Soliton solutions and lump solutions of the generalized multi-dimensional Estevez–Mansfield–Clarkson equation with time-varying coefficients, and learning of the equation based on augmented GPINN

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Abstract
To address the limitations of the standard Estevez–Mansfield–Clarkson (EMC) equation in describing practical physical scenarios-such as inhomogeneous media and multi-dimensional flow fields-this paper also aims to accurately capture the dynamic evolution characteristics of non-autonomous systems. Building on the basic framework of the standard equation, it proposes the generalized multi-dimensional EMC equation with time-varying coefficients (TVC-GM-EMC) through dimensional extension and time-varying coefficient modification. This paper analyzes the TVC-GM-EMC. It investigates the equation’s analytical solutions, including soliton solutions, interaction solutions, and lump solutions. It also explores the learning and prediction of these solutions using physics-informed neural network (PINN) and augmented graph PINN (AGPINN). First, the Hirota bilinear method is used to successfully construct soliton solutions for the equation. The regulatory mechanisms of TVC on soliton amplitude, propagation velocity and waveform stability are analyzed in depth. Second, lump solutions of the equation are derived via the long-wave limit method. Finally, PINN and AGPINN are respectively applied to learn the analytical solutions of the equation. A comparative analysis is conducted on the prediction accuracy of the two networks in handling solutions to multi-dimensional equations with TVC. The research findings not only deepen the theoretical understanding of the solution space structure of the TVC-GM-EMC. They also verify the scientific validity and practicality of the equation’s extension. The equation demonstrates broad application prospects in cutting-edge fields, including multi-dimensional wave propagation in plasma physics, multi-scale coupling mechanisms in fluid turbulence and multi-dimensional pulse control in nonlinear optics.
Funding Information
  • National Natural Science Foundation of China (11505090)

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