An Artificial Neural Network Method for Simulating Soliton Propagation Based on the Rosenau-KdV-RLW Equation on Unbounded Domains

Laurence Finch, Weizhong Dai () and Aniruddha Bora
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Laurence Finch: Program in Computational Analysis and Modeling, Louisiana Tech University, Ruston, LA 71272, USA
Weizhong Dai: Mathematics & Statistics, College of Engineering & Science, Louisiana Tech University, Ruston, LA 71272, USA
Aniruddha Bora: Division of Applied Mathematics, Brown University, Providence, RI 02906, USA

Mathematics, 2025, vol. 13, issue 7, 1-21

Abstract: The simulation of wave propagation, such as soliton propagation, based on the Rosenau-KdV-RLW equation on unbounded domains requires a bounded computational domain. Therefore, a special boundary treatment, such as an absorbing boundary condition (ABC) or a perfectly matched layer (PML), is necessary to minimize the reflections of outgoing waves at the boundary, preventing interference with the simulation’s accuracy. However, the presence of higher-order partial derivatives, such as u x x t and u x x x x t in the Rosenau-KdV-RLW equation, raises challenges in deriving accurate artificial boundary conditions. To address this issue, we propose an artificial neural network (ANN) method that enables soliton propagation through the computational domain without imposing artificial boundary conditions. This method randomly selects training points from the bounded computational space-time domain, and the loss function is designed based solely on the initial conditions and the Rosenau-KdV-RLW equation itself, without any boundary conditions. We analyze the convergence of the ANN solution theoretically. This new ANN method is tested in three examples. The results indicate that the present ANN method effectively simulates soliton propagation based on the Rosenau-KdV-RLW equation in unbounded domains or over extended periods.

Keywords: Rosenau-KdV-RLW; wave propagation; artificial neural network; unbounded domain (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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