Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds

This paper is dedicated to proving the complete integrability of Benjamin--Ono (BO) equation on line when restricted every $N$-soliton manifold, denoted by $\mathcal{U}_N$. We construct generalized action--angle coordinates which establish a real analytic symplectomorphism from $\mathcal{U}_N$ onto some open convex subset $\mathbb{R}^{2N}$ and allow solve quadrature for any such initial datum. As consequence, universal covering manifold $N$-gap potentials BO torus as described G\'erard--Kappeler $[19]$. The global well-posedness in given polynomial characterization spectral Besides analysis Lax operator shift semigroup acting Hardy spaces, construction also relies use generating functional, encodes entire hierarchy.