A numerical study of the large-period limit of a Zakharov–Shabat eigenvalue problem with periodic potentials

Abstract

Deconinck and Kutz (2006 J. Comput. Phys. 219 296–321) developed an efficient algorithm for solving the Zakharov–Shabat eigenvalue problem with periodic potentials numerically. It is natural to use the same algorithm for solving the problem for non-periodic potential (decaying potentials defined over the whole real line) using large periods. In this paper, we propose the use of a specific value of the Floquet exponent. Our numerical results indicate that it can produce accurate results long before the period becomes large enough for the analytical convergence results of Gardner (1997 J. Reine Angew. Math. 491 149–81) to be valid. We also illustrate the rather complicated path to convergence of some nonlinear Schrödinger potentials.