Two-phase periodic solutions to the AKNS hierarchy equations

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