Model Order Reduction for Nonlinear Schr\"odinger Equation

TL;DR

This paper develops a reduced order modeling approach for the nonlinear Schrödinger equation using proper orthogonal decomposition, ensuring accurate low-dimensional approximations that preserve key dynamics like energy and soliton solutions.

Contribution

It introduces a POD-based reduction method for NLS equations with error estimates and demonstrates its effectiveness through numerical experiments.

Findings

POD accurately captures NLS dynamics

Reduced models preserve energy and solitons

Effective in 1D and 2D cases

Abstract

We apply the proper orthogonal decomposition (POD) to the nonlinear Schr\"odinger (NLS) equation to derive a reduced order model. The NLS equation is discretized in space by finite differences and is solved in time by structure preserving symplectic mid-point rule. A priori error estimates are derived for the POD reduced dynamical system. Numerical results for one and two dimensional NLS equations, coupled NLS equation with soliton solutions show that the low-dimensional approximations obtained by POD reproduce very well the characteristic dynamics of the system, such as preservation of energy and the solutions.