The Benjamin–Ono (BO) equation, which describes internal long waves of deep stratified fluids, has multi-soliton solutions. I shall prove the invariance of every multi-soliton manifold under the BO flow and construct global (generalized) action–angle coordinates in order to solve this equation by quadrature for any such initial datum. The complete integrability of the BO equation on every N-soliton manifold constitutes a first step towards the soliton resolution conjecture of the BO equation on the line. The construction of such coordinates relies on the Lax pair structure, the inverse spectral transform and 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 shift semigroup acting on some Hardy spaces. Furthermore, the N-soliton manifold of the BO equation on the line can be interpreted as the universal covering of the manifold of N-gap potentials for the space-periodic BO equation as described by Gérard–Kappeler.

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