Semiclassical eigenvalue distribution of the Zakharov-Shabat eigenvalue problem

In this paper we consider the semi-classical limit of the non-self-adjoint Zakharov-Shabat eigenvalue problem. We conduct a series of careful numerical experiments which provide strong evidence that the number of eigenvalues scales like epsilon(-1), just as in the self-adjoint case, and that the eigenvalues appear to approach a limiting curve. One general choice of potential functions produces a Y-shaped spectrum. We give an asymptotic argument which predicts a critical value for the phase for which the straight line spectra bifurcates to produce the Y-shaped spectra. This asymptotic prediction agrees quite well with numerical experiments. The asymptotic argument also predicts a symmetry breaking in the eigenfunctions, which we are able to observe numerically. We also show that the number of eigenvalues living away from the real axis for a restricted class of potentials is bounded by c epsilon(-1), where c is an explicit constant. A complete theory of the shape of the eigenvalue curve and a general bound on the number of eigenvalues is still lacking.