Data-driven soliton manifold approximations for dark and bright waves: Some prototypical 1D case examples.

Area of Science:

• Nonlinear dynamics
• Soliton physics
• Computational physics

Background:

• Solitary-wave interactions in the nonlinear Schrödinger model are typically studied using approximate methods like variational approaches.
• These methods yield ordinary differential equations (ODEs) for bright and dark-soliton dynamics, relying on expert theoretical insights.
• A gap exists in leveraging rich partial differential equation (PDE) data for understanding these complex interactions.

Purpose of the Study:

• To reconstruct approximate soliton dynamics using a data-driven approach, specifically sparse identification of nonlinear dynamics.
• To verify the accuracy and robustness of established ODE approximations for soliton interactions.
• To explore a complementary methodology that relies more on PDE data and less on theoretical assumptions.

Main Methods:

• Utilizing time-series data from partial differential equation (PDE) simulations of solitary-wave interactions.
• Applying the sparse identification of nonlinear dynamics (SINDy) algorithm to reconstruct governing equations.
• Comparing data-driven ODEs with established analytical and numerical solutions.

Main Results:

• Successfully reconstructed approximate ordinary differential equations (ODEs) governing solitary-wave dynamics from PDE data.
• Demonstrated the robustness of existing ODE models through data-driven verification.
• Showcased the potential of sparse identification for discovering underlying dynamics without prior theoretical knowledge.

Conclusions:

• The data-driven sparse identification of nonlinear dynamics is a viable method for studying soliton interactions.
• This approach can complement traditional theoretical methods, offering new insights from complex PDE data.
• Future work can expand this methodology to more complex nonlinear systems and potentials.