Scholarlink

<p>In this work, we present a comprehensive study of the (3+1)-dimensional nonlinear Korteweg–de Vries (KdV)-type hierarchy equation using a hybrid methodology that integrates the bilinear neural network method (BNNM) and symbolic computation. Leveraging the powerful characteristics of BNNM, we first construct a single-hidden-layer network to obtain exact breather, lump, and interaction wave solutions. The model’s architecture effectively encodes the bilinear form of the presenting equation, allowing effective extraction of nonlinear wave structures. To extend the representational capacity and capture more intricate dynamical behaviors, we then implement a deeper neural architecture particularly, a "4-2-2-1" multi-layer network model. This setup presents superior flexibility in handling complex interactions between solitary waves and localized structures. In contrast to the traditional analytical techniques, the BNNM framework suggests a dual advantage: it preserves the mathematical rigor of bilinear soliton theory while harnessing the universal approximation capabilities of neural networks. Through symbolic computation integrated with the neural outputs, we ensure exact algebraic satisfaction of the target PDE. This technique not only enhances accuracy and computational efficiency but also allows the discovery of diverse wave phenomena including breathing solitons, interference waves, and hybrid lump-stripe structures that are difficult to detect via classical techniques. The findings advance the understanding of KdV-type systems and provide a promising direction for data-driven soliton theory and high-dimensional nonlinear modeling.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Lump–Breather Interactions and Inelastic Wave Dynamics in KdV Hierarchy Systems via the Bilinear Neural Network-Based Approach

  • 期刊论文

  • 发表日期：

    2025年10月21日

查看全文
  • Sonia Akram,
  • Mati ur Rahman,
  • Laila A. AL-Essa

摘要

In this work, we present a comprehensive study of the (3+1)-dimensional nonlinear Korteweg–de Vries (KdV)-type hierarchy equation using a hybrid methodology that integrates the bilinear neural network method (BNNM) and symbolic computation. Leveraging the powerful characteristics of BNNM, we first construct a single-hidden-layer network to obtain exact breather, lump, and interaction wave solutions. The model’s architecture effectively encodes the bilinear form of the presenting equation, allowing effective extraction of nonlinear wave structures. To extend the representational capacity and capture more intricate dynamical behaviors, we then implement a deeper neural architecture particularly, a "4-2-2-1" multi-layer network model. This setup presents superior flexibility in handling complex interactions between solitary waves and localized structures. In contrast to the traditional analytical techniques, the BNNM framework suggests a dual advantage: it preserves the mathematical rigor of bilinear soliton theory while harnessing the universal approximation capabilities of neural networks. Through symbolic computation integrated with the neural outputs, we ensure exact algebraic satisfaction of the target PDE. This technique not only enhances accuracy and computational efficiency but also allows the discovery of diverse wave phenomena including breathing solitons, interference waves, and hybrid lump-stripe structures that are difficult to detect via classical techniques. The findings advance the understanding of KdV-type systems and provide a promising direction for data-driven soliton theory and high-dimensional nonlinear modeling.