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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 ›› 2020, Vol. 37 ›› Issue (1): 56-66.doi: 10.3969/j.issn.1005-3085.2020.01.005

• • 上一篇    下一篇

广义Tanh函数法中Riccati方程和sine-Gordon方程的新解及其新应用

林府标,  张千宏   

  1. 贵州财经大学数统学院，贵阳  550025
  • 收稿日期:2017-12-13 接受日期:2018-10-11 出版日期:2020-02-15 发布日期:2020-04-15
  • 基金资助:
    国家自然科学基金(11761018; 11361012)；贵州省科技计划基金项目([2019]1051)；贵州财经大学创新探索及学术新苗项目([2017]5736-020).

A New Application of the Extended Tanh-function Method and New Solutions of the Riccati Equation and sine-Gordon Equation

LIN Fu-biao,  ZHANG Qian-hong   

  1. School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025
  • Received:2017-12-13 Accepted:2018-10-11 Online:2020-02-15 Published:2020-04-15
  • Supported by:
    The National Natural Science Foundation of China (11761018; 11361012); the Science and Technology Program Fund Project of Guizhou Province ([2019]1051); the Innovation Exploration and Academic New Seedling Project of Guizhou University of Finance and Economics ([2017]5736-020).

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摘要： 非线性偏微分方程的显式解析解，特别是行波解，蕴含了方程的丰富信息，对于描述各种现象的发展规律起着至关重要的作用．本文尝试构造 KdV 方程多种形式的新显式行波解．首先，利用试探函数法和 Matlab 计算给出了 Riccat 方程的许多新显式解析解．其次，运用广义 Tanh 函数法以及 Riccati 方程的新解得到了 sine-Gordon 方程的许多新显式解析解．最后，作为新的应用，把三角函数法结合 sine-Gordon 方程的新显式解析解并利用简化的变换形式进一步找到了 KdV 方程的许多新显式行波解．这些结果推广和补充了以往的相关研究成果，特别地，这些方法和新的结果可以用于求解许多非线性偏微分方程的新显式行波解．

关键词: Riccati 方程, 一维 sine-Gordon 方程, KdV 方程, 广义 Tanh 函数法, 正余弦函数法；行波解

Abstract: The explicit analytical solutions of nonlinear partial differential equations, in particular, the travelling wave solutions, contain rich information about the equations, and they are very important for describing the development of various phenomena. In the paper, many types of new explicit travelling wave solutions are presented for the KdV equation. First, many new explicit analytical solutions of the Riccati equation are given by using the trial function method and Matlab software. Second, many types of new explicit analytical solutions of the sine-Gordon equation are obtained by using the extended tanh-function method and new solutions of Riccati equation. Finally, as a new application, a great many new travelling wave solutions of the KdV equation are provided by using the sine-cosine function method, new solutions of the sine-Gordon equation and its simplified transformation forms. The obtained results extend and complement some relevant existing works. In particular, these methods and obtained new results can be applied to find explicit new travelling wave solutions of many nonlinear partial differential equations.

Key words: Riccati equation, one dimensional sine-Gordon equation, KdV equation; , extended Tanh-function method, sine-cosine function method, travelling wave solution

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