A Space-Time Discontinuous Petrov-Galerkin Finite Element Formulation for a Modified Schr\"odinger Equation for Laser Pulse Propagation in Waveguides

TL;DR

This paper introduces a modified nonlinear Schr"odinger equation for optical pulse propagation in waveguides, employing a space-time discontinuous Petrov-Galerkin finite element method for stable numerical solutions.

Contribution

It develops a novel stable first-order system model that bifurcates into elliptic and hyperbolic equations, and applies a space-time discontinuous Petrov-Galerkin method for discretization.

Findings

Model bifurcates into elliptic and hyperbolic systems.

Proves stability of the proposed model.

Numerical examples demonstrate stability on space-time meshes.

Abstract

In this article, we propose a modified nonlinear Schr\"odinger equation for modeling pulse propagation in optical waveguides. The proposed model bifurcates into a system of elliptic and hyperbolic equations depending on waveguide parameters. The proposed model leads to a stable first-order system of equations, distinguishing itself from the canonical nonlinear Schr\"odinger equation. We have employed the space-time discontinuous Petrov-Galerkin finite element method to discretize the first-order system of equations. We present a stability analysis for both the elliptic and hyperbolic systems of equations and demonstrate the stability of the proposed model through several numerical examples on space-time meshes.