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

Abstract

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.