Parametric Reduced Models for the Nonlinear Schr\"odinger Equation

TL;DR

This paper develops reduced parametric models for the nonlinear Schrödinger equation using Mori-Zwanzig formalism, rational and colored noise approximations, and ensemble Kalman filtering for parameter inference, validated across temperature regimes.

Contribution

It introduces novel reduced models based on rational and colored noise approximations, combined with ensemble Kalman filtering for parameter estimation, enhancing forecasting accuracy.

Findings

Models accurately predict moments up to order four

Effective across different temperature regimes

Validated by correlation functions and marginal densities

Abstract

Reduced models for the (defocusing) nonlinear Schr\"odinger equation are developed. In particular, we develop reduced models that only involve the low-frequency modes given noisy observations of these modes. The ansatz of the reduced parametric models are obtained by employing a rational approximation and a colored noise approximation, respectively, on the memory terms and the random noise of a generalized Langevin equation that is derived from the standard Mori-Zwanzig formalism. The parameters in the resulting reduced models are inferred from noisy observations with a recently developed ensemble Kalman filter-based parameterization method. The forecasting skill across different temperature regimes are verified by comparing the moments up to order four, a two-time correlation function statistics, and marginal densities of the coarse-grained variables.