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A Space-Time Discontinuous Petrov-Galerkin Finite Element Formulation for a Modified Schrödinger Equation for Laser Pulse Propagation in Waveguides

Ankit Chakraborty, Judit Munoz-Matute, Leszek Demkowicz, Jake Grosek

TL;DR

This paper addresses ill-posedness and conditioning in discretizing the nonlinear Schrödinger equation for laser pulse propagation in optical waveguides. It introduces a modified model that retains a second-order spatial derivative and yields a stable first-order system that can be hyperbolic or elliptic depending on the sign of $\beta_2$. A space-time discontinuous Petrov-Galerkin (DPG) discretization with optimal test functions is developed for both regimes, supported by stability analyses and a residual-based error estimator. Numerical results on space-time meshes demonstrate stability, optimal convergence rates, and effective mesh adaptivity; conditioning in the elliptic case can be controlled via a scaling parameter. The framework enables robust, mesh-independent simulations of nonlinear pulse propagation with potential impact on waveguide design and photonics applications.

Abstract

In this article, we propose a modified nonlinear Schrödinger 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 Schrödinger 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.

A Space-Time Discontinuous Petrov-Galerkin Finite Element Formulation for a Modified Schrödinger Equation for Laser Pulse Propagation in Waveguides

TL;DR

This paper addresses ill-posedness and conditioning in discretizing the nonlinear Schrödinger equation for laser pulse propagation in optical waveguides. It introduces a modified model that retains a second-order spatial derivative and yields a stable first-order system that can be hyperbolic or elliptic depending on the sign of . A space-time discontinuous Petrov-Galerkin (DPG) discretization with optimal test functions is developed for both regimes, supported by stability analyses and a residual-based error estimator. Numerical results on space-time meshes demonstrate stability, optimal convergence rates, and effective mesh adaptivity; conditioning in the elliptic case can be controlled via a scaling parameter. The framework enables robust, mesh-independent simulations of nonlinear pulse propagation with potential impact on waveguide design and photonics applications.

Abstract

In this article, we propose a modified nonlinear Schrödinger 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 Schrödinger 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.

Paper Structure

This paper contains 18 sections, 98 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Pulsed laser propagation model
  3. Nonlinear Pulse Propagation
  4. Moving frame of reference:
  5. Nondimensionalization of the propagation model
  6. Well-Posedness Analysis: Hyperbolic Case
  7. Well-Posedness Analysis for the Elliptic Case
  8. Second-order problem and classical variational formulation
  9. First-order system and ultraweak variational formulation
  10. DPG discretization
  11. Hyperbolic System
  12. Ultraweak formulation:
  13. Elliptic System
  14. Ultraweak formulation:
  15. Implementation of DPG method:
  16. ...and 3 more sections

Figures (11)

  • Figure 1: Domain of interest.
  • Figure 2: Boundary conditions for the hyperbolic system of equations.
  • Figure 3: A tensor product element $K$ for the hyperbolic problem.
  • Figure 4: First-order soliton: Contour plots of the (a) real and (b) imaginary part of the solution $u$ for the hyperbolic system.
  • Figure 5: Second-order soliton: Contour plots of the (a) real and (b) imaginary part of the solution $u$ for the hyperbolic system.
  • ...and 6 more figures