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Parametric reduced models for the nonlinear Schrödinger equation.

John Harlim1,2, Xiantao Li1

  • 1Department of Mathematics, the Pennsylvania State University, University Park, Pennsylvania 16802-6400, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 13, 2015
PubMed
Summary

We developed reduced models for the nonlinear Schrödinger equation using low-frequency modes and noisy data. These models accurately forecast system behavior across different temperature regimes.

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Area of Science:

  • Nonlinear dynamics
  • Statistical physics

Background:

  • The nonlinear Schrödinger equation (NLSE) describes various physical phenomena.
  • Reduced models are essential for efficiently analyzing complex NLSE dynamics.
  • Extracting information from low-frequency modes with noisy observations presents challenges.

Purpose of the Study:

  • To develop and validate reduced parametric models for the defocusing NLSE.
  • To incorporate noisy observations of low-frequency modes into model development.
  • To assess the forecasting capabilities of the reduced models.

Main Methods:

  • Derivation of a generalized Langevin equation using the Mori-Zwanzig formalism.
  • Application of rational and colored-noise approximations for model ansatz.
  • Parameter inference using an ensemble Kalman filter-based method.

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  • Validation through comparison of moments, correlation functions, and marginal densities.

    • Main Results:

    • Successfully developed reduced models capturing essential dynamics from low-frequency modes.
    • Demonstrated effective parameter inference from noisy observational data.
    • Validated forecasting skill across various temperature regimes.

    Conclusions:

    • The developed reduced models offer an efficient and accurate approach for analyzing NLSE.
    • The methodology effectively integrates noisy, low-frequency observational data.
    • The study provides a robust framework for reduced-order modeling in nonlinear systems.