Scholarlink

<p>In recent years, special types of analytical solutions for nonlinear evolution equations, including soliton solutions and lump solutions, have once again become a research focus. Recent developments have revealed novel soliton structures such as circular solitons, breathing solitons, elliptic solitons, and twisted solitons, demonstrating increasingly complex dynamic behaviors in high-dimensional wave systems. The studies on circular solitons, breathing solitons, and the latest elliptic solitons have received more attention. This study focuses on the exact solutions and numerical solutions of the (2+1)-dimensional Boiti-Leon-Pempinelli (BLP) equation. Firstly, based on Hirota’s bilinear method, the multi-soliton solutions of this equation were constructed, including single soliton solutions, double soliton solutions and triple soliton solutions, and the details of the phase shift of soliton interactions were provided. At the same time, single and double lump solutions were obtained through the long-wave limit method, and visual analysis showed that the amplitude, waveform and moving direction of the lump solutions remained stable during propagation. Secondly, the physical information neural network (PINN) and extended physical information graph neural network (EGPINN) were introduced to learn some of the soliton solutions obtained. The EGPINN method achieves learning through four steps: data generation, model design, training optimization and result verification. Its prediction results are highly consistent with the analytical solutions, and it outperforms the traditional PINN method in local feature modeling, computational speed, model fusion degree and prediction accuracy, providing an efficient tool for the rapid numerical simulation of shallow water wave fields. The obtained analytical solutions and numerical methods have potential applications in coastal engineering for predicting flood wave propagation, analyzing wind-driven wave patterns in large lakes, and assessing nearshore transient disturbances caused by underwater terrain changes. Furthermore, the EGPINN framework can be extended to real-time wave forecasting and early warning systems for shallow water regions, contributing to disaster prevention and water resource management.</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

Soliton solutions and lump solutions of the (2+1) dimensional Boiti-Leon-Pempinelli equation, and learning of the equation based on extended GPINN

  • 期刊论文

  • 发表日期：

    2026年3月25日

查看全文
  • Xiaoshu Xie,
  • Xiaohan Gong,
  • Yingying Zhang,
  • Xiangpeng Xin

摘要

In recent years, special types of analytical solutions for nonlinear evolution equations, including soliton solutions and lump solutions, have once again become a research focus. Recent developments have revealed novel soliton structures such as circular solitons, breathing solitons, elliptic solitons, and twisted solitons, demonstrating increasingly complex dynamic behaviors in high-dimensional wave systems. The studies on circular solitons, breathing solitons, and the latest elliptic solitons have received more attention. This study focuses on the exact solutions and numerical solutions of the (2+1)-dimensional Boiti-Leon-Pempinelli (BLP) equation. Firstly, based on Hirota’s bilinear method, the multi-soliton solutions of this equation were constructed, including single soliton solutions, double soliton solutions and triple soliton solutions, and the details of the phase shift of soliton interactions were provided. At the same time, single and double lump solutions were obtained through the long-wave limit method, and visual analysis showed that the amplitude, waveform and moving direction of the lump solutions remained stable during propagation. Secondly, the physical information neural network (PINN) and extended physical information graph neural network (EGPINN) were introduced to learn some of the soliton solutions obtained. The EGPINN method achieves learning through four steps: data generation, model design, training optimization and result verification. Its prediction results are highly consistent with the analytical solutions, and it outperforms the traditional PINN method in local feature modeling, computational speed, model fusion degree and prediction accuracy, providing an efficient tool for the rapid numerical simulation of shallow water wave fields. The obtained analytical solutions and numerical methods have potential applications in coastal engineering for predicting flood wave propagation, analyzing wind-driven wave patterns in large lakes, and assessing nearshore transient disturbances caused by underwater terrain changes. Furthermore, the EGPINN framework can be extended to real-time wave forecasting and early warning systems for shallow water regions, contributing to disaster prevention and water resource management.