长水波近似方程的Painlevé分析与精确解  

The PainlevéAnalysis and Solution for the Approximate Equations of Long Water Wave

作　　者：陈南 CHEN Nan(College of computer and artificial intelligence,Xiamen Institute of Technology,Xiamen 361021,China)

机构地区：[1]厦门工学院计算机与人工智能学院,福建厦门361021

出　　处：《厦门理工学院学报》2020年第3期91-95,共5页Journal of Xiamen University of Technology

基　　金：福建省中青年教师教育科研项目(JAT190958);厦门工学院校级科研基金项目(KYT2019021)。

摘　　要：利用Painlevé分析方法,假设长水波近似方程具有洛朗级数形式的解,对其主导项进行分析;将假设的洛朗级数形式的解代入方程,比较φ的同次幂系数,利用一般项表达式计算调谐因子项,将方程进行有限项“截断”,证明长水波近似方程具有Painlevé可积性。在此基础上,导出长水波近似方程的Bäcklund变换和奇异流形满足的Schwarz导数方程,通过研究相关的Schwarz导数方程的性质求出该方程的精确解,该精确解可以用双曲三角函数表示。In this paper,the approximate equation for long water wave is studied using Painlevéanalysis.First,supposing the equation has a solution in the form of Laurent series,analyze its dominant term.Next,substitute the solution in the form of Laurent series into the equation,compare the same power coefficient,use the general term expression to calculate the tuning factor term,and“truncate”the finite term to prove that it has Painlevéintegrability.The Schwarz derivative equation satisfied by Bäcklund transformation and singular manifold is derived.The exact solution is obtained by studying the properties of the related Schwarz derivative equation,and the exact solution can be expressed by hyperbolic trigonometric function.

关 键 词：长水波近似方程 PAINLEVÉ分析 精确解 主导项分析 调谐因子 Bäcklund变换 SCHWARZ导数 

分 类 号：O174.55[理学—数学]