This paper is dedicated to proving the complete integrability of the Benjamin–Ono (BO) equation on the line when restricted to 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 of \({\mathbb {R}}^{2N}\) and allow to solve the equation by quadrature for any such initial datum. As a consequence, \(\mathcal {U}_N\) is the universal covering of the manifold of N-gap potentials for the BO equation on the torus as described by Gérard–Kappeler (Commun Pure Appl Math, 2020. https://doi.org/10.1002/cpa.21896. arXiv:1905.01849). The global well-posedness of the BO equation on \(\mathcal {U}_N\) is given by a polynomial characterization and a spectral characterization of the manifold \(\mathcal {U}_N\). Besides the spectral analysis of the Lax operator of the BO equation and the shift semigroup acting on some Hardy spaces, the construction of such coordinates also relies on the use of a generating functional, which encodes the entire BO hierarchy. The inverse spectral formula of an N-soliton provides a spectral connection between the Lax operator and the infinitesimal generator of the very shift semigroup.

本文致力于证明当限于每个N孤子流形时，本杰明·奥诺（BO）方程在行上的完全可积性，用\（\ mathcal {U} _N \）表示。我们构造了广义的动作角坐标，该坐标从\（\ mathcal {U} _N \）到\（{\ mathbb {R}} ^ {2N} \）的某个开放凸子集上建立了一个真正的解析同构，并允许求解对于任何这样的初始基准，通过正交方程求解。结果，\（\ mathcal {U} _N \）是N的流形的通用覆盖如Gérard–Kappeler所描述的那样，圆环上的BO方程具有-gap电位（Commun Pure Appl Math，2020年。https：//doi.org/10.1002/cpa.21896。arXiv：1905.01849）。BO （\ mathcal {U} _N \）上的BO方程的整体适定性由流形\（\ mathcal {U} _N \）的多项式表征和谱表征给出。除了对BO方程的Lax算符和作用于某些Hardy空间的移位半群进行频谱分析外，此类坐标的构造还依赖于生成函数的使用，该函数对整个BO层次进行编码。N孤立子的逆谱公式提供了Lax算子与极移半群的无穷小生成器之间的谱连接。