On the Inverse Scattering Transform for the Benjamin-Ono Equation

We extend the IST for the Benjamin-Ono (BO) equation 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 also extend the analysis of the asymptotics of the Jost functions and the scattering data to include the non-generic 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 functions beta(lambda) and f(lambda). 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 on possible bound states that can be contained in the initial data. We also study the structure of the scattering data and the Jost functions for pure soliton solutions, and obtain remarkably simple solutions for these Jost functions. Since they are examples of non-generic potentials, they demonstrate the asymptotics for non-generic potentials. We then carefully detail the asymptotics in the limit A lambda 0(+) for generic and non-generic potentials. Lastly, we show how to obtain the infinity of conserved quantities from one of the Jost functions of the BO equation, and also how to obtain these conserved quantities in terms of the various moments of the scattering data.