Two-Phase Periodic Solutions to the AKNS Hierarchy Equations

In this paper, we investigate genus 2 algebro-geometric solutions of the AKNS hierarchy equations strictly periodic with respect to the space variable x. In general, genus 2 solutions, which are expressed in terms of two-dimensional Riemann theta functions, are not strictly periodic in x. We show that x-periodic solutions can be obtained by an appropriate choice of a hyperelliptic spectral curve having a structure of a covering of an elliptic curve. For odd-numbered members of the AKNS hierarchy, these solutions can be made periodic also with respect to the corresponding time variables of the AKNS hierarchy, by imposing further restrictions on the structure of the spectral curve. The corresponding solutions are especially interesting from the point of view of potential applications to the study of signal propagation in nonlinear optical fibers.