Parametric Reduced Models for the Nonlinear Schr\"odinger Equation
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.
Cite
@article{arxiv.1502.04310,
title = {Parametric Reduced Models for the Nonlinear Schr\"odinger Equation},
author = {John Harlim and Xiantao Li},
journal= {arXiv preprint arXiv:1502.04310},
year = {2015}
}