PERIODIC SOLUTIONS OF THE DISPERSION-MANAGED NONLINEAR SCHRODINGER EQUATION

Abstract

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 $T_{(s)}=T_{D(s)}= e^{-iD(s)\partial_x^2}$. 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 $T_{D(s)}$ to prove global well-posedness for initial data in $H^r$ 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 $\alpha>0,$ the convergence follows from the Rellich-Kondrachov Theorem; in the case $\alpha=0,$ we use a concentration-compactness argument due to Kunze, but with significant modifications.