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<p>A variable-coefficient bilinear neural network method is proposed for deriving analytical solutions to the variable-coefficient Kadomtsev–Petviashvili equation and the (2+1)-dimensional variable-coefficient Sawada–Kotera equation. By constructing adaptive 3-2-2-1 and 3-3-2-1 neural network models, we present abundant analytical solutions for these equations. Additionally, a variable-coefficient positive quadratic function method is introduced to obtain abundant lump-type solutions for the Sawada–Kotera equation. Dynamic properties of the derived solutions, including amplitude evolution and spatial interactions, are visualized through three-dimensional surface plots and density graphics, revealing their nonlinear wave behaviors.</p>

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Variable-Coefficient Bilinear Neural Network Method for Solving Kadomtsev–Petviashvili and Sawada–Kotera Equations and New Lump-Type Solutions

  • 期刊论文

  • 发表日期：

    2026年8月3日

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  • Jian-Guo Liu,
  • Qing Ye,
  • Xin-Yi Gao,
  • Gang-Wei Wang

摘要

A variable-coefficient bilinear neural network method is proposed for deriving analytical solutions to the variable-coefficient Kadomtsev–Petviashvili equation and the (2+1)-dimensional variable-coefficient Sawada–Kotera equation. By constructing adaptive 3-2-2-1 and 3-3-2-1 neural network models, we present abundant analytical solutions for these equations. Additionally, a variable-coefficient positive quadratic function method is introduced to obtain abundant lump-type solutions for the Sawada–Kotera equation. Dynamic properties of the derived solutions, including amplitude evolution and spatial interactions, are visualized through three-dimensional surface plots and density graphics, revealing their nonlinear wave behaviors.