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Data-driven Soliton Manifold Approximations for Dark and Bright Waves: Some Prototypical 1d Case Examples

Su Yang, Shaoxuan Chen, Wei Zhu, Panayotis G. Kevrekidis

TL;DR

The paper addresses learning reduced-order ODE models for soliton interactions in the 1D Gross-Pitaevskii/nonlinear Schrödinger equation, covering both dark and bright solitons and the role of parabolic trapping. It employs Sparse Identification of Nonlinear Dynamics (SINDy) and SINDy with control to identify governing equations directly from PDE time-series, using carefully constructed libraries and multiple initial conditions. The authors show that SINDy can accurately recover known reduced dynamics in many two- and four-soliton configurations, while also revealing limitations in higher-dimensional cases and boundary/inhomogeneous settings; physics-informed variants improve accuracy and coherence with Newton's third law. Overall, the work demonstrates that data-driven projections onto the soliton manifold can complement traditional variational and perturbative approaches, potentially guiding analytical derivations and extending to larger soliton ensembles.

Abstract

In this paper, we revisit the investigation of solitary-wave interactions in the nonlinear Schrödinger model, both in the presence and absence of a parabolic trapping potential. While approximate dynamics, based on variational or similar methods, governed by a system of ordinary differential equations (ODEs) for both bright and dark-soliton interactions have been well established in the literature based on physical expert considerations, this study focuses on a data-driven approach, the so-called Sparse Identification of Nonlinear Dynamics (SINDy). Accordingly, our purpose is to use PDE time-series of select waveform diag- nostics in order to numerically reconstruct such approximate dynamics, without prior knowledge thereof. The purpose is not only to verify the robustness of the dynamical approximated ODEs, but also to shed light on the application of such a data-driven methodology in the study of soliton interactions and to formu- late a complementary approach, more reliant on the wealth of PDE data and less so on expert theoretical constructs.

Data-driven Soliton Manifold Approximations for Dark and Bright Waves: Some Prototypical 1d Case Examples

TL;DR

The paper addresses learning reduced-order ODE models for soliton interactions in the 1D Gross-Pitaevskii/nonlinear Schrödinger equation, covering both dark and bright solitons and the role of parabolic trapping. It employs Sparse Identification of Nonlinear Dynamics (SINDy) and SINDy with control to identify governing equations directly from PDE time-series, using carefully constructed libraries and multiple initial conditions. The authors show that SINDy can accurately recover known reduced dynamics in many two- and four-soliton configurations, while also revealing limitations in higher-dimensional cases and boundary/inhomogeneous settings; physics-informed variants improve accuracy and coherence with Newton's third law. Overall, the work demonstrates that data-driven projections onto the soliton manifold can complement traditional variational and perturbative approaches, potentially guiding analytical derivations and extending to larger soliton ensembles.

Abstract

In this paper, we revisit the investigation of solitary-wave interactions in the nonlinear Schrödinger model, both in the presence and absence of a parabolic trapping potential. While approximate dynamics, based on variational or similar methods, governed by a system of ordinary differential equations (ODEs) for both bright and dark-soliton interactions have been well established in the literature based on physical expert considerations, this study focuses on a data-driven approach, the so-called Sparse Identification of Nonlinear Dynamics (SINDy). Accordingly, our purpose is to use PDE time-series of select waveform diag- nostics in order to numerically reconstruct such approximate dynamics, without prior knowledge thereof. The purpose is not only to verify the robustness of the dynamical approximated ODEs, but also to shed light on the application of such a data-driven methodology in the study of soliton interactions and to formu- late a complementary approach, more reliant on the wealth of PDE data and less so on expert theoretical constructs.
Paper Structure (23 sections, 43 equations, 4 figures)

This paper contains 23 sections, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model description
  3. Dark soliton
  4. Bright soliton
  5. Multi-soliton construction
  6. Data-driven methods
  7. Sparse identification of nonlinear dynamics
  8. SINDy with controls
  9. Numerical results
  10. Data generation
  11. Interaction of dark solitons
  12. Two dark solitons without parabolic trap potential
  13. Four dark solitons without parabolic trap potential (no interaction across boundary)
  14. Four dark solitons without parabolic trap potential (with interaction across boundary)
  15. One dark soliton with parabolic trap potential
  16. ...and 8 more sections

Figures (4)

  • Figure 1: The spatio-temporal plot of dark-soliton interaction dynamics without a parabolic trap potential: the left panel shows the interaction of two dark solitons, while the right panel presents that of four dark solitons. Notice that the white solid curves and blue asterisks refer to theoretical prediction of the soliton positions based on system \ref{['eq: dark solitary wave dynamics ODE']} and on the SINDy prediction based on \ref{['eq: SINDy prediction for 2 ds']} (two dark solitons) and \ref{['eq: dynamics of 4 ds']} (four dark solitons), respectively.
  • Figure 2: The comparisons of the SINDy predicted dark-soliton positions (based on Eq. \ref{['eq: dynamics for 4 ds with parabolic trap']}) and of the Physics-informed predicted dark-soliton positions (based on Eq. \ref{['eq: physics-informed learing results']}) with that of ground-truth dark-soliton positions sampled from the PDE \ref{['eq: nonlinear schrodinger model']}. Notice that the dashed red curves depict the SINDy and Physics-informed predicted trajectories of the dark-soliton positions, while the solid blue curve represents the ground-truth dark-soliton positions sampled from the PDE.
  • Figure 3: The spatio-temporal plot of the bright-soliton interaction in the presence of (Left panel) and without (Right panel) a parabolic trap potential. Notice that the solid white curves and the blue asterisks depict the theoretical prediction of the positions of the two bright solitons based on system \ref{['eq: dynamics for BS with parabolic trap']} and the SINDy prediction on the soliton positions based on \ref{['eq: SINDy dynamics for 2BS with trap']} and \ref{['eq: SINDy prediction for [-3,3]']}. The underlying contour map stems from the simulation of the original NLS PDE with the parabolic confinement.
  • Figure 4: The comparison of the four soliton parameters in the four degrees of freedom system: we notice that in each panel above, the solid black, dashed red, and dashed-dotted blue curves depict the time-series data generated from the PDE simulation, the theoretical prediction based on system \ref{['eq: BS dynamical ODEs BSs']}, and SINDy prediction based on the system \ref{['eq: SINDy prediction on 4-degrees of freedom system']} for each of the two solitons.