Continuum limit related to dispersion managed nonlinear Schr\"{o}dinger equations

TL;DR

This paper proves that solutions of a discrete dispersion managed nonlinear Schrödinger equation converge to the continuous version as the step size approaches zero, establishing a rigorous connection between discrete and continuous models.

Contribution

It demonstrates the strong convergence of discrete solutions to the continuous dispersion managed NLS in the $L^2$ space as the discretization parameter tends to zero.

Findings

Strong $L^2$ convergence of discrete to continuous solutions

Global well-posedness of the discrete equations established

Convergence holds for all step sizes in (0,1]

Abstract

We consider the dispersion managed nonlinear Schr\"odinger 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.