On the Integrability of the<scp>Benjamin‐Ono</scp>Equation on the Torus
Patrick Gérard, Thomas Kappeler
Abstract
In this paper we prove that the Benjamin‐Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: this equation admits global Birkhoff coordinates on the space of real‐valued, 2 π ‐periodic, L 2 ‐integrable functions of mean 0. These are coordinates that allow us to integrate it by quadrature and hence are also referred to as nonlinear Fourier coefficients. As a consequence, all the solutions of the Benjamin‐Ono equation are almost periodic functions of the time variable. The construction of such coordinates relies on the spectral study of the Lax operator in the Lax pair formulation of the Benjamin‐Ono equation and on the use of a generating functional, which encodes the entire Benjamin‐Ono hierarchy. © 2020 Wiley Periodicals, Inc.