THE INVERSE SCATTERING TRANSFORM FOR THE BENJAMIN-ONO-EQUATION

We extend the inverse scattering transform (IST) for the Benjamin-One (BO) equation, given by A. S. Fokas and M. J. Ablowitz (Stud. Appl. Ma th. 68:1, 1983), in two important ways. First, we restrict the IST to purely real potentials, in which case the scattering data and the inve rse scattering equations simplify. Second, we extend the analysis of t he asymptotics of the Jest functions and the scattering data to includ e 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 beta(lambda) and f(lambda) ) introduced by Fokas and Ablowitz. Furthermore, we show that the refl ection coefficient also defines a phase shift, which can be interprete d as the phase shift between the left Jest function and the right Jest 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 c an be contained in the initial data. For both generic and nongeneric p otentials, we detail the asymptotics of the Jest functions and the sca ttering data. In particular, we are able to give improved asymptotics for nongeneric potentials in the limit of a vanishing spectral paramet er. We also study the structure of the scattering data and the Jest fu nctions for pure soliton solutions, which are examples of nongeneric p otentials. We obtain remarkably simple solutions for these Jest functi ons, and they demonstrate the different asymptotics that nongeneric po tentials possess. Last, we show how to obtain the infinity of conserve d quantities from one of the Jest functions of the BO equation and how to obtain these conserved quantities in terms of the various moments of the scattering data.