Shape-morphing reduced-order models for nonlinear Schrodinger equations

TL;DR

This paper compares two reduced-order modeling methods for nonlinear Schrödinger equations, revealing their connection and demonstrating RONS's superior accuracy in predicting wave velocities and solutions.

Contribution

It uncovers the relationship between the reduced Lagrangian and RONS methods and shows RONS's effectiveness when the Lagrangian approach fails.

Findings

RONS predicts the correct group velocity for NLS.

The two methods derive from a single complex master equation.

RONS accurately approximates solutions for modified NLS.

Abstract

We consider reduced-order modeling of nonlinear dispersive waves described by a class of nonlinear Schrodinger (NLS) equations. We compare two nonlinear reduced-order modeling methods: (i) The reduced Lagrangian approach which relies on the variational formulation of NLS and (ii) The recently developed method of reduced-order nonlinear solutions (RONS). First, we prove the surprising result that, although the two methods are seemingly quite different, they can be obtained from the real and imaginary parts of a single complex-valued master equation. Furthermore, for the NLS equation in a stationary frame, we show that the reduced Lagrangian method fails to predict the correct group velocity of the waves whereas RONS predicts the correct group velocity. Finally, for the modified NLS equation, where the reduced Lagrangian approach is inapplicable, the RONS reduced-order model accurately approximates the true solutions.