华东师范大学学报（自然科学版） >

2021 , Vol. 2021 >Issue 3: 56 - 64

DOI: https://doi.org/10.3969/j.issn.1000-5641.2021.03.007

数学

确定有限级数解的阶数上界的一种n阶展开方法

  • 宋宸苇 ,
  • 柳银萍
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  • 1. 华东师范大学 计算机科学与技术学院, 上海　200062
    2. 华东师范大学 数学科学学院, 上海　200241

收稿日期: 2020-03-12

  网络出版日期: 2021-05-26

基金资助

国家自然科学基金 (11871328); 上海市科委重点项目 (18511103105); 上海市自然科学基金 (19ZR1414000)

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An n-order expansion method for determining the upper bound of the order of finite series solutions

  • Chenwei SONG ,
  • Yinping LIU
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  • 1. School of Computer Science and Technology, East China Normal University, Shanghai　200062, China
    2. School of Mathematical Sciences, East China Normal University, Shanghai　200241, China

Received date: 2020-03-12

  Online published: 2021-05-26

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摘要

构造非线性演化方程有限级数解的直接代数方法都是基于齐次平衡原则发展起来的, 如双曲正切方法、Jacobi椭圆函数法、Painlevé截断展开法、CRE方法等. 这些方法中所求解的阶数都是由齐次平衡原则确定的. 本文进一步拓展了齐次平衡原则, 考虑了更多的平衡可能性, 提出了一种n阶展开方法来确定有限级数解可能的更高阶数. 通过将该方法应用到双曲正切方法中, 对所考虑的非线性演化方程确实获得了一些新的更高的阶数和新的解.

关键词： 非线性演化方程; 齐次平衡原则; n阶展开方法; 有限级数解

本文引用格式

宋宸苇 , 柳银萍 . 确定有限级数解的阶数上界的一种n阶展开方法[J]. 华东师范大学学报（自然科学版）, 2021 , 2021(3) : 56 -64 . DOI: 10.3969/j.issn.1000-5641.2021.03.007

Abstract

A number of algebraic methods used for constructing exact finite series solutions of nonlinear evolution equations are based on the homogeneous balance principle, such as the tanh function method, the Jacobi elliptic function method, the Painlevé truncated expansion method, the CRE method, etc. In each of these methods, the order of required solutions is determined by the homogeneous balance principle. In this paper, the homogeneous balance principle is further extended by considering additional balance possibilities. An n-order expansion method is proposed to determine possible new orders of required solutions. By applying the proposed method to several examples, we show that higher orders and new solutions can be obtained.

Key words： nonlinear evolution equation; homogeneous balance principle; n-order expansion method; finite series solution

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