Coupling constant behavior of eigenvalues of Zakharov-Shabat systems

Abstract

We consider the eigenvalues of the non-self-adjoint Zakharov-Shabat systems as the coupling constant of the potential is varied. In particular, we are interested in eigenvalue collisions and eigenvalue trajectories in the complex plane. We identify shape features in the potential that are responsible for the occurrence of collisions and we prove asymptotic formulas for large coupling constants that tell us where eigenvalues collide or where they emerge from the continuous spectrum. Some examples are provided which show that the asymptotic methods yield results that compare well with exact numerical computations. (c) 2007 American Institute of Physics.