Continuum limit related to dispersion managed nonlinear Schrödinger equations

We consider the dispersion managed nonlinear Schrödinger equation with power-law nonlinearity and its discrete version of equations with step size $h\in (0,1]$. We prove that the solutions of the discrete equations strongly converge in $L^2(\mathbb{R})$ to the solution of the dispersion managed NLS as $h\to 0$ after showing the global well-posedness of the discrete equations.