Averaging of dispersion managed nonlinear Schr\"odinger equations

We consider the dispersion managed power-law nonlinear Schr\"odinger(DM NLS) equations with a small parameter $\varepsilon > 0$ and the averaged equation, which are used in optical fiber communications. We prove that the solutions of DM NLS equations converge to the solution of the averaged equation in $H^1(\mathbb{R})$ as $\varepsilon$ goes to zero. Meanwhile, in the positive average dispersion, we obtain the global existence of the solution to DM NLS equation in $H^1(\mathbb{R})$ for sufficiently small $\varepsilon > 0$, even when the exponent of the nonlinearity is beyond the mass-critical power.