Methods for Exact Solutions of Nonlinear Ordinary Differential Equations  

非线性常微分方程精确解的求解方法

作　　者：Robert CONTE Micheline MUSETTE Tuen Wai NG WU Chengfa Robert CONTE;Micheline MUSETTE;Tuen Wai NG;吴成发(巴黎萨克雷大学,巴黎-萨克雷高等师范学校,法国国家科学研究中心,Borelli研究中心,法国Gif-sur-Yvette,F-91190;香港大学数学系,香港薄扶林999077;布鲁塞尔自由大学理论物理学部,比利时Pleinlaan2,布鲁塞尔,B-1050;深圳大学高等研究院,广东深圳518060)

机构地区：[1]UniversitéParis-Saclay,ENS Paris-Saclay,CNRS,Centre Borelli,Gif-sur-Yvette,F-91190,France [2]Department of Mathematics,The University of Hong Kong,Pokfulam,Hong Kong,999077,P.R.China [3]Dienst Theoretische Natuurkunde,Vrije Universiteit Brussel,Pleinlaan 2,Brussels,B-1050,Belgium [4]Institute for Advanced Study,Shenzhen University,Shenzhen,Guangdong,518060,P.R.China

出　　处：《数学进展》2025年第2期379-389,共11页Advances in Mathematics(China)

基　　金：partially supported by RGC(No.17307420);supported by NSFC(No.12471077)。

摘　　要：In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations,numerous tricks have been proposed.The goal of this short review is to explain how a theorem of Eremenko on meromorphic solutions of some nonlinear ODEs together with some classical,19th-century results,can be turned into algorithms(thus avoiding ad hoc assumptions)which provide all(as opposed to some)solutions in a precise class.To illustrate these methods,we present some new such exact solutions,physically relevant.为了寻找非可积非线性常微分方程的闭合形式解,人们已经提出了许多技巧.这篇短综述的目标是解释如何利用Eremenko关于某些非线性常微分方程亚纯解的定理,以及一些19世纪的经典结果,将其转化为算法(从而避免依赖特定假设),以求得某一特定类别中的所有(而非部分)解.为了说明这些方法,我们给出了一些新的精确解,这些解在物理上具有重要意义.

关 键 词：elliptic solution complex Ginzburg-Landau equation Closed-form solution Nevanlinna theory 

分 类 号：O174.52[理学—数学] O174.54[理学—基础数学] O175.14