Parametric reduced models for the nonlinear Schrödinger equation

Reduced models for the (defocusing) nonlinear Schrödinger 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 parametrization 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.

Figure 1

State estimates $u$ and $f$ for the parametric model (3.16).

Figure 2

Deterministic parameter estimates, $a, b, c$, and $d$.

Figure 3

Stochastic parameter estimates, $\sigma ,R$.

Figure 4

Solutions of the reduced model in (3.16), integrated with the estimated parameters.

Figure 5

The marginal distribution (left) and the time correlation function (right) predicted by the reduced mode (3.16) for low temperature, $\beta ={10}^4$. For comparison, the statistics of the true solution is also shown.

Figure 6

Predicted marginal distribution and correlation function for $\beta =10$ using the first reduced model (3.16).

Figure 7

Predicted marginal distribution and correlation function for $\beta =10$ using the reduced model (3.19).

Figure 8

Solutions of the reduced model in (3.23) compared to those of the full model.

Figure 9

Predicted marginal distributions and correlation functions for $\beta =1/20$ from the reduced model in (3.23)