benjamin-Ono equation with infinite-order multisolitons in the sharp well-posedness space

We prove the soliton resolution conjecture for infinite-order multisoliton solutions in the sharp well-posedness space $H^s(\mathbb R)$, $s>-\frac12$, and construct soliton gases in $H^{s}(\mathbb R)$ with arbitrary $s\in\mathbb R$, for the Benjamin--Ono equation. In particular, our work addresses two important open problems about inverse spectral for Lax operator, including the problem of spectral characterization of infinite-order multisolitons and that of determining the range of their asymptotic parameters. A central ingredient of our work is a spectral decomposition adapted to rough initial data. This extends the global soliton--radiation description to weakly weighted $H^s$ data while allowing infinite negative eigenvalues. Another important ingredient is a continuous norming action that extends finite inverse spectral shifts to gas limits while retaining a full continuous norming weight. Our framework provides a basis for studying infinite-order soliton resolution and the construction and asymptotics of soliton gases for nonlocal equations.

math.AP↗

arXiv2026

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

We prove the soliton resolution conjecture for the Benjamin-Ono equation in the sharp well-posedness space $H^s\left(\mathbb{R}\right)$ for $s>-\frac12$. 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.

math.AP↗