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<p>The Estevez–Mansfield–Clarkson (EMC) equation is a model equation that can be used to study and explain shallow water waves in fluid dynamics. This paper analyzes the Estevez–Mansfield–Clarkson equation, discusses its analytical solutions (including soliton solutions and lump solutions), and uses the Physics-Informed Neural Network (PINN) and the proposed Extension Physics-Informed Graph Neural Network (EGPINN) to learn the solved soliton solutions, respectively, with a comparison of the prediction results. Firstly, the multi-soliton solutions of the equation, including one-soliton, two-soliton, and three-soliton structures, are successfully derived using the Hirota bilinear method. The formation mechanism and propagation characteristics of the soliton solutions are analyzed in detail, and the laws governing the interaction between solitons are revealed. Then, the lump solutions of the equation are obtained using the positive quadratic function method, with in-depth analysis of the morphological evolution laws of the lump solutions under multivariate conditions and their physical connotations. Finally, PINN and EGPINN are used to learn the soliton solutions derived by the Hirota bilinear method. The research conclusions deepen the theoretical understanding of the solution space characteristics of the Estevez–Mansfield–Clarkson equation, provide a theoretical basis for understanding nonlinear phenomena such as the propagation, interaction, and waveform evolution of shallow water waves, and show application prospects in cutting-edge fields such as plasma physical processes, fluid turbulence mechanisms, and nonlinear optical effects.</p>

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Soliton and lump solutions of the Esteves–Mansfield–Clarkson equation and learning the equation based on extended GPINN

  • 期刊论文

  • 发表日期：

    2026年7月3日

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  • Yingying Zhang,
  • Xiaohan Gong,
  • Xiaoshu Xie,
  • Xiangpeng Xin

摘要

The Estevez–Mansfield–Clarkson (EMC) equation is a model equation that can be used to study and explain shallow water waves in fluid dynamics. This paper analyzes the Estevez–Mansfield–Clarkson equation, discusses its analytical solutions (including soliton solutions and lump solutions), and uses the Physics-Informed Neural Network (PINN) and the proposed Extension Physics-Informed Graph Neural Network (EGPINN) to learn the solved soliton solutions, respectively, with a comparison of the prediction results. Firstly, the multi-soliton solutions of the equation, including one-soliton, two-soliton, and three-soliton structures, are successfully derived using the Hirota bilinear method. The formation mechanism and propagation characteristics of the soliton solutions are analyzed in detail, and the laws governing the interaction between solitons are revealed. Then, the lump solutions of the equation are obtained using the positive quadratic function method, with in-depth analysis of the morphological evolution laws of the lump solutions under multivariate conditions and their physical connotations. Finally, PINN and EGPINN are used to learn the soliton solutions derived by the Hirota bilinear method. The research conclusions deepen the theoretical understanding of the solution space characteristics of the Estevez–Mansfield–Clarkson equation, provide a theoretical basis for understanding nonlinear phenomena such as the propagation, interaction, and waveform evolution of shallow water waves, and show application prospects in cutting-edge fields such as plasma physical processes, fluid turbulence mechanisms, and nonlinear optical effects.