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The Inverse Scattering Transform for the Benjamin–Ono Equation

We extend the inverse scattering transform (IST) for the Benjamin–Ono (BO) equation, given by A. S. Fokas and M. J. Ablowitz (Stud. Appl. Math. 68:1, 1983), in two important ways. First, we restrict the IST to purely real potentials, in which case the scattering data and the inverse scattering equations simplify. Second, we extend the analysis of the asymptotics of the Jost functions and the scattering data to include the nongeneric classes of potentials, which include, but may not be limited to, all N‐soliton solutions. In the process, we also study the adjoint equation of the eigenvalue problem for the BO equation, from which, for real potentials, we find a very simple relation between the two reflection coefficients (the functions β(λ) and f(λ)) introduced by Fokas and Ablowitz. Furthermore, we show that the reflection coefficient also defines a phase shift, which can be interpreted as the phase shift between the left Jost function and the right Jost function. This phase shift leads to an analogy of Levinson's theorem, as well as a condition on the number of possible bound states that can be contained in the initial data. For both generic and nongeneric potentials, we detail the asymptotics of the Jost functions and the scattering data. In particular, we are able to give improved asymptotics for nongeneric potentials in the limit of a vanishing spectral parameter. We also study the structure of the scattering data and the Jost functions for pure soliton solutions, which are examples of nongeneric potentials. We obtain remarkably simple solutions for these Jost functions, and they demonstrate the different asymptotics that nongeneric potentials possess. Last, we show how to obtain the infinity of conserved quantities from one of the Jost functions of the BO equation and how to obtain these conserved quantities in terms of the various moments of the scattering data.