Solutions of a nonlinear integral equation for high-energy scattering. II. Numerical solutions

Solutions of the Ball-Zachariasen equation for high-energy scattering are calculated unmerically. For small values of the parameter c, which measures the strength of particle production, the solutions coincide with those obtained in the existence proof of paper I. When $c{\sigma }_{el}$ (${\sigma }_{el}=elasticcrosssection$) is much smaller than the physical value, the solutions are obtained successfully by either plain iteration or Newton-Kantorovich iteration, but they disagree qualitatively with experiment. A continuation to larger c is attempted by using the result of a successful iteration as the starting point for an iteration with slightly larger c. The continuation proceeds only a short distance before coming to a halt, because of a singularity of the Fr\'echet derivative of the nonlinear integral operator. The singularity is circumvented by an excursion into the complex c plane. On return to the real axis the solutions are complex, however, which is not allowed physically. A proposed approximate solution of Ball and Zachariasen, quite different from those of paper I, is also investigated. Plain or Newton-Kantorovich iteration starting with this function fails to converge. The failure is explained again in terms of a nearby singularity of the Fr\'echet derivative. The singularity makes it difficult to decide whether the Ball-Zachariasen function is close to a true solution.