在线咨询
中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2024, Vol. 41 ›› Issue (4): 757-768.doi: 10.3969/j.issn.1005-3085.2024.04.011

• • 上一篇    下一篇

与第三类潘勒韦方程相关的广义代数微分方程的亚纯解

古勇毅,  张欣茹,  孔荫莹   

  1. 广东财经大学统计与数学学院，广州 510320
  • 收稿日期:2021-11-22 接受日期:2022-12-29 出版日期:2024-08-15
  • 基金资助:
    国家自然科学基金 (11901111)；广东省基础与应用基础研究基金 (2022A1515012429)；广东省教育厅创新团队项目 (2022WCXTD009)；广东省高等教育教学改革项目 (2023680).

Meromorphic Solutions of a Generalized Algebraic Differential Equation Related to the Third Painlevé Equation

GU Yongyi,  ZHANG Xinru,  KONG Yinying   

  1. School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320
  • Received:2021-11-22 Accepted:2022-12-29 Online:2024-08-15
  • Supported by:
    The National Natural Science Foundation of China (11901111); the Basic and App-lied Basic Research Foundation of Guangdong Province (2022A1515012429); the Project of Guangdong Province Innovative Team (2022WCXTD009); the Higher Education Teaching Reform Project of Guangdong Province (2023680).

摘要：

潘勒韦方程是典型的非线性微分方程，应用广泛。复方法结合了复分析和微分方程理论，是求解非线性微分方程亚纯解的有效方法。利用复方法证明了与第三类潘勒韦方程相关的广义代数微分方程的亚纯解属于$W$类，并得到了该方程的亚纯解。通过适当的变换以及应用所得的结果，得到了修正的double sine-Gordon方程和修正的Kortweg-de-Vries (mKdV) 方程的$W$类亚纯精确解，并得到了相关非线性偏微分方程的非$W$类亚纯精确解。众所周知，使用基于行波变换等变换以及与第三类潘勒韦方程相关的广义代数微分方程的亚纯解，便于寻求其他相关的数学物理方程的亚纯精确解。在今后的工作中，可进一步对其他类别的潘勒韦方程相关的广义代数微分方程的亚纯解及应用进行研究。

关键词: 微分方程, 潘勒韦方程, 修正的double sine-Gordon方程, mKdV方程, 亚纯函数

Abstract:

The Painlevé equation is a typical nonlinear differential equation and has a wide range of applications. The complex method combines complex analysis and differential equation theory, and is an effective method for solving meromorphic solutions of nonlinear differential equations. The complex method is used to prove that meromorphic solutions of generalized algebraic differential equations related to the third Painlevé equation belong to class $W$, and meromorphic solutions of the mentioned equation were obtained. Through appropriate transformations and application of the obtained results, class $W$ meromorphic exact solutions for the modified double sine Gordon equation and modified Kortweg de Vries (mKdV) equation were derived, and non class $W$ meromorphic exact solutions for the relevant nonlinear partial differential equations were obtained. It is well known that based on the suitable transformations and these results, it is convenient to seek meromorphic exact solutions of other related mathematical and physical equations. In future work, further research can be conducted on the meromorphic solutions and applications of generalized algebraic differential equations related to other categories of the Painlevé equation.

Key words: differential equation, Painlevé equation, modified double sine-Gordon equation, mKdV equation, meromorphic function

中图分类号: