Averaging of dispersion managed nonlinear Schr\"odinger equations

TL;DR

This paper proves that solutions of dispersion managed nonlinear Schrödinger equations in optical fibers converge to an averaged model as the dispersion management parameter diminishes, ensuring global solutions under certain conditions.

Contribution

It establishes convergence of solutions to the averaged equation and proves global existence for small dispersion parameters even with supercritical nonlinearities.

Findings

Solutions converge in H^1 as epsilon approaches zero

Global existence of solutions for small epsilon in positive dispersion case

Applicable to nonlinearities beyond mass-critical power

Abstract

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.