We study the existence, uniqueness and stability of solutions to the initial-value problem for the periodic dispersion-managed nonlinear Schr"{o}dinger (DMNLS) equation, an equation that models the propagation of periodic, nonlinear, quasi-monochromatic electromagnetic pulses in a dispersion-managed fiber. The periodic DMNLS equation we derive is the same as the non-periodic DMNLS equation (\ref{eq:1.2}), except with a subtle difference in the operator T(s)=TD(s)=e−iD(s)∂x2. The periodic function D(s) still controls the dispersive properties of the optical fiber.

With respect to the Cauchy problem for the periodic DMNLS equation, under certain assumptions on the variable dispersion, we use a Strichartz estimate (Theorem \ref{th:3.2}) on the family of operators TD(s) to prove global well-posedness for initial data in Hr for non negative integer values of r.

Lastly, we prove results on the existence and stability of ground state solutions by considering the convergence of minimizing sequences for certain variational problems. In the case α>0, the convergence follows from the Rellich-Kondrachov Theorem; in the case α=0, we use a concentration-compactness argument due to Kunze, but with significant modifications.