Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds
Ruoci Sun
Abstract
Abstract 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 $$\mathcal {U}_N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>U</mml:mi><mml:mi>N</mml:mi></mml:msub></mml:math> . We construct generalized action–angle coordinates which establish a real analytic symplectomorphism from $$\mathcal {U}_N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>U</mml:mi><mml:mi>N</mml:mi></mml:msub></mml:math> onto some open convex subset of $${\mathbb {R}}^{2N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:math> and allow to solve the equation by quadrature for any such initial datum. As a consequence, $$\mathcal {U}_N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>U</mml:mi><mml:mi>N</mml:mi></mml:msub></mml:math> 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. 10.1002/cpa.21896 . arXiv:1905.01849 ). The global well-posedness of the BO equation on $$\mathcal {U}_N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>U</mml:mi><mml:mi>N</mml:mi></mml:msub></mml:math> is given by a polynomial characterization and a spectral characterization of the manifold $$\mathcal {U}_N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>U</mml:mi><mml:mi>N</mml:mi></mml:msub></mml:math> . 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.